A Novel Algorithm for Studying the Effects of Squeezing

Research ArticleA Novel Algorithm for Studying the Effects of Squeezing Flow ofa Casson Fluid between Parallel Plates on Magnetic Field

Abdul-Sattar J A Al-Saif1 and Abeer Majeed Jasim 12

1Department of Mathematics College of Education for Pure Science University of Basrah Basrah Iraq2Department of Mathematics College of Science University of Basrah Basrah Iraq

Correspondence should be addressed to Abeer Majeed Jasim abeerjassemyahoocom

Received 4 December 2018 Revised 30 January 2019 Accepted 5 March 2019 Published 26 March 2019

Academic Editor Oluwole D Makinde

Copyright copy 2019 Abdul-Sattar J A Al-Saif and Abeer Majeed Jasim This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

In this paper themagneto hydrodynamic (MHD) squeezing flow of a non-Newtonian namely Casson fluid between parallel platesis studied The suitable one of similarity transformation conversion laws is proposed to obtain the governing MHD flow nonlinearordinary differential equationThe resulting equation has been solved by a novel algorithm Comparisons between the results of thenovel algorithm technique and other analytical techniques and one numerical Range-Kutta fourth-order algorithm are providedThe results are found to be in excellent agreement Also a novel convergence proof of the proposed algorithm based on propertiesof convergent series is introduced Flow behavior under the changing involved physical parameters such as squeeze number Cassonfluid parameter and magnetic number is discussed and explained in detail with help of tables and graphs

1 Introduction

Squeezing flow between parallel walls occurs in many indus-trial and biological systems The unsteady squeezing flowof a viscous fluid between parallel plates in motion normalto their own surfaces is a great interest in hydrodynamicalmachines The pioneer work and the basic formulation ofsqueezing flows under lubricationwere assumed by Stefan [1]In past researches over few decades Reynolds [2] analyzedthe squeezing flow between elliptic plates while Archibald[3] suggested the same problem for rectangular plates TheReynolds equation was studied for squeezing flows whichwere not sufficient for some cases as was demonstrated byJackson [4] and Usha and Sridharan [5] The study on themotion of electrically conducting fluid in the presence ofa magnetic field is called magneto hydrodynamics (MHD)In engineering the application of MHD can be seen inthe electromagnetic pump The pumping motion of thisdevice is caused by the Lorentz force This force is producedwhen mutually perpendicular magnetic fields and electriccurrents are applied in the direction perpendicular to theaxis of a pipe containing conducting fluid [6] The laws of

conservations under the similarity transformation for thesqueezing flow of an electrically conducting Casson fluidwhich was suggested by Wang [7] have been used to extracta highly nonlinear ordinary differential equation governingthe magneto hydrodynamic (MHD) Many researchers haveconducted numerous research attempts for the purposeof understanding and analyzing squeezing flows [8ndash14]Duwairi et al [15] investigated the effects of squeezing on heattransfer of a viscous fluid between parallel plates Mohyud-Din et al [16] have studied heat and mass transfer analysisfor the flow of a nanofluid between rotating parallel plateswhile Mohyud-Din and Khan [17] analyzed the nonlinearradiation effects on squeezing flow of a Casson fluid betweenparallel disks Ahmed et al [18] assumed MHD flow ofan incompressible fluid through porous medium betweendilating and squeezing permeable plates Khan et al [19] haveexplainedMHDsqueezing flowbetween twofinite plates andHayat et al [20] have demonstrated the effect of squeezingflow of second grade fluid between two parallel disks Naveedet al [21] studied effects on magnetic field in squeezing flowof a Casson fluid between parallel plates The main scopeof this paper is to implement novel algorithm to obtain

HindawiJournal of Applied MathematicsVolume 2019 Article ID 3679373 19 pageshttpsdoiorg10115520193679373

2 Journal of Applied Mathematics

analytical-approximate solution which depends mainly onthe coefficients of powers seriesThis is close to the numericalsolution for the squeezing flow of a Casson fluid betweenparallel plates although the resulting series of the novelalgorithm which contains initial conditions are known theothers are unknown and their values can be found throughthe boundary conditions that are defined These series alsocontain physical parameters which can be compensated byconstants and the geometrical effects of these parameterswould be investigated depending on the resulting solutionsIn this work it can be clearly seen that a novel algorithmis successfully applied to solve the equations of an unsteadysqueeze of flow between parallel plates as well as to find ananalytical-approximate solution In contrast it was found thatour results from algorithm completely agree with the resultsobtained from thenumerical solution through the applicationof Range-Kutta fourth-order method [RK-4] the numericalsolutions of Wang [7] and variation of parameter method(VPM) [21] The organization of this paper is as followsThe governing equations are derived in Section 2 Details ofderivation of the novel algorithm have been written as stepsin Section 3 The performance of the novel algorithm forthe squeezing flow is applied in Section 4 In Section 5 theconvergence analysis is presented Results and discussions aregiven in Section 6 Finally the conclusions are indicated inSection 7

2 Governing Equations

Weconsider an incompressible flowof aCassonfluid betweentwo parallel plates separated by a distance y = plusmn119897(1 minus 120572119905)12 =plusmnℎ(119905) where 119897 is the initial gap between the plates (at a time119905 = 0) Additionally 120572 gt 0 corresponds to a squeezingmotionof both plates until they touch each other at 119905 = 1120572 for120572 lt 0the plates bear a receding motion and dilate as described inFigure 1

Rheological equation for Casson fluid is defined [22] as

120591119894119895 =

2[120583119861 + (1199011199102120587)] 119890119894119895 120587 gt 1205871198882 [120583119861 + ( 1199011199102120587119888)] 119890119894119895 120587119888 gt 120587

(1)

where 120587 = 119890119894119895119890119894119895 and 119890119894119895 is the (119894 119895)119905ℎ component of thedeformation rate 120587 is the product of the component ofdeformation rate with itself 120587119888 is the critical value of thesaid product 120583119861 is the plastic dynamic viscosity of the non-Newtonian fluid and 119901119910 is the yield stress of slurry fluidWe are also applying the following assumptions on the flowmodel

(i) The effects of induced magnetic and electric fieldproduced due to the flow of electrically conductingfluid are negligible

(ii) No external electric field is present

Under aforementioned constraints the conservation equa-tions for the flow are120597119906120597119909 + 120597V120597119910 = 0 (2)

120597119906120597119905 + 119906120597119906120597119909 + V120597119906120597119910

= minus1120588 120597119901120597119909 + 120583120588 (1 + 1120574)(212059721199061205971199092 + 12059721199061205971199102 + 1205972V120597119910120597119909)

minus 1205901205732120588 119906(3)

120597V120597119905 + 119906 120597V120597119909 + V120597V120597119910

= minus1120588 120597119901120597119910 + 120583120588 (1 + 1120574)(2 1205972V1205971199092 + 1205972V1205971199102 + 1205972119906120597119910120597119909)

(4)

where 119906 and V are the velocity components in 119909 and 119910directions respectively 119901 is the pressure 120583120588 is the dynamicviscosity of the fluid (ratio of Kinematic viscosity and den-sity) 120574 = 120583119861radic2120587119888119901119910 is the Casson fluid parameter and120573 is the magnitude of imposed magnetic field Supportingconditions for the problem are as follows

119906 = 0V = V119908 = 119889ℎ119889119905

119886119905 119910 = ℎ (119905) 120597119906120597119910 = 0V = 0

119886119905 119910 = 0

(5)

and we can simplify the above system of equations byeliminating the pressure term equations (3)-(4) and using (2)After cross differentiation and introducing vorticity 120596

120596 = 120597V120597119909 minus 120597119906120597119910 (6)

we get

120597120596120597119905 + 119906120597120596120597119909 + V120597120596120597119910 = 120583120588 (1 + 1120574)(120597

21205961205971199092 + 12059721205961205971199102 )minus 1205901205732120588 120597119906120597119910

(7)

Transform introduced by Wang [7] for a two dimensionalflow is stated as

119906 = 120572119909[2 (1 minus 120572119905)12]1198911015840 (120578) (8)

V = minus120572119897[2 (1 minus 120572119905)12]119891 (120578) (9)

Journal of Applied Mathematics 3

y

ux

D

2l(1 minus t)12

Figure 1 Schematic diagram for the flow problem

where

120578 = 119910[119897 (1 minus 120572)12] (10)

Substituting (8)-(10) in (7) and using (6) yield a nonlinearordinary differential equation describing the Casson fluidflow as

(1 + 1120574)119891119894V minus 119878 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)minus 119891 (120578) 119891101584010158401015840 (120578)) minus119872211989110158401015840 (120578) = 0

(11)

where 119878 = 12057211989721205882120583 denotes the nondimensional Squeezenumber and 119872 = 119897120573radic120590(1 minus 120572119905)120583 is magnetic numberBoundary conditions for the problembyusing (8)-(10) reduceto

119891 (0) = 011989110158401015840 (0) = 0119891 (1) = 11198911015840 (1) = 0

(12)

and squeezing number 119878describes themovement of the plates(119878 gt 0 corresponds to the plates moving apart while 119878 lt0 corresponds to collapsing movement of the plates) It ispertinent to mention here that for119872 = 0 and 120574 997888rarr infin ourstudy reduces to the one obtained by Wang [7] Skin frictioncoefficient is defined as

11989721199092 (1 minus 120572119905)119877119890119909119862119891 = (1 + 1120574)11989110158401015840 (1) (13)

where

119862119891 = 120583120588(120597119906120597119905)119910=ℎ(119905)

V2119908 (14)

119877119890119909 = 2119897V2119908V119909 (1 minus 120572119905)12 (15)

3 The Basic Steps of the Novel Algorithm

This section describes how to obtain a novel algorithm tocalculate the coefficients of the power series solution result-ing from solving nonlinear ordinary differential equations

resulting from using transforms ((8)-(10)) to find analytical-approximate solution These coefficients are important basesto construct the solution formula therefore they can becomputed recursively by differentiationways To illustrate thecomputation of these coefficients and derivation of the novelalgorithm we summarized the detailed new outlook in thefollowing steps

Step 1 Consider the nonlinear differential equation as fol-lows

119867(119891 (120578) 1198911015840 (120578) 11989110158401015840 (120578) 119891101584010158401015840 (120578) 119891(119899minus1) (120578) 119891(119899) (120578))= 0 (16)

and integrating (16) with respect to 120578 on [0 120578] yields119891 (120578) = 119891 (0) + 1198911015840 (0) 120578 + 11989110158401015840 (0) 12057822

+ 119891(119899minus1) (0) 120578119899minus1(119899 minus 1) + 119871minus1119866 [119891 (120578)]

(17)

where

119866 [119891 (120578)]= 119867(119891 (120578) 1198911015840 (120578) 11989110158401015840 (120578) 119891(119899minus1) (120578))

119871minus1 = int1205780int1205780sdot sdot sdot int1205780(119889120578)119899

(18)

Step 2 Assume that

119866 [119891 (120578)] = infinsum119899=1

119889119899minus1119866 (1198910 (120578))119889120578119899minus1 (19)

rewriting (19)

119866 [119891 (120578)] = 119866 [1198910 (120578)] + 1198661015840 [1198910 (120578)] + 11986610158401015840 [1198910 (120578)]+ 119866101584010158401015840 [1198910 (120578)] + 1198661015840101584010158401015840 [1198910 (120578)] + sdot sdot sdot (20)

and substituting (20) in (17) we obtain

119891 (120578) = 1198910 + 1198911 + 1198912 + 1198913 + 1198914 + sdot sdot sdot (21)


where

1198910 = 119891 (0) + 1198911015840 (0) 120578 + 11989110158401015840 (0) 12057822 sdot sdot sdot+ 119891(119899minus1) (0) 120578(119899minus1)

(119899 minus 1) 1198911 = 119871minus1119866 [1198910 (120578)] 1198912 = 119871minus11198661015840 [1198910 (120578)] 1198913 = 119871minus111986610158401015840 [1198910 (120578)] 1198914 = 119871minus1119866101584010158401015840 [1198910 (120578)]

(22)

Step 3 We focus on computing the derivatives of 119866 withrespect to 120578 which is the crucial part of the proposedmethod Let us start calculating119866[119891(120578)] 1198661015840[119891(120578)] 11986610158401015840[119891(120578)]119866101584010158401015840[119891(120578)] 119866 [119891 (120578)] = 119867 (119891 (120578) 1198911015840 (120578) 11989110158401015840 (120578) 119891101584010158401015840 (120578) 1198911015840101584010158401015840 (120578)

119891(119899minus1) (120578)) (23)

1198661015840 [119891 (120578)] = 119889119866 [119891 (120578)]119889120578 = 119866119891119891120578 + 1198661198911015840 (119891120578)1015840 + sdot sdot sdot

+ 119866119891(119899minus1) (119891120578)(119899minus1) (24)

11986610158401015840 [119891 (120578)] = 1198892119866 [119891 (120578)]1198891205782 = 119866119891119891 (119891120578)2 + 1198661198911198911015840 (119891120578)1015840

sdot 119891120578 + 11986611989111989110158401015840 119891120578 (119891120578)10158401015840 + sdot sdot sdot + 119866119891119891(119899minus1) (119891120578) (119891120578)(119899minus1)+ 119866119891119891120578120578 + 1198661198911015840119891 (119891120578)1015840 119891120578 + 11986611989110158401198911015840 (119891120578)10158402 + sdot sdot sdot+ 1198661198911015840119891(119899minus1) (119891120578)1015840 (119891120578)(119899minus1) + 1198661198911015840 (119891120578120578)1015840 + 11986611989110158401015840119891(119891120578)10158401015840 119891120578 + 119866119891101584010158401198911015840 (119891120578)1015840 (119891120578)10158401015840 + 1198661198911015840101584011989110158401015840 (119891120578)101584010158402+ 11986611989110158401015840119891(119899minus1) (119891120578)10158401015840 (119891120578)(119899minus1) + 11986611989110158401015840 (119891120578120578)10158401015840 + sdot sdot sdot+ 119866119891(119899minus1)119891 (119891120578)(119899minus1) 119891120578 + 119866119891(119899minus1)1198911015840 (119891120578)(119899minus1) (119891120578)1015840+ sdot sdot sdot + 119866119891(119899minus1)119891(119899minus1) (119891120578)(119899minus1)2 + 119866119891(119899minus1) (119891120578120578)(119899minus1)

(25)

119866101584010158401015840 [119891 (120578)] = 1198893119866 [119891 (120578)]1198891205783 = 119866119891119891119891 (119891120578)3 + 1198661198911198911198911015840

(119891120578)2 (119891120578)1015840 + sdot sdot sdot + 119866119891119891119891(119899minus1) (119891120578)2 (119891120578)(119899minus1) + 1198661198911198912 (119891120578) 119891120578120578 + 1198661198911198911015840119891 (119891120578)1015840 (119891120578)2 + 11986611989111989110158401198911015840 (119891120578)10158402sdot (119891120578) + sdot sdot sdot + 1198661198911198911015840119891(119899minus1) (119891120578)1015840 (119891120578) (119891120578)(119899minus1)

+ 1198661198911198911015840 [(119891120578120578)1015840 119891120578 + (119891120578)1015840 119891120578120578] + 11986611989111989110158401015840119891 (119891120578)10158401015840sdot (119891120578)2 + 119866119891119891101584010158401198911015840 (119891120578)10158401015840 (119891120578) (119891120578)1015840 + sdot sdot sdot+ 11986611989111989110158401015840119891(119899minus1) (119891120578)10158401015840 (119891120578) (119891120578)(119899minus1) + 11986611989111989110158401015840 [119891120578120578(119891120578)10158401015840 + 119891120578 (119891120578120578)10158401015840] + sdot sdot sdot + 119866119891119891(119899minus1)119891 (119891120578)2 (119891120578)(119899minus1) + 119866119891119891(119899minus1)1198911015840 (119891120578) (119891120578)1015840 (119891120578)(119899minus1) + sdot sdot sdot+ 119866119891119891(119899minus1)119891(119899minus1) (119891120578) (119891120578)(119899minus1)2 + 119866119891119891(119899minus1) [(119891120578120578) (119891120578)(119899minus1) + (119891120578) (119891120578120578)(119899minus1)] + 119866119891119891119891120578120578 (119891120578)+ 1198661198911198911015840 119891120578120578 (119891120578)1015840 + sdot sdot sdot + 119866119891119891(119899minus1) 119891120578120578 (119891120578)(119899minus1)+ 119866120578119891120578120578120578 + 1198661198911015840119891119891 (119891120578)1015840 (119891120578)2 + 11986611989110158401198911015840119891 (119891120578)10158402sdot (119891120578) + sdot sdot sdot + 1198661198911015840119891119891(119899minus1) (119891120578)1015840 (119891120578) (119891120578)(119899minus1)+ 1198661198911015840119891 [(119891120578120578)1015840 119891120578 + (119891120578)1015840 119891120578120578] + 11986611989110158401198911015840119891 (119891120578)10158402 119891119911 + 119866119891101584011989110158401198911015840 (119891120578)10158403 + sdot sdot sdot + 11986611989110158401198911015840119891(119899minus1) (119891120578)10158402 (119891120578)(119899minus1) + 11986611989110158401198911015840 2 (119891120578)1015840 (119891120578120578)1015840 + sdot sdot sdot+ 119866119891(119899minus1)119891(119899minus1)119891 (119891120578)(119899minus1)2 119891120578 + 119866119891(119899minus1)119891(119899minus1)1198911015840 (119891120578)(119899minus1)2 (119891120578)1015840 + sdot sdot sdot + 119866119891(119899minus1)119891(119899minus1)119891(119899minus1) (119891120578)(119899minus1)3+ 119866119891(119899minus1)119891(119899minus1) 2 (119891120578)(119899minus1) (119891120578120578)(119899minus1) + 119866119891(119899minus1)119891(119891120578120578)(119899minus1) 119891120578 + 119866119891(119899minus1)1198911015840 (119891120578120578)(119899minus1) (119891120578)1015840 + sdot sdot sdot+ 119866119891(119899minus1)119891(119899minus1) (119891120578120578)(119899minus1) (119891120578)(119899minus1) + 119866119891(119899minus1) (119891120578120578120578)(119899minus1)

(26)

We see that the calculations become more complicated in thesecond and third derivatives because of the numerous calcu-lations Consequently the systematic structure on calculationis extremely important Fortunately due to the assumptionthat the operator 119866 and the solution119891 are analytic functionsthe mixed derivatives are equivalent

We note that the derivatives function to119891 is unknown sowe suggest the following hypothesis

119891120578 = 1198911 = 119871minus1119866 [1198910 (120578)] 119891120578120578 = 1198912 = 119871minus11198661015840 [1198910 (120578)]


119891120578120578120578 = 1198913 = 119871minus111986610158401015840 [1198910 (120578)] 119891120578120578120578120578 = 1198914 = 119871minus1119866101584010158401015840 [1198910 (120578)] 119891120578120578120578120578120578 = 1198915 = 119871minus11198661015840101584010158401015840 [1198910 (120578)]

(27)

Therefore (23)-(26) are evaluated by

119866 [1198910 (120578)]= 119867 (1198910 (120578) 11989110158400 (120578) 119891101584010158400 (120578) 119891(119899minus1)0 (120578)) (28)

1198661015840 [1198910 (120578)] = 1198661198910 1198911 + 11986611989110158400 (1198911)1015840 + 119866119891101584010158400 (1198911)10158401015840 + sdot sdot sdot+ 119866119891(119899minus1)0 (1198911)(119899minus1)

(29)

11986610158401015840 [1198910 (120578)] = 11986611989101198910 (1198911)2 + 119866119891011989110158400 (1198911)1015840 1198911+ 1198661198910119891101584010158400 1198911 (1198911)10158401015840 + sdot sdot sdot + 1198661198910119891(119899minus1)0 (1198911) (1198911)(119899minus1)+ 1198661198910 1198912 + 119866119891101584001198910 (1198911)1015840 119891 (1) + 1198661198911015840011989110158400 (1198911)10158402 + sdot sdot sdot+ 11986611989110158400119891(119899minus1)0 (1198911)1015840 (1198911)(119899minus1) + 11986611989110158400 (1198912)1015840

+ 119866119891101584010158400 1198910 (1198911)10158401015840 1198911 + 119866119891101584010158400 11989110158400 (1198911)1015840 (1198911)10158401015840+ 119866119891101584010158400 119891101584010158400 (1198911)101584010158402 + sdot sdot sdot + 119866119891101584010158400 119891(119899minus1)0 (1198911)10158401015840 (1198911)(119899minus1)

+ 119866119891101584010158400 (1198912)10158401015840 + 119866119891(119899minus1)119891 (1198911)(119899minus1) 1198911+ 119866119891(119899minus1)0 11989110158400 (1198911)(119899minus1) (1198911)1015840 + sdot sdot sdot+ 119866119891(119899minus1)0 119891(119899minus1)0 (1198911)(119899minus1)2 + sdot sdot sdot + 119866119891(119899minus1)0 (1198912)(119899minus1)

(30)

119866101584010158401015840 [1198910 (120578)] = 119866119891011989101198910 (1198911)3 + 1198661198910119891011989110158400 (1198911)2 (1198911)1015840+ sdot sdot sdot + 11986611989101198910119891(119899minus1)0 (1198911)2 (1198911)(119899minus1)

+ 11986611989101198910 2 (1198911) 1198912 + 1198661198910119891101584001198910 (1198911)1015840 (1198911)2+ 11986611989101198911015840011989110158400 (1198911)10158402 (1198911) + sdot sdot sdot+ 119866119891011989110158400119891(119899minus1)0 (1198911)1015840 (1198911) (1198911)(119899minus1)

+ 119866119891011989110158400 [(1198912)1015840 119891119911 + (1198911)1015840 1198912]+ 1198661198910119891101584010158400 1198910 (1198911)10158401015840 (1198911)2 + 1198661198910119891101584010158400 11989110158400 (1198911)10158401015840 (1198911) (1198911)1015840+ sdot sdot sdot + 1198661198910119891101584010158400 119891(119899minus1)0 (1198911)10158401015840 (1198911) (1198911)(119899minus1)

+ 1198661198910119891101584010158400 [1198912 (1198911)10158401015840 + 1198911 (1198912)10158401015840] + sdot sdot sdot+ 1198661198910119891(119899minus1)0 1198910 (1198911)2 (1198911)(119899minus1)

+ 1198661198910119891(119899minus1)0 11989110158400 (1198911) (1198911)1015840 (1198911)(119899minus1) + sdot sdot sdot+ 1198661198910119891(119899minus1)0 119891(119899minus1)0 (1198911) (1198911)2(119899minus1)

+ 1198661198910119891(119899minus1)0 [(1198912) (1198912)(119899minus1) + (1198911) (1198912)(119899minus1)]+ 11986611989101198910 1198912 (1198911) + 119866119891011989110158400 1198912 (1198911)1015840 + + 1198661198910119891(119899minus1)0 1198912 (1198911)(119899minus1) + 1198661198910 1198913+ 1198661198911015840011989101198910 (1198911)1015840 (1198911)2 + 11986611989110158400119891101584001198910 (1198911)10158402 (1198911) + sdot sdot sdot+ 119866119891101584001198910119891(119899minus1)0 (1198911)1015840 (1198911) (1198911)(119899minus1)

+ 119866119891101584001198910 [(1198912)1015840 1198911 + (1198911)1015840 1198911] + 11986611989110158400119891101584001198910 (1198911)10158402 1198911+ 119866119891101584001198911015840011989110158400 (1198911)10158403 + sdot sdot sdot+ 1198661198911015840011989110158400119891(119899minus1)0 (1198911)10158402 (1198911)(119899minus1)+ 1198661198911015840011989110158400 2 (1198911)1015840 (1198912)1015840 + sdot sdot sdot+ 119866119891(119899minus1)0 119891(119899minus1)0 1198910 (1198911)(119899minus1)2 1198911+ 119866119891(119899minus1)0 119891(119899minus1)0 11989110158400 (1198911)(119899minus1)2 (1198911)1015840 + sdot sdot sdot+ 119866119891(119899minus1)0 119891(119899minus1)0 119891(119899minus1)0 (1198911)(119899minus1)3+ 119866119891(119899minus1)0 119891(119899minus1)0 2 (1198911)(119899minus1) (1198912)(119899minus1)

+ 119866119891(119899minus1)0 1198910 (119891(119899minus1)2 1198911 + 119866119891(119899minus1)0 11989110158400 (1198912)(119899minus1) (1198911)1015840+ sdot sdot sdot + 119866119891(119899minus1)0 119891(119899minus1)0 (1198912)(119899minus1) (1198911)(119899minus1)

+ 119866119891(119899minus1)0 (1198913)(119899minus1)

(31)

Step 4 Substituting (28)-(31) in (21) we will get the requiredanalytical-approximate solution for (16)

4 Application

The novel algorithm described in the previous section canbe used as a powerful solver to the nonlinear differential


equations of squeezing flow between two parallel plates (11)- (12) in order to find a new analytical-approximate solutionFrom Step 1 we have

119891 (120578) = 119891 (0) + 1198911015840 (0) 120578 + 11989110158401015840 (0) 12057822 + 119891101584010158401015840 (0) 1205783

3+ 119871minus1 [ 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578)]

(32)

and we rewrite (32) as follows

119891 (120578) = 1198601 + 1198602120578 + 1198603 12057822 + 1198604 1205783

3 + 119871minus1119866 [119891 (120578)] (33)

where

1198601 = 119891 (0) 1198602 = 1198911015840 (0) 1198603 = 11989110158401015840 (0) 1198604 = 119891101584010158401015840 (0) 119866 [119891] = 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)

minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578) 119886119899119889 119871minus1 () = int120578

0int1205780int1205780int1205780(119889120578)4

(34)

From the boundary conditions (33) becomes

119891 (120578) = 1198602120578 + 1198604 12057833 + 119871minus1 [119866 [119891 (120578)] (35)

and from Step 2 we have

1198910 = 1198602120578 + 1198604 12057833 1198911 = 119871minus1119866 [1198910 (120578)] 1198912 = 119871minus11198661015840 [1198910 (120578)] 1198913 = 119871minus111986610158401015840 [1198910 (120578)] 1198914 = 119871minus1119866101584010158401015840 [1198910 (120578)]

(36)

Step 3 yields

119866 [119891 (120578])] = 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578)

+ 1198911015840 (120578) 11989110158401015840 (120578) minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578)] (37)

1198661015840 [119891 (120578)] = 119889119866 [119891 (120578)]119889120578 = 119866119891119891120578 + 1198661198911015840 (119891120578)1015840 + 11986611989110158401015840

(119891120578)10158401015840 + 119866119891101584010158401015840 (119891120578)101584010158401015840 (38)

11986610158401015840 [119891 (120578)] = 1198892119866 [119891 (120578)]1198891205782 = 119866119891119891 (119891120578)2 + 21198661198911198911015840

119891120578 (119891120578)1015840 + 211986611989111989110158401015840 119891120578 (119891120578)10158401015840 + 2119866119891119891101584010158401015840 119891120578 (119891120578)101584010158401015840+ 11986611989110158401198911015840 (119891120578)10158402 + 2119866119891101584011989110158401015840 (119891120578)1015840 (119891120578)10158401015840 + 21198661198911015840119891101584010158401015840 (119891120578)1015840 (119891120578)101584010158401015840 + 1198661198911015840101584011989110158401015840 (119891120578)101584010158402 + 211986611989110158401015840119891101584010158401015840 (119891120578)10158401015840sdot (119891120578)101584010158401015840 + 119866119891101584010158401015840119891101584010158401015840 (119891120578)1015840101584010158402 + 119866119891119891120578120578 + 1198661198911015840 (119891120578120578)1015840+ 11986611989110158401015840 (119891120578120578)10158401015840 + 119866119891101584010158401015840 (119891120578120578)101584010158401015840

(39)

119866101584010158401015840 [119891 (120578)] = 1198893119866 [119891 (120578)]1198891205783 = 119866119891119891119891 (119891120578)3 + 3

1198661198911198911198911015840 (119891120578)2 (119891120578)1015840 + 311986611989111989111989110158401015840 (119891120578)2 (119891120578)10158401015840 + 3119866119891119891119891101584010158401015840 (119891120578)2 (119891120578)101584010158401015840 + 311986611989111989110158401198911015840 (119891120578) (119891120578)10158402 + 4119866119891119891101584011989110158401015840 (119891120578) (119891120578)1015840 (119891120578)10158401015840 + 41198661198911198911015840119891101584010158401015840 (119891120578) (119891120578)1015840 (119891120578)101584010158401015840 + 31198661198911198911015840101584011989110158401015840 (119891120578) (119891120578)101584010158402 + 411986611989111989110158401015840119891101584010158401015840 (119891120578) (119891120578)10158401015840 (119891120578)101584010158401015840 + 3119866119891119891101584010158401015840119891101584010158401015840 (119891120578) (119891120578)1015840101584010158402 + 119866119891101584011989110158401198911015840 (119891120578)10158403 + 31198661198911015840119891101584011989110158401015840 (119891120578)10158402 (119891120578)10158401015840 + 11986611989110158401198911015840119891101584010158401015840 (119891120578)10158402 (119891120578)101584010158401015840 + 4119866119891101584011989110158401015840119891 (119891120578)1015840 119891120578 (119891120578)10158401015840 + 311986611989110158401198911015840101584011989110158401015840 (119891120578)1015840 (119891120578)101584010158402 + 4119866119891101584011989110158401015840119891101584010158401015840 (119891120578)1015840 (119891120578)10158401015840 (119891120578)101584010158401015840+ 41198661198911015840119891101584010158401015840119891 (119891120578)1015840 119891120578 (119891120578)101584010158401015840 + 4119866119891101584011989110158401015840101584011989110158401015840 (119891120578)1015840 (119891120578)10158401015840 (119891120578)101584010158401015840 + 31198661198911015840119891101584010158401015840119891101584010158401015840 (119891120578)1015840 (119891120578)1015840101584010158402+ 119866119891101584010158401198911015840101584011989110158401015840 (119891120578)101584010158403 + 31198661198911015840101584011989110158401015840101584011989110158401015840 (119891120578)101584010158402 (119891120578)101584010158401015840+ 311986611989110158401015840101584011989110158401015840101584011989110158401015840 (119891120578)1015840101584010158402 (119891120578)10158401015840 + 119866119891101584010158401015840119891101584010158401015840119891101584010158401015840 (119891120578)1015840101584010158403+ 3119866119891119891119891120578120578119891120578 + 31198661198911198911015840 119891120578120578 (119891120578)1015840 + 311986611989111989110158401015840 119891120578120578(119891120578)10158401015840 + 3119866119891119891101584010158401015840 119891120578120578 (119891120578)101584010158401015840 + 119866119891119891120578120578120578 + 31198661198911015840119891


(119891120578120578)1015840 (119891120578) + 311986611989110158401198911015840 (119891120578120578)1015840 (119891120578)1015840 + 3119866119891101584011989110158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 31198661198911015840119891101584010158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 1198661198911015840 (119891120578120578120578)1015840 + 311986611989110158401015840119891 (119891120578120578)10158401015840 (119891120578) + 3119866119891101584010158401198911015840 (119891120578120578)10158401015840 (119891120578)1015840 + 31198661198911015840101584011989110158401015840 (119891120578120578)10158401015840 (119891120578)10158401015840 + 311986611989110158401015840119891101584010158401015840 (119891120578120578)10158401015840 (119891120578)101584010158401015840 + 11986611989110158401015840 (119891120578120578120578)10158401015840 + 3119866119891101584010158401015840119891 (119891120578120578)101584010158401015840 (119891120578)+ 31198661198911015840101584010158401198911015840 (119891120578120578)101584010158401015840 (119891120578)1015840 + 311986611989110158401015840101584011989110158401015840 (119891120578120578)101584010158401015840 (119891120578)10158401015840+ 3119866119891101584010158401015840119891101584010158401015840 (119891120578120578)101584010158401015840 (119891120578)101584010158401015840 + 119866119891101584010158401015840 (119891120578120578120578)101584010158401015840

(40)

We note that the derivatives of 119891 with respect to 120578 that aregiven in (27) can be computed by (37)-(40) as

119866 [1198910 (120578)] = 1205741198781 + 120574 ((1205781198911015840101584010158400 (120578) + 311989110158400 (120578)+ 11989110158400 (120578) 119891101584010158400 (120578) minus 1198910 (120578) 1198911015840101584010158400 (120578)) +1198722119891101584010158400 (120578))

(41)

1198661015840 [1198910 (120578)] = 1198661198910 1198911 + 11986611989101015840 (1198911)1015840 + 119866119891101584010158400 (1198911)10158401015840+ 1198661198911015840101584010158400 (1198911)101584010158401015840

(42)

11986610158401015840 [1198910 (120578)] = 11986611989101198910 (1198911)2 + 2119866119891011989110158400 1198911 (1198911)1015840 + 21198661198910119891101584010158400 1198911 (1198911)10158401015840 + 211986611989101198911015840101584010158400 1198911 (1198911)101584010158401015840 + 1198661198911015840011989110158400 (1198911)10158402+ 211986611989110158400119891101584010158400 (1198911)1015840 (1198911)10158401015840 + 2119866119891101584001198911015840101584010158400 (1198911)1015840 (1198911)101584010158401015840+ 119866119891101584010158400 119891101584010158400 (1198911)101584010158402 + 2119866119891101584010158400 1198911015840101584010158400 (1198911)10158401015840 (1198911)101584010158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158402 + 1198661198910 1198912 + 11986611989110158400 (1198912)1015840 + 119866119891101584010158400 (1198912)10158401015840 + 1198661198911015840101584010158400 (1198912)101584010158401015840

(43)

119866101584010158401015840 [1198910 (120578)] = 119866119891011989101198910 (1198911)3 + 31198661198910119891011989110158400 (1198911)2 (1198911)1015840+ 311986611989101198910119891101584010158400 (1198911)2 (1198911)10158401015840 + 3119866119891011989101198911015840101584010158400 (1198911)2 (1198911)101584010158401015840+ 311986611989101198911015840011989110158400 (1198911) (1198911)10158402 + 4119866119891011989110158400119891101584010158400 (1198911) (1198911)1015840 (1198911)10158401015840 + 41198661198910119891101584001198911015840101584010158400 (1198911) (1198911)1015840 (1198911)101584010158401015840 + 31198661198910119891101584010158400 119891101584010158400 (1198911) (1198911)101584010158402 + 41198661198910119891101584010158400 1198911015840101584010158400 (1198911) (1198911)10158401015840 (1198911)101584010158401015840 + 311986611989101198911015840101584010158400 1198911015840101584010158400 (1198911) (1198911)1015840101584010158402 + 119866119891101584001198911015840011989110158400 (1198911)10158403 + 31198661198911015840011989110158400119891101584010158400 (1198911)10158402 (1198911)10158401015840 + 11986611989110158400119891101584001198911015840101584010158400 (1198911)10158402 (1198911)101584010158401015840 + 411986611989110158400119891101584010158400 1198910 (1198911)1015840 1198911 (1198911)10158401015840 + 311986611989110158400119891101584010158400 119891101584010158400 (1198911)1015840 (1198911)101584010158402

+ 411986611989110158400119891101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 1198910 (1198911)1015840 1198911 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 119891101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 3119866119891101584001198911015840101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)1015840101584010158402 + 119866119891101584010158400 119891101584010158400 119891101584010158400 (1198911)101584010158403 + 3119866119891101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)101584010158402 (1198911)101584010158401015840 + 31198661198911015840101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)1015840101584010158402 (1198911)10158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158403 + 311986611989101198910 11989121198911 + 3119866119891011989110158400 1198912(1198911)1015840 + 31198661198910119891101584010158400 1198912 (1198911)10158401015840 + 311986611989101198911015840101584010158400 1198912 (1198911)101584010158401015840+ 1198661198910 1198913 + 3119866119891101584001198910 (1198912)1015840 (1198911) + 31198661198911015840011989110158400 (1198912)1015840 (1198911)1015840+ 311986611989110158400119891101584010158400 (1198912)1015840 (1198911)10158401015840 + 3119866119891101584001198911015840101584010158400 (1198912)1015840 (1198911)10158401015840+ 11986611989110158400 (1198913)1015840 + 3119866119891101584010158400 1198910 (1198912)10158401015840 (1198911) + 3119866119891101584010158400 11989110158400 (1198912)10158401015840 (1198911)1015840 + 3119866119891101584010158400 119891101584010158400 (1198912)10158401015840 (1198911)10158401015840 + 3119866119891101584010158400 1198911015840101584010158400 (1198912)10158401015840 (1198911)101584010158401015840 + 119866119891101584010158400 (1198913)10158401015840 + 31198661198911015840101584010158400 1198910 (1198912)101584010158401015840 (1198911)+ 31198661198911015840101584010158400 11989110158400 (1198912)101584010158401015840 (1198911)1015840 + 31198661198911015840101584010158400 119891101584010158400 (1198912)101584010158401015840 (1198911)10158401015840+ 31198661198911015840101584010158400 1198911015840101584010158400 (1198912)101584010158401015840 (1198911)101584010158401015840 + 1198661198911015840101584010158400 (1198913)101584010158401015840

(44)

Now we need to extract the first derivatives of G as follows

1198661198910 = minus 1205741198781 + 1205741198911015840101584010158400 (120578) 11986611989101198910 = 119866119891011989110158400 = 1198661198910119891101584010158400 = 011986611989101198911015840101584010158400 = minus 1205741198781 + 120574 119866119891011989101198910 = 1198661198910119891101584001198910 = 11986611989101198911015840011989110158400 = 1198661198910119891101584010158400 11989110158400 = 1198661198910119891011989110158400

= 11986611989101198911015840101584010158400 1198911015840101584010158400 = 1198661198910119891101584010158400 119891101584010158400 = 011986611989110158400 = 1205741198781 + 120574 = (3 + 119891101584010158400 (120578))

119866119891101584001198910 = 1198661198911015840011989110158400 = 119866119891101584001198911015840101584010158400 = 011986611989110158400119891101584010158400 = 1205741198781 + 120574 1198661198911015840011989101198910 = 11986611989110158400119891101584001198910 = 119866119891101584001198911015840011989110158400 = 11986611989110158400119891101584010158400 11989110158400 = 119866119891101584001198911015840101584010158400 11989110158400

= 119866119891101584001198911015840101584010158400 1198911015840101584010158400 = 11986611989110158400119891101584010158400 119891101584010158400 = 0119866119891101584010158400 = 1205741198781 + 1205741198911015840 (120578) + 12057411987221 + 120574

119866119891101584010158400 1198910 = 119866119891101584010158400 119891101584010158400 = 119866119891101584010158400 1198911015840101584010158400 = 0


119866119891101584010158400 11989110158400 = 1205741198781 + 120574 119866119891101584010158400 11989101198910 = 119866119891101584010158400 119891101584001198910 = 119866119891101584010158400 1198911015840011989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400

= 119866119891101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 119866119891101584010158400 119891101584010158400 119891101584010158400 = 01198661198911015840101584010158400 = 1205741198781205781 + 120574 minus 1205741198781 + 1205741198910 (120578)

1198661198911015840101584010158400 11989110158400 = 1198661198911015840101584010158400 119891101584010158400 = 1198661198911015840101584010158400 1198911015840101584010158400 = 01198661198911015840101584010158400 1198910 = minus 1205741198781 + 120574

1198661198911015840101584010158400 11989101198910 = 1198661198911015840101584010158400 119891101584001198910 = 1198661198911015840101584010158400 1198911015840011989110158400 = 1198661198911015840101584010158400 119891101584010158400 11989110158400 = 1198661198911015840101584010158400 1198911015840101584010158400 11989110158400= 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 0

(45)

and from (36) by using (41)-(45) we obtain

1198910 = 1198602120578 + 1611986041205783 (46)

1198911 = 12520119903119878119860241205787 + 1120119903 (41198781198604 +11987221198604) 1205785 (47)

1198912 = ( 1210119903211987821198604 + 1504119903211987811987221198604 minus 16301199032119860211987821198604minus 125201199032119878119860211987221198604 + 15040119903211987241198604) 1205787minus ( 14536011990321198782119860241198602 minus 1604801199032119872211987811986024minus 1113401199032119878211986024) 1205789 minus 12494800119903211987821198603412057811

(48)

1198913 = ( 11890119903311987831198604 + 13453601199033119878211986041198722+ minus122681199033119860211987831198604 + minus1567011990331198782119860211987221198604+ 120160119903311987811987241198604 + 11134011990331198602211987831198604+ 145360119903311986022119878211987221198604 + minus1604801199033119878119860211987241198604+ 1362880119903311987261198604) 1205789 + ( 1693001199033119878311986042minus 1324948001199033119905111198783119860421198602 + 59997920011990331198722119878211986042minus 11425600119903311986042119878211986021198722 + 13199584001199033119878119860421198724+ 1831600119903311986022119878311986042) 12057811 + ( minus188452001199033119878311986043

+ 11389960011990331198783119860431198602 + minus19729720011990331198722119878211986043)sdot 12057813 + 8340864824000119903311987831198604412057815

(49)

From Step 4 substituting (46)-(49) in (21) we get theanalytical-approximate solution

119891 (120578) = 1198602120578 + 1611986041205783 + 1120119903 (41198781198604 +11987221198604) 1205785+ ( 1210119903211987821198604 + 1504119903211987811987221198604 minus 16301199032119860211987821198604minus 125201199032119878119860211987221198604 + 15040119903211987221198604+ 12520119903S11986024) 1205787 + ( minus14536011990321198782119860241198602+ 1604801199032119872211987811986024 + 11890119903311987831198604+ 13199584001199033119878119860241198602 + minus122681199033119860211987831198604+ minus1567011990331198782119860211987221198604 + 120160119903311987811987241198604+ 11134011990331198602211987831198604 + 145360119903311986022119878211987221198604+ minus1604801199033119878119860211987241198604 + 1362880119903311987261198604) 1205789 +

(50)

5 Convergence Analysis

Here we study the analysis of convergence for the analytical-approximate solution (50) resulting from the application ofpower series novel algorithm for solving the problem of thesqueezing flow between two parallel plates

Definition 1 Suppose that 119867 is Banach space 119877 is the realnumber and 119866[119865] is a nonlinear operator defined by 119866[119865] 119867 997888rarr 119877 Then the sequence of the solutions generated by anew approach can be written as

119865119899+1 = 119866 [119865119899] 119865119899 = 119899sum119896=0

119891119896119899 = 0 1 2 3

(51)

where 119866[119865] satisfies Lipschitz condition such that for 120572 gt 0120572 isin 119877 we have1003817100381710038171003817119866 [119865119899] minus 119866 [119865119899minus1] ⩽ 120572 1003817100381710038171003817119865119899 minus 119865119899minus11003817100381710038171003817 (52)


Theorem 2 e series of the analytical-approximate solution119891(120578) = suminfin119896=0 119891119896(120578) generated by novel approach converge if thefollowing condition is satisfied

1003817100381710038171003817119865119899 minus 1198651198981003817100381710038171003817 997888rarr 0 119886119904 119898 997888rarr infin119891119900119903 0 ⩽ 120572 lt 1 (53)

Proof From the above definition we have

1003817100381710038171003817119865119899 minus 1198651198981003817100381710038171003817 =1003817100381710038171003817100381710038171003817100381710038171003817119899sum119896=0

119891119896 minus 119898sum119896=0

1198911198961003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817[1198910 + 119871minus1119899sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]

minus [1198910 + 119871minus1 119898sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]1003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817119871minus1119866[119899minus1sum

119896=0

119891119896] minus 119871minus1119866[119898minus1sum119896=0

119891119896]1003817100381710038171003817100381710038171003817100381710038171003817

(119904119894119899119888119890 119865119899 = 119866 [119865119899minus1])⩽ 10038161003816100381610038161003816Łminus110038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119866[119899minus1sum119896=0

119891119896] minus 119866[119898minus1sum119896=0

119891119896]1003817100381710038171003817100381710038171003817100381710038171003817

⩽ 10038161003816100381610038161003816119871minus110038161003816100381610038161003816 1003817100381710038171003817119866 [119865119899minus1] minus 119866 [119865119898minus1]1003817100381710038171003817 ⩽ 120572 1003817100381710038171003817119865119899minus1 minus 119865119898minus11003817100381710038171003817

(54)

since G[F] satisfies Lipschitz condition Let 119899 = 119898 + 1 then1003817100381710038171003817119865119898+1 minus 1198651198981003817100381710038171003817 ⩽ 120572 1003817100381710038171003817119865119898 minus 119865119898minus11003817100381710038171003817 (55)

and hence

1003817100381710038171003817119865119898 minus 119865119898minus11003817100381710038171003817 ⩽ 120572 1003817100381710038171003817119865119898minus1 minus 119865119898minus21003817100381710038171003817 ⩽ sdot sdot sdot⩽ 120572119898minus1 10038171003817100381710038171198651 minus 11986501003817100381710038171003817 (56)

From (56) we get

10038171003817100381710038171198652 minus 11986511003817100381710038171003817 ⩽ 120572 10038171003817100381710038171198651 minus 11986501003817100381710038171003817 10038171003817100381710038171198653 minus 11986521003817100381710038171003817 ⩽ 1205722 10038171003817100381710038171198651 minus 11986501003817100381710038171003817 10038171003817100381710038171198654 minus 11986531003817100381710038171003817 ⩽ 1205723 10038171003817100381710038171198651 minus 11986501003817100381710038171003817

1003817100381710038171003817119865119898 minus 119865119898minus11003817100381710038171003817 ⩽ 120572119898minus1 10038171003817100381710038171198651 minus 11986501003817100381710038171003817

(57)

Using triangle inequality

1003817100381710038171003817119865119899 minus 1198651198981003817100381710038171003817 = 1003817100381710038171003817119865119899 minus 119865119899minus1 minus 119865119899minus2 minus sdot sdot sdot minus 119865119898+1 minus 1198651198981003817100381710038171003817 ⩽ 1003817100381710038171003817119865119899 + 119865119899minus11003817100381710038171003817 + 1003817100381710038171003817119865119899minus1 minus 119865119899minus21003817100381710038171003817 + sdot sdot sdot + 1003817100381710038171003817119865119898+1 minus 1198651198981003817100381710038171003817 ⩽ [120572119899minus1 + 120572119899minus2 + sdot sdot sdot + 120572119898] 10038171003817100381710038171198651 minus 11986501003817100381710038171003817 = 120572119898 [120572119899minus119898minus1 + 120572119899minus119898minus2 + sdot sdot sdot + 1] 10038171003817100381710038171198651 minus 11986501003817100381710038171003817 ⩽ 1205721198981 minus 120572 10038171003817100381710038171198651 minus 11986501003817100381710038171003817

(58)

as119898 997888rarr infin we have 119865119899minus119865119898 997888rarr 0 and then119865119899 is a Cauchysequence in Banach space119867

Theorem 3 e solution 119891(120578) = suminfin119896=0 119891119896(120578) converges isanalytic solution will be close to the solution of problem (11)-(12) when the following property is achieved

119871minus1119866 [119891] = lim119899997888rarrinfin

119871minus1 [119866 [119865119899]] (59)

Proof For 119865 isin 119867 define an operator for119867 to119867 by

119869 (119865) = 1198650 + 119871minus1119866 [119865] (60)

and let 1198651 1198652 isin 119867 we have

1003817100381710038171003817119869 (1198651) minus 119869 (1198652)1003817100381710038171003817= 10038171003817100381710038171003817[1198650 + 119871minus1119866 [1198651]] minus [1198650 + 119871minus1119866 [1198652]]10038171003817100381710038171003817 = 10038171003817100381710038171003817119871minus1119866 [1198651] minus 119871minus1119866 [1198652]10038171003817100381710038171003817 ⩽ 10038161003816100381610038161003816119871minus110038161003816100381610038161003816 1003817100381710038171003817119866 [1198651] minus 119866 [1198652]1003817100381710038171003817 ⩽ 120574 10038171003817100381710038171198651 minus 11986521003817100381710038171003817

(61)

so the mapping 119869 is contraction and by the Banach fixed-point theorem for contraction there is a unique solution ofproblem (11)-(12) Now we prove that that the series solution119891(120578) satisfies problem (11)-(12)

119871minus1119866 [119891] = 119871minus1119866[infinsum119896=0

119891119896]

= 119871minus1119866[ lim119899997888rarrinfin

119899sum119896=0

119891119896] = 119871minus1119866[ lim

119899997888rarrinfin119865119899]

= lim119899997888rarrinfin

119871minus1119866 [119865119899]

(62)

In practice Theorems 2 and 3 suggest computing thevalue of 120572 as described in the following definition


Table 1 Comparison between analytical-approximate solution andnumerical solution for 119891(120578) when119872 = 0 and 120574 997888rarr infin

S Wang [7] Present results097800 -219100 -219900000000 -300000 -300000-049770- -261930 -261910-009998 -292770 -292770009403 -306630 -306630043410 -329400 -329400012240 -370800 -370500

Definition 4 For 119896 = 1 2 3

120572119896 =

1003817100381710038171003817119865119896+1 minus 119865119896100381710038171003817100381710038171003817100381710038171198651 minus 11986501003817100381710038171003817 = 1003817100381710038171003817119891119896+11003817100381710038171003817100381710038171003817100381711989111003817100381710038171003817 100381710038171003817100381711989111003817100381710038171003817 = 00 100381710038171003817100381711989111003817100381710038171003817 = 0 (63)

and now we will apply Definition 4 on the squeezing flowbetween two parallel plates to find convergence then weobtain for example the following

if we put 119878 = minus1 119872 = 1 1198602 = 1511619152 1198604 =minus3143837395 120574 = 02 the value of 120572 will be

10038171003817100381710038171198652 minus 119865110038171003817100381710038172 ⩽ 120572 10038171003817100381710038171198651 minus 119865010038171003817100381710038172 997904rArr120572 = 00078755538 lt 1



10038171003817100381710038171198652 minus 11986511003817100381710038171003817+infin ⩽ 120572 10038171003817100381710038171198651 minus 11986501003817100381710038171003817+infin 997904rArr

120572 = 00078442924 lt 110038171003817100381710038171198653 minus 11986521003817100381710038171003817+infin ⩽ 1205722 10038171003817100381710038171198651 minus 11986501003817100381710038171003817+infin 997904rArr


1205723 = 00000001976 lt 1

(64)

Also if we choose 119878 = 1119872 = 1 120574 = 02 1198602 = 14811866591198604 = minus2772867578 then we obtain







1205723 = 00000006301 lt 1

(65)

Then suminfin119896=0 119891119896(120578) converges to the solution 119891(120578) when 0 ⩽120572119896 lt 1 119896 = 1 2 6 Results and Discussions

In this section the influences of the squeeze number 119878 Cassonfluid parameter 120574 and the magnetic number119872 on the axial119891(120578) and radial1198911015840(120578) velocities have been represented Table 1shows a comparison between the present results obtainedfrom a novel algorithm and the numerical results introducedby Wang [7] while Table 2 shows a comparison between thecollected results with variation of parameter method (VPM)[21] It was noted that the solution of the present algorithmis in agreement with these methods The convergence of theof values 1198911015840(0) and 119891101584010158401015840(0) is indicated in Table 3 Table 4illustrates the numerical values of skin friction coefficientwhich can be observed from an increase of all the parametersleading to an increase of a magnitude of the skin frictioncoefficient Subsequently there is also a comparison withvariation parametermethod (VPM) so that the results appearrelatively convergent A comparison between the analytical-approximate solution and the numerical solution for the axialand radial velocities was shown in Tables 5 and 6 Thesetables clearly demonstrated that the solution has approximatemanner In Figures 2ndash7 there are two cases with regard to thesqueeze number 119878 three figures show the casewhen the platesare moving apart 119878 gt 0 In the other case there are three


Table 2 Comparison of 119891(120578) for119872 = 1 and 120574 = 04120578 119878 = 5 119878 = minus5

Present results VPM [21] 119877119870 minus 4 Present results VPM [21] 119877119870 minus 401 0139103 0139081 0139104 0166665 0166839 016666302 0276402 0276358 0276405 0328112 0328444 032810703 0409981 0409918 0409984 0479402 0479861 047939604 0537705 0537628 0537709 0616144 0616685 061613705 0657098 0537628 0657105 0734719 0735286 073471306 0765208 0765125 0765217 0832465 0832992 083246107 0858455 0858383 0858467 0907792 0908218 090779708 0932458 0932408 0932471 0960237 0960506 096027009 0981839 0981819 0981843 0990434 0990529 0990550

Table 3 Convergence of analytical-approximate solution for119872 = 1 and 120574 = 02order of approximations 119878 = 1 119878 = minus11198602 1198604 1198602 11986042 terms 1481730942 -2777709528 1511797484 -31459605803 terms 1481190867 -2772901421 1511619398 -31438389234 terms 1481186624 -2772867340 1511619138 -31438372875 terms 1481186658 -2772867569 1511619153 -31438373986 terms 1481186659 -2772867578 1511619152 -31438373957 terms 1481186659 -2772867578 1511619152 -31438373958 terms 1481186659 -2772867578 1511619152 -3143837395

Table 4 Comparison of the values for skin friction coefficient (1 +1120574)11989110158401015840(1)119878 120574 119872 present results VPM [19]-5 04 1 -63949610 -62987080-3 04 1 -83207270 -83207270-1 04 1 -99705750 -997037601 04 1 -11375837 -113762403 04 1 -12604612 -126106695 04 1 -13697927 -13718095-3 01 1 -30991843 -30991005-3 03 1 -10851771 -10873387-3 05 1 -67896240 -677154903 01 1 -35260196 -352601963 03 1 -15145259 -151495773 05 1 -11071084 -11078736-3 04 2 -90435300 -90381960-3 04 4 -11532951 -11531981-3 04 6 -14819321 -148193213 04 2 -13092241 -131015723 04 4 -14908219 -149082193 04 6 -17471534 -17501183

figures when dilating plates 119878 lt 0 These cases explain theeffects of physical parameters and studying the path of thecurves 119891(120578) and 1198911015840(120578) between parallel plates The results ofthese cases can be summarized in the following points

(i) 119879ℎ119890 119891119894119903119904119905 119888119886119904119890 119878 gt 0 Figure 2 shows the effects ofincreasing values of squeeze number 119878 on the axialvelocity and average of radial velocityThis Figure alsoindicates that an increase of 119878 leads to a decrease of119891(120578) In addition the curves of 119891(120578) are started fromthe top plate towards the bottom plate according tothe increase in 119878 for 0 lt 120578 lt 1 while for 1198911015840(120578) thesame case occurred until 120578 le 05 and the reversesituation of its curves happened for 05 lt 120578 le 1Themagnetic number119872 and Casson fluid parameter120574 on 119891(120578) and 1198911015840(120578) were described in Figures 3 and4 respectively It can be observed that increasing 119872and 120574 leads to decrease of 119891(120578) for the positive valueof squeeze number 119878 Moreover the increase of119872 and120574 on 1198911015840(120578) for the positive value of squeeze number 119878proves the similar behavior in case of the increase in119878

(ii) 119879ℎ119890 119904119890119888119900119899119889 119888119886119904119890 119878 lt 0 Figure 5 illustrates the effectsof the negative values for the squeezing number 119878showing that the decrease of 119878 values causes increas-ing in axial velocity This figure demonstrates thatthe curves of 119891(120578) which starts from the bottomplate towards the top plate tend to decrease 119878 for0 lt 120578 lt 1 while for 1198911015840(120578) it was found that thesame case happened when 120578 le 04 and the reversesituation of their curves happened for 04 lt 120578 le 1Figure 6 indicates an effect ofmagnetic number119872 foraxial and radial velocity This effect is similar to theeffect of 119872 in the positive value of squeeze number


Table 5 Comparison between the solution of present algorithm and 119877119870 minus 4 algorithm for119872 = 1 120574 = 02 119878 = 1119891(120578) 1198911015840(120578)120578 Present results 119877119870 minus 4 Present results 119877119870 minus 4

01 0147656 0147656 1467313 146731302 0292534 0292534 1425575 142557503 0431831 0431832 1355629 135562904 0562701 0562701 1256898 125689905 0682225 0682226 1128583 112858306 0787397 0787398 0969661 096966107 0875096 0875097 0778889 077888108 0942065 0942067 0554821 055482109 0984895 0984897 0295804 0295804

Table 6 Comparison between the solution of present algorithm and 119877119870 minus 4 algorithm for119872 = 1 120574 = 02 119878 = minus1119891(120578) 1198911015840(120578)120578 Present results 119877119870 minus 4 Present results 119877119870 minus 4

01 0150638 0150638 1485907 149590702 0298136 0298136 1448847 144884703 0439360 0439361 1370673 137067304 0571246 0571246 1261767 126176705 0690716 0690716 1122650 112265006 0794791 0794790 0953971 095397107 0880550 0880541 0756487 075648708 0945156 0945156 0531047 053104709 0985858 0985876 0278565 0278566

on 119891(120578) and 1198911015840(120578) but the point of intersection ofthe curves 1198911015840(120578) is 120578 = 04 Figure 7 demonstratesthe effects of Casson fluid parameter 120574 for axial andradial velocity it was also noted from this figurethat the increasing of 120574 causes increasing of 119891(120578)Subsequently the movement of the curves of 119891(120578)occurs from the bottom plate across the top plate dueto an increase of 120574 for 0 lt 120578 lt 1 whereas thesame situation of 1198911015840(120578) occurs when 120578 le 04 and thereverse case of their curves occurs when 04 lt 120578 le 1

From the above two cases it can be seen that the curves of119891(120578) and 1198911015840(120578) are different in distance between them wherein 119878 gt 0 it is wide then 119878 lt 0 However they have thesame behavior Moreover the bifurcation of 1198911015840(120578) in 119878 gt 0happened when 120578 = 05 and for 119878 lt 0 happened when120578 = 04 Generally it seems that the curve of119891(120578) is increasingwith increasing of 120578 for all values of 119878 119872 and 120574 while thecurve of 1198911015840(120578) is decreasing with increasing of 120578 for all abovevalues As for the effects of different physical parameters (thesqueezing number 119878 Casson fluid parameter 120574 the magneticnumber 119872) and the values 120572 119897 on the velocity components119906(119909 119910 119905) V(119909 119910 119905) and the vorticity 120596(119909 119910 119905) the followingcan be concluded

(i) In the case that the physical parameters and the values120572 119897 are constants (when there is a change in thesqueezing number) the surfaces are almost identical

except for the surface V(119909 119910 119905) This case shows aslight change which is a little curvature as indicatedin Figures 8 and 9 However there is a change in thesurfaces when 120572 lt 0 and 119878 ge 11 with a change inthe surface of119908(119909 119910 119905) while the surfaces of 119906(119909 119910 119905)and V(119909 119910 119905) remain similar

(ii) There is no change of other fixations values on thesurfaces when a change happened in the parameter119872or 120574 as shown in Figures 10ndash14 But there is change inthe surfaces 119906(119909 119910 119905) 119908(119909 119910 119905) and V(119909 119910 119905) when120572 lt 0 119878 le minus14 119872 le 4 and 120574 ge 04 with observingthat the surface V(119909 119910 119905) simply changes when119872 = 4in Figure 11 and 120574 = 04 in Figure 14

(iii) All the physical parameters are constants except achange in their values 120572 even if they are positive ornegative Each value of 120572 shows that the surface isdifferent from the other values of 119906(119909 119910 119905) V(119909 119910 119905)and 120596(119909 119910 119905) It was also noticed that the surfaceV(119909 119910 119905) takes the negative values of 120572 inversely withthe surface V(119909 119910 119905) that has a positive value of 120572Thiscan be seen in Figures 15 and 16

(iv) Finally Figures 17ndash19 show that there is no changein other fixations values on the surfaces 119906(119909 119910 119905)120596(119909 119910 119905) and V(119909 119910 119905) when a change happened invalue 119897 But there is change in the surfaces when120572 lt 0 minus15 lt 119897 lt 15 for 119906(119909 119910 119905) 120596(119909 119910 119905)


1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10

Figure 2 119891(120578) and 1198911015840(120578) for positive values 119878 when119872 = 1 120574 = 041

08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10

Figure 3 119891(120578) and 1198911015840(120578) for the values119872 when 119878 = 3 120574 = 03

and V(119909 119910 119905) with noticing that the surface V(119909 119910 119905)slightly changes when 119897 = 1 in Figure 18

Physically magneto field parameter 119872 is increased thisleads to stronger Lorentz force along the vertical directionwhich offers more resistance to the flowThe velocity profilesexponentially decay to zero at shorter distances when either119872 or 120574 is increased This indicates that increase in either119872 or 120574 leads to thinner boundary layer Casson fluid

between parallel plates is a non-Newtonian fluid and itsparameter 120574 corresponds to viscosity in Newtonian fluidThus the increasing of 120574 leads to increase of viscous forceand decrease of the velocity of fluid The squeezing number119878 includes the flow which results from a movement ofthe two parallel plates with a fluid apart The geometricexplanation can be summarized as the magnetic field isapplied to electrical conducting fluid which accompaniesan increase of 119878 leading to increase of the velocity of the


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

Figure 4 119891(120578) and 1198911015840(120578) for the values 120574 when119872 = 2 119878 = 4

02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10

Figure 5 119891(120578) and 1198911015840(120578) for negative values 119878 when119872 = 1 120574 = 04

fluid This helps to generate a magnetic normal force tothe direction of the magnetic field Current geometry inthis case comprises the magnetic field which is applied in aperpendicular direction to generate a force in 119909-directionThis force can reduce the movement of the fluid thereforethe magnetic field is used to control or reduce the movementof the fluid

7 Conclusions

In this paper a novel algorithm for analytic techniquebased on the coefficients of power series resulting fromintegrating nonlinear differential equations with appro-priate conditions is proposed and it is employed toobtain a new analytical-approximate solution for unsteady


1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10

Figure 6 119891(120578) and 1198911015840(120578) for the values119872 when 119878 = minus10 120574 = 04

1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04

Figure 7 119891(120578) and 1198911015840(120578) for the values 120574 when119872 = 2 119878 = minus5

two-dimensional nonlinear squeezing flow between twoparallel plates successfully It has been found that the con-struction of the novel algorithm possessed good convergentseries and the convergence of the results is shown explicitlyGraphs and tables are presented to investigate the influence ofphysical parameters on the velocity Convergence analysis iscarried out to back up the validity of the novel algorithm andcheck its computational efficiency Analysis of the converge

confirms that the novel algorithm is an efficient techniqueas compared to other methods presented in this work Ascan be seen from the comparison the computational resultsof the present solution and the results of other solutionsare identical with 5 or 6 decimal places It is noted thata magnetic field is a control phenomenon in many flowsand it could be employed to normalize the flow behaviorMoreover a squeeze number plays an important role in these


002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109

Figure 8 Surface of 119906 V and 120596 for119872 = 1 S = 5 120574 = 05 119910 = 1 120572 = 3 119897 = 2

002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200

Figure 10 Surface of 119906 V and 120596 forM = 1 119878 = minus15 120574 = 05 119910 = 1 120572 = minus4 119897 = 2

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1

Figure 13 Surface of 119906 V and 120596 for119872 = 1 119878 = minus14 120574 = 01 119910 = 1 120572 = minus3 119897 = 2

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-

Figure 15 Surface of 119906 V and 120596 for119872 = 1 119878 = 5 120574 = 05 119910 = 1 120572 = minus6 119897 = 2


002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109

Figure 16 Surface of 119906 V and 120596 for119872 = 1 119878 = 5 120574 = 05 119910 = 1 120572 = 6 119897 = 2

002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10

Figure 17 Surface of 119906 V and 120596 for119872 = 2 119878 = minus10 120574 = 03 119910 = 1 120572 = minus2 l = 08

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-



types of problems as increasing the squeeze number leadsto increase in the velocity profile The graphs which have astrong magnetic field were used to enhance the flow whenthe plates came together In addition the squeeze numberincreased the velocity profile when the plates become closerand going apart

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

We would like to thank Assistant Professor Mahdi MohsinMohammed and Dr Ahmed Majeed Jassem for Englishproofreading

References

[1] M J Stefan ldquoVersuch Uber die scheinbare adhesionrdquo AkadWissensch Wien Math Natur vol 69 p 713 1874

[2] O Reynolds ldquoOn the theory of lubrication and its application toMr Beauchamp towerrsquos experiments including an experimentaldetermination of the viscosity of olive oilrdquo Philosophical Trans-actions of the Royal Society A vol 177 pp 157ndash234 1886

[3] F R Archibald ldquoLoad capacity and time relations for squeezefilmsrdquo Journal of LubricationTechnology vol 78 ppA231ndashA2451956

[4] J D Jackson ldquoA study of squeezing flowrdquo Applied ScientificResearch vol 11 no 1 pp 148ndash152 1963

[5] RUsha andR Sridharan ldquoArbitrary squeezing of a viscous fluidbetween elliptic platesrdquo Fluid Dynamics Research vol 18 no 1pp 35ndash51 1996

[6] R I Yahaya N M Arifin and S S P M Isa ldquoStability analysison magnetohydrodynamic flow of casson fluid over a shrinkingsheet with homogeneous-heterogeneous reactionsrdquo Entropyvol 20 no 9 2018

[7] C Y Wang ldquoThe squeezing of a fluid between two platesrdquoJournal of Applied Mechanics pp 579ndash583 1979

[8] Y Bouremel ldquoExplicit series solution for the Glauert-jet prob-lem by means of the homotopy analysis methodrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 12 no5 pp 714ndash724 2007

[9] M M Rashidi H Shahmohamadi and S Dinarvand ldquoAna-lytic approximate solutions for unsteady two-dimensional andaxisymmetric squeezing flows between parallel platesrdquo Mathe-matical Problems in Engineering vol 2008 Article ID 93509513 pages 2008

[10] G Domairry and A Aziz ldquoApproximate analysis of MHDsqueeze flow between two parallel disk with sunction orinjection by homotopy perturbuation methodrdquo MathematicalProblems in Engineering vol 2009 Article ID 603916 19 pages2009

[11] X J Ran Q Y Zhu and Y Li ldquoAn explicit series solution ofthe squeezing flow between two infinite plates by means of

the homotopy analysis methodrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 1 pp 119ndash1322009

[12] Z Makukula S Motsa and P Sibanda ldquoOn a new solutionfor the viscoelastic squeezing flow between two parallel platesrdquoJournal of Advanced Research in AppliedMathematics vol 2 no4 pp 31ndash38 2010

[13] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013

[14] U Khan S I Khan N Ahmed S Bano and S T Mohyud-Din ldquoHeat transfer analysis for squeezing flow of a Casson fluidbetween parallel platesrdquo Ain Shams Engineering Journal vol 7no 1 pp 497ndash504 2016

[15] HMDuwairi B Tashtoush andRADamseh ldquoOnheat trans-fer effects of a viscous fluid squeezed and extruded between twoparallel platesrdquoHeat andMass Transfer vol 41 no 2 pp 112ndash1172004

[16] S T Mohyud-Din Z A Zaidi U Khan and N Ahmed ldquoOnheat and mass transfer analysis for the flow of a nanofluidbetween ratating parallel platesrdquoAerospace Science and Technol-ogy vol 46 Article ID 4692014 pp 514ndash522 2014

[17] S T Mohyud-Din and S I Khan ldquoNonlinear radiation effectson squeezing flow of a Casson fluid between parallel disksrdquoAerospace Science and Technology vol 48 pp 186ndash192 2016

[18] N Ahmed U Khan X J Yang Z A Zaidi and S T Mohyud-Din ldquoMagneto hydrodynamic (MHD) squeezing flow of aCasson fluid between parallel disksrdquo International Journal ofPhysical Sciences vol 8 pp 1788ndash1799 2013

[19] U Khan N Ahmed Z A Zaidi M Asadullah and S TMohyud-Din ldquoMHD squeezing flow between two infiniteplatesrdquo Ain Shams Engineering Journal vol 5 no 1 pp 187ndash1922014

[20] T Hayat A Yousaf M Mustafa and S Obaidat ldquo(MHD)squeezing flow of second-grade fluid between two paralleldisksrdquo International Journal for Numerical Methods in Fluidsvol 69 no 2 pp 399ndash410 2012

[21] N Ahmed U Khan S I Khan S Bano and S T Mohyud-Din ldquoEffects on magnetic field in squeezing flow of a Cassonfluid between parallel platesrdquo Journal of King Saud University -Science vol 29 no 1 pp 119ndash125 2017

[22] S Nadeem R Mehmood and N S Akbar ldquoOptimizedanalytical solution for oblique flow of a Casson-nano fluidwith convective boundary conditionsrdquo International Journal ofermal Sciences vol 78 pp 90ndash100 2014

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analytical-approximate solution which depends mainly onthe coefficients of powers seriesThis is close to the numericalsolution for the squeezing flow of a Casson fluid betweenparallel plates although the resulting series of the novelalgorithm which contains initial conditions are known theothers are unknown and their values can be found throughthe boundary conditions that are defined These series alsocontain physical parameters which can be compensated byconstants and the geometrical effects of these parameterswould be investigated depending on the resulting solutionsIn this work it can be clearly seen that a novel algorithmis successfully applied to solve the equations of an unsteadysqueeze of flow between parallel plates as well as to find ananalytical-approximate solution In contrast it was found thatour results from algorithm completely agree with the resultsobtained from thenumerical solution through the applicationof Range-Kutta fourth-order method [RK-4] the numericalsolutions of Wang [7] and variation of parameter method(VPM) [21] The organization of this paper is as followsThe governing equations are derived in Section 2 Details ofderivation of the novel algorithm have been written as stepsin Section 3 The performance of the novel algorithm forthe squeezing flow is applied in Section 4 In Section 5 theconvergence analysis is presented Results and discussions aregiven in Section 6 Finally the conclusions are indicated inSection 7

2 Governing Equations

Weconsider an incompressible flowof aCassonfluid betweentwo parallel plates separated by a distance y = plusmn119897(1 minus 120572119905)12 =plusmnℎ(119905) where 119897 is the initial gap between the plates (at a time119905 = 0) Additionally 120572 gt 0 corresponds to a squeezingmotionof both plates until they touch each other at 119905 = 1120572 for120572 lt 0the plates bear a receding motion and dilate as described inFigure 1

Rheological equation for Casson fluid is defined [22] as

120591119894119895 =

2[120583119861 + (1199011199102120587)] 119890119894119895 120587 gt 1205871198882 [120583119861 + ( 1199011199102120587119888)] 119890119894119895 120587119888 gt 120587

(1)

where 120587 = 119890119894119895119890119894119895 and 119890119894119895 is the (119894 119895)119905ℎ component of thedeformation rate 120587 is the product of the component ofdeformation rate with itself 120587119888 is the critical value of thesaid product 120583119861 is the plastic dynamic viscosity of the non-Newtonian fluid and 119901119910 is the yield stress of slurry fluidWe are also applying the following assumptions on the flowmodel

(i) The effects of induced magnetic and electric fieldproduced due to the flow of electrically conductingfluid are negligible

(ii) No external electric field is present

Under aforementioned constraints the conservation equa-tions for the flow are120597119906120597119909 + 120597V120597119910 = 0 (2)

120597119906120597119905 + 119906120597119906120597119909 + V120597119906120597119910

= minus1120588 120597119901120597119909 + 120583120588 (1 + 1120574)(212059721199061205971199092 + 12059721199061205971199102 + 1205972V120597119910120597119909)

minus 1205901205732120588 119906(3)

120597V120597119905 + 119906 120597V120597119909 + V120597V120597119910

= minus1120588 120597119901120597119910 + 120583120588 (1 + 1120574)(2 1205972V1205971199092 + 1205972V1205971199102 + 1205972119906120597119910120597119909)

(4)

where 119906 and V are the velocity components in 119909 and 119910directions respectively 119901 is the pressure 120583120588 is the dynamicviscosity of the fluid (ratio of Kinematic viscosity and den-sity) 120574 = 120583119861radic2120587119888119901119910 is the Casson fluid parameter and120573 is the magnitude of imposed magnetic field Supportingconditions for the problem are as follows

119906 = 0V = V119908 = 119889ℎ119889119905

119886119905 119910 = ℎ (119905) 120597119906120597119910 = 0V = 0

119886119905 119910 = 0

(5)

and we can simplify the above system of equations byeliminating the pressure term equations (3)-(4) and using (2)After cross differentiation and introducing vorticity 120596

120596 = 120597V120597119909 minus 120597119906120597119910 (6)

we get

120597120596120597119905 + 119906120597120596120597119909 + V120597120596120597119910 = 120583120588 (1 + 1120574)(120597

21205961205971199092 + 12059721205961205971199102 )minus 1205901205732120588 120597119906120597119910

(7)

Transform introduced by Wang [7] for a two dimensionalflow is stated as

119906 = 120572119909[2 (1 minus 120572119905)12]1198911015840 (120578) (8)

V = minus120572119897[2 (1 minus 120572119905)12]119891 (120578) (9)


y

ux

D

2l(1 minus t)12


where

120578 = 119910[119897 (1 minus 120572)12] (10)



(11)


119891 (0) = 011989110158401015840 (0) = 0119891 (1) = 11198911015840 (1) = 0

(12)


11989721199092 (1 minus 120572119905)119877119890119909119862119891 = (1 + 1120574)11989110158401015840 (1) (13)

where

119862119891 = 120583120588(120597119906120597119905)119910=ℎ(119905)

V2119908 (14)

119877119890119909 = 2119897V2119908V119909 (1 minus 120572119905)12 (15)





119867(119891 (120578) 1198911015840 (120578) 11989110158401015840 (120578) 119891101584010158401015840 (120578) 119891(119899minus1) (120578) 119891(119899) (120578))= 0 (16)



(17)

where

119866 [119891 (120578)]= 119867(119891 (120578) 1198911015840 (120578) 11989110158401015840 (120578) 119891(119899minus1) (120578))


(18)

Step 2 Assume that

119866 [119891 (120578)] = infinsum119899=1

119889119899minus1119866 (1198910 (120578))119889120578119899minus1 (19)

rewriting (19)

119866 [119891 (120578)] = 119866 [1198910 (120578)] + 1198661015840 [1198910 (120578)] + 11986610158401015840 [1198910 (120578)]+ 119866101584010158401015840 [1198910 (120578)] + 1198661015840101584010158401015840 [1198910 (120578)] + sdot sdot sdot (20)


119891 (120578) = 1198910 + 1198911 + 1198912 + 1198913 + 1198914 + sdot sdot sdot (21)


where



(22)


119891(119899minus1) (120578)) (23)

1198661015840 [119891 (120578)] = 119889119866 [119891 (120578)]119889120578 = 119866119891119891120578 + 1198661198911015840 (119891120578)1015840 + sdot sdot sdot

+ 119866119891(119899minus1) (119891120578)(119899minus1) (24)

11986610158401015840 [119891 (120578)] = 1198892119866 [119891 (120578)]1198891205782 = 119866119891119891 (119891120578)2 + 1198661198911198911015840 (119891120578)1015840


(25)

119866101584010158401015840 [119891 (120578)] = 1198893119866 [119891 (120578)]1198891205783 = 119866119891119891119891 (119891120578)3 + 1198661198911198911198911015840



(26)



119891120578 = 1198911 = 119871minus1119866 [1198910 (120578)] 119891120578120578 = 1198912 = 119871minus11198661015840 [1198910 (120578)]



(27)


119866 [1198910 (120578)]= 119867 (1198910 (120578) 11989110158400 (120578) 119891101584010158400 (120578) 119891(119899minus1)0 (120578)) (28)


(29)




(30)









+ 119866119891(119899minus1)0 (1198913)(119899minus1)

(31)


4 Application




119891 (120578) = 119891 (0) + 1198911015840 (0) 120578 + 11989110158401015840 (0) 12057822 + 119891101584010158401015840 (0) 1205783

3+ 119871minus1 [ 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578)]

(32)


119891 (120578) = 1198601 + 1198602120578 + 1198603 12057822 + 1198604 1205783

3 + 119871minus1119866 [119891 (120578)] (33)

where

1198601 = 119891 (0) 1198602 = 1198911015840 (0) 1198603 = 11989110158401015840 (0) 1198604 = 119891101584010158401015840 (0) 119866 [119891] = 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)

minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578) 119886119899119889 119871minus1 () = int120578

0int1205780int1205780int1205780(119889120578)4

(34)


119891 (120578) = 1198602120578 + 1198604 12057833 + 119871minus1 [119866 [119891 (120578)] (35)



(36)

Step 3 yields

119866 [119891 (120578])] = 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578)

+ 1198911015840 (120578) 11989110158401015840 (120578) minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578)] (37)

1198661015840 [119891 (120578)] = 119889119866 [119891 (120578)]119889120578 = 119866119891119891120578 + 1198661198911015840 (119891120578)1015840 + 11986611989110158401015840

(119891120578)10158401015840 + 119866119891101584010158401015840 (119891120578)101584010158401015840 (38)

11986610158401015840 [119891 (120578)] = 1198892119866 [119891 (120578)]1198891205782 = 119866119891119891 (119891120578)2 + 21198661198911198911015840

119891120578 (119891120578)1015840 + 211986611989111989110158401015840 119891120578 (119891120578)10158401015840 + 2119866119891119891101584010158401015840 119891120578 (119891120578)101584010158401015840+ 11986611989110158401198911015840 (119891120578)10158402 + 2119866119891101584011989110158401015840 (119891120578)1015840 (119891120578)10158401015840 + 21198661198911015840119891101584010158401015840 (119891120578)1015840 (119891120578)101584010158401015840 + 1198661198911015840101584011989110158401015840 (119891120578)101584010158402 + 211986611989110158401015840119891101584010158401015840 (119891120578)10158401015840sdot (119891120578)101584010158401015840 + 119866119891101584010158401015840119891101584010158401015840 (119891120578)1015840101584010158402 + 119866119891119891120578120578 + 1198661198911015840 (119891120578120578)1015840+ 11986611989110158401015840 (119891120578120578)10158401015840 + 119866119891101584010158401015840 (119891120578120578)101584010158401015840

(39)

119866101584010158401015840 [119891 (120578)] = 1198893119866 [119891 (120578)]1198891205783 = 119866119891119891119891 (119891120578)3 + 3

1198661198911198911198911015840 (119891120578)2 (119891120578)1015840 + 311986611989111989111989110158401015840 (119891120578)2 (119891120578)10158401015840 + 3119866119891119891119891101584010158401015840 (119891120578)2 (119891120578)101584010158401015840 + 311986611989111989110158401198911015840 (119891120578) (119891120578)10158402 + 4119866119891119891101584011989110158401015840 (119891120578) (119891120578)1015840 (119891120578)10158401015840 + 41198661198911198911015840119891101584010158401015840 (119891120578) (119891120578)1015840 (119891120578)101584010158401015840 + 31198661198911198911015840101584011989110158401015840 (119891120578) (119891120578)101584010158402 + 411986611989111989110158401015840119891101584010158401015840 (119891120578) (119891120578)10158401015840 (119891120578)101584010158401015840 + 3119866119891119891101584010158401015840119891101584010158401015840 (119891120578) (119891120578)1015840101584010158402 + 119866119891101584011989110158401198911015840 (119891120578)10158403 + 31198661198911015840119891101584011989110158401015840 (119891120578)10158402 (119891120578)10158401015840 + 11986611989110158401198911015840119891101584010158401015840 (119891120578)10158402 (119891120578)101584010158401015840 + 4119866119891101584011989110158401015840119891 (119891120578)1015840 119891120578 (119891120578)10158401015840 + 311986611989110158401198911015840101584011989110158401015840 (119891120578)1015840 (119891120578)101584010158402 + 4119866119891101584011989110158401015840119891101584010158401015840 (119891120578)1015840 (119891120578)10158401015840 (119891120578)101584010158401015840+ 41198661198911015840119891101584010158401015840119891 (119891120578)1015840 119891120578 (119891120578)101584010158401015840 + 4119866119891101584011989110158401015840101584011989110158401015840 (119891120578)1015840 (119891120578)10158401015840 (119891120578)101584010158401015840 + 31198661198911015840119891101584010158401015840119891101584010158401015840 (119891120578)1015840 (119891120578)1015840101584010158402+ 119866119891101584010158401198911015840101584011989110158401015840 (119891120578)101584010158403 + 31198661198911015840101584011989110158401015840101584011989110158401015840 (119891120578)101584010158402 (119891120578)101584010158401015840+ 311986611989110158401015840101584011989110158401015840101584011989110158401015840 (119891120578)1015840101584010158402 (119891120578)10158401015840 + 119866119891101584010158401015840119891101584010158401015840119891101584010158401015840 (119891120578)1015840101584010158403+ 3119866119891119891119891120578120578119891120578 + 31198661198911198911015840 119891120578120578 (119891120578)1015840 + 311986611989111989110158401015840 119891120578120578(119891120578)10158401015840 + 3119866119891119891101584010158401015840 119891120578120578 (119891120578)101584010158401015840 + 119866119891119891120578120578120578 + 31198661198911015840119891


(119891120578120578)1015840 (119891120578) + 311986611989110158401198911015840 (119891120578120578)1015840 (119891120578)1015840 + 3119866119891101584011989110158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 31198661198911015840119891101584010158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 1198661198911015840 (119891120578120578120578)1015840 + 311986611989110158401015840119891 (119891120578120578)10158401015840 (119891120578) + 3119866119891101584010158401198911015840 (119891120578120578)10158401015840 (119891120578)1015840 + 31198661198911015840101584011989110158401015840 (119891120578120578)10158401015840 (119891120578)10158401015840 + 311986611989110158401015840119891101584010158401015840 (119891120578120578)10158401015840 (119891120578)101584010158401015840 + 11986611989110158401015840 (119891120578120578120578)10158401015840 + 3119866119891101584010158401015840119891 (119891120578120578)101584010158401015840 (119891120578)+ 31198661198911015840101584010158401198911015840 (119891120578120578)101584010158401015840 (119891120578)1015840 + 311986611989110158401015840101584011989110158401015840 (119891120578120578)101584010158401015840 (119891120578)10158401015840+ 3119866119891101584010158401015840119891101584010158401015840 (119891120578120578)101584010158401015840 (119891120578)101584010158401015840 + 119866119891101584010158401015840 (119891120578120578120578)101584010158401015840

(40)


119866 [1198910 (120578)] = 1205741198781 + 120574 ((1205781198911015840101584010158400 (120578) + 311989110158400 (120578)+ 11989110158400 (120578) 119891101584010158400 (120578) minus 1198910 (120578) 1198911015840101584010158400 (120578)) +1198722119891101584010158400 (120578))

(41)

1198661015840 [1198910 (120578)] = 1198661198910 1198911 + 11986611989101015840 (1198911)1015840 + 119866119891101584010158400 (1198911)10158401015840+ 1198661198911015840101584010158400 (1198911)101584010158401015840

(42)

11986610158401015840 [1198910 (120578)] = 11986611989101198910 (1198911)2 + 2119866119891011989110158400 1198911 (1198911)1015840 + 21198661198910119891101584010158400 1198911 (1198911)10158401015840 + 211986611989101198911015840101584010158400 1198911 (1198911)101584010158401015840 + 1198661198911015840011989110158400 (1198911)10158402+ 211986611989110158400119891101584010158400 (1198911)1015840 (1198911)10158401015840 + 2119866119891101584001198911015840101584010158400 (1198911)1015840 (1198911)101584010158401015840+ 119866119891101584010158400 119891101584010158400 (1198911)101584010158402 + 2119866119891101584010158400 1198911015840101584010158400 (1198911)10158401015840 (1198911)101584010158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158402 + 1198661198910 1198912 + 11986611989110158400 (1198912)1015840 + 119866119891101584010158400 (1198912)10158401015840 + 1198661198911015840101584010158400 (1198912)101584010158401015840

(43)

119866101584010158401015840 [1198910 (120578)] = 119866119891011989101198910 (1198911)3 + 31198661198910119891011989110158400 (1198911)2 (1198911)1015840+ 311986611989101198910119891101584010158400 (1198911)2 (1198911)10158401015840 + 3119866119891011989101198911015840101584010158400 (1198911)2 (1198911)101584010158401015840+ 311986611989101198911015840011989110158400 (1198911) (1198911)10158402 + 4119866119891011989110158400119891101584010158400 (1198911) (1198911)1015840 (1198911)10158401015840 + 41198661198910119891101584001198911015840101584010158400 (1198911) (1198911)1015840 (1198911)101584010158401015840 + 31198661198910119891101584010158400 119891101584010158400 (1198911) (1198911)101584010158402 + 41198661198910119891101584010158400 1198911015840101584010158400 (1198911) (1198911)10158401015840 (1198911)101584010158401015840 + 311986611989101198911015840101584010158400 1198911015840101584010158400 (1198911) (1198911)1015840101584010158402 + 119866119891101584001198911015840011989110158400 (1198911)10158403 + 31198661198911015840011989110158400119891101584010158400 (1198911)10158402 (1198911)10158401015840 + 11986611989110158400119891101584001198911015840101584010158400 (1198911)10158402 (1198911)101584010158401015840 + 411986611989110158400119891101584010158400 1198910 (1198911)1015840 1198911 (1198911)10158401015840 + 311986611989110158400119891101584010158400 119891101584010158400 (1198911)1015840 (1198911)101584010158402

+ 411986611989110158400119891101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 1198910 (1198911)1015840 1198911 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 119891101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 3119866119891101584001198911015840101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)1015840101584010158402 + 119866119891101584010158400 119891101584010158400 119891101584010158400 (1198911)101584010158403 + 3119866119891101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)101584010158402 (1198911)101584010158401015840 + 31198661198911015840101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)1015840101584010158402 (1198911)10158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158403 + 311986611989101198910 11989121198911 + 3119866119891011989110158400 1198912(1198911)1015840 + 31198661198910119891101584010158400 1198912 (1198911)10158401015840 + 311986611989101198911015840101584010158400 1198912 (1198911)101584010158401015840+ 1198661198910 1198913 + 3119866119891101584001198910 (1198912)1015840 (1198911) + 31198661198911015840011989110158400 (1198912)1015840 (1198911)1015840+ 311986611989110158400119891101584010158400 (1198912)1015840 (1198911)10158401015840 + 3119866119891101584001198911015840101584010158400 (1198912)1015840 (1198911)10158401015840+ 11986611989110158400 (1198913)1015840 + 3119866119891101584010158400 1198910 (1198912)10158401015840 (1198911) + 3119866119891101584010158400 11989110158400 (1198912)10158401015840 (1198911)1015840 + 3119866119891101584010158400 119891101584010158400 (1198912)10158401015840 (1198911)10158401015840 + 3119866119891101584010158400 1198911015840101584010158400 (1198912)10158401015840 (1198911)101584010158401015840 + 119866119891101584010158400 (1198913)10158401015840 + 31198661198911015840101584010158400 1198910 (1198912)101584010158401015840 (1198911)+ 31198661198911015840101584010158400 11989110158400 (1198912)101584010158401015840 (1198911)1015840 + 31198661198911015840101584010158400 119891101584010158400 (1198912)101584010158401015840 (1198911)10158401015840+ 31198661198911015840101584010158400 1198911015840101584010158400 (1198912)101584010158401015840 (1198911)101584010158401015840 + 1198661198911015840101584010158400 (1198913)101584010158401015840

(44)


1198661198910 = minus 1205741198781 + 1205741198911015840101584010158400 (120578) 11986611989101198910 = 119866119891011989110158400 = 1198661198910119891101584010158400 = 011986611989101198911015840101584010158400 = minus 1205741198781 + 120574 119866119891011989101198910 = 1198661198910119891101584001198910 = 11986611989101198911015840011989110158400 = 1198661198910119891101584010158400 11989110158400 = 1198661198910119891011989110158400

= 11986611989101198911015840101584010158400 1198911015840101584010158400 = 1198661198910119891101584010158400 119891101584010158400 = 011986611989110158400 = 1205741198781 + 120574 = (3 + 119891101584010158400 (120578))

119866119891101584001198910 = 1198661198911015840011989110158400 = 119866119891101584001198911015840101584010158400 = 011986611989110158400119891101584010158400 = 1205741198781 + 120574 1198661198911015840011989101198910 = 11986611989110158400119891101584001198910 = 119866119891101584001198911015840011989110158400 = 11986611989110158400119891101584010158400 11989110158400 = 119866119891101584001198911015840101584010158400 11989110158400

= 119866119891101584001198911015840101584010158400 1198911015840101584010158400 = 11986611989110158400119891101584010158400 119891101584010158400 = 0119866119891101584010158400 = 1205741198781 + 1205741198911015840 (120578) + 12057411987221 + 120574

119866119891101584010158400 1198910 = 119866119891101584010158400 119891101584010158400 = 119866119891101584010158400 1198911015840101584010158400 = 0


119866119891101584010158400 11989110158400 = 1205741198781 + 120574 119866119891101584010158400 11989101198910 = 119866119891101584010158400 119891101584001198910 = 119866119891101584010158400 1198911015840011989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400

= 119866119891101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 119866119891101584010158400 119891101584010158400 119891101584010158400 = 01198661198911015840101584010158400 = 1205741198781205781 + 120574 minus 1205741198781 + 1205741198910 (120578)

1198661198911015840101584010158400 11989110158400 = 1198661198911015840101584010158400 119891101584010158400 = 1198661198911015840101584010158400 1198911015840101584010158400 = 01198661198911015840101584010158400 1198910 = minus 1205741198781 + 120574

1198661198911015840101584010158400 11989101198910 = 1198661198911015840101584010158400 119891101584001198910 = 1198661198911015840101584010158400 1198911015840011989110158400 = 1198661198911015840101584010158400 119891101584010158400 11989110158400 = 1198661198911015840101584010158400 1198911015840101584010158400 11989110158400= 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 0

(45)


1198910 = 1198602120578 + 1611986041205783 (46)

1198911 = 12520119903119878119860241205787 + 1120119903 (41198781198604 +11987221198604) 1205785 (47)


(48)


+ 11389960011990331198783119860431198602 + minus19729720011990331198722119878211986043)sdot 12057813 + 8340864824000119903311987831198604412057815

(49)



(50)




119865119899+1 = 119866 [119865119899] 119865119899 = 119899sum119896=0

119891119896119899 = 0 1 2 3

(51)






1003817100381710038171003817119865119899 minus 1198651198981003817100381710038171003817 =1003817100381710038171003817100381710038171003817100381710038171003817119899sum119896=0

119891119896 minus 119898sum119896=0

1198911198961003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817[1198910 + 119871minus1119899sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]

minus [1198910 + 119871minus1 119898sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]1003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817119871minus1119866[119899minus1sum

119896=0


119891119896]1003817100381710038171003817100381710038171003817100381710038171003817

(119904119894119899119888119890 119865119899 = 119866 [119865119899minus1])⩽ 10038161003816100381610038161003816Łminus110038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119866[119899minus1sum119896=0

119891119896] minus 119866[119898minus1sum119896=0

119891119896]1003817100381710038171003817100381710038171003817100381710038171003817


(54)


and hence


From (56) we get



(57)



(58)




119871minus1 [119866 [119865119899]] (59)


119869 (119865) = 1198650 + 119871minus1119866 [119865] (60)



(61)



119891119896]


119899sum119896=0

119891119896] = 119871minus1119866[ lim

119899997888rarrinfin119865119899]


119871minus1119866 [119865119899]

(62)






120572119896 =

1003817100381710038171003817119865119896+1 minus 119865119896100381710038171003817100381710038171003817100381710038171198651 minus 11986501003817100381710038171003817 = 1003817100381710038171003817119891119896+11003817100381710038171003817100381710038171003817100381711989111003817100381710038171003817 100381710038171003817100381711989111003817100381710038171003817 = 00 100381710038171003817100381711989111003817100381710038171003817 = 0 (63)









1205723 = 00000001976 lt 1

(64)








1205723 = 00000006301 lt 1

(65)













01 0147656 0147656 1467313 146731302 0292534 0292534 1425575 142557503 0431831 0431832 1355629 135562904 0562701 0562701 1256898 125689905 0682225 0682226 1128583 112858306 0787397 0787398 0969661 096966107 0875096 0875097 0778889 077888108 0942065 0942067 0554821 055482109 0984895 0984897 0295804 0295804


01 0150638 0150638 1485907 149590702 0298136 0298136 1448847 144884703 0439360 0439361 1370673 137067304 0571246 0571246 1261767 126176705 0690716 0690716 1122650 112265006 0794791 0794790 0953971 095397107 0880550 0880541 0756487 075648708 0945156 0945156 0531047 053104709 0985858 0985876 0278565 0278566









1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10


08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10






02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









y

ux

D

2l(1 minus t)12


where

120578 = 119910[119897 (1 minus 120572)12] (10)



(11)


119891 (0) = 011989110158401015840 (0) = 0119891 (1) = 11198911015840 (1) = 0

(12)


11989721199092 (1 minus 120572119905)119877119890119909119862119891 = (1 + 1120574)11989110158401015840 (1) (13)

where

119862119891 = 120583120588(120597119906120597119905)119910=ℎ(119905)

V2119908 (14)

119877119890119909 = 2119897V2119908V119909 (1 minus 120572119905)12 (15)





119867(119891 (120578) 1198911015840 (120578) 11989110158401015840 (120578) 119891101584010158401015840 (120578) 119891(119899minus1) (120578) 119891(119899) (120578))= 0 (16)



(17)

where

119866 [119891 (120578)]= 119867(119891 (120578) 1198911015840 (120578) 11989110158401015840 (120578) 119891(119899minus1) (120578))


(18)

Step 2 Assume that

119866 [119891 (120578)] = infinsum119899=1

119889119899minus1119866 (1198910 (120578))119889120578119899minus1 (19)

rewriting (19)

119866 [119891 (120578)] = 119866 [1198910 (120578)] + 1198661015840 [1198910 (120578)] + 11986610158401015840 [1198910 (120578)]+ 119866101584010158401015840 [1198910 (120578)] + 1198661015840101584010158401015840 [1198910 (120578)] + sdot sdot sdot (20)


119891 (120578) = 1198910 + 1198911 + 1198912 + 1198913 + 1198914 + sdot sdot sdot (21)


where



(22)


119891(119899minus1) (120578)) (23)

1198661015840 [119891 (120578)] = 119889119866 [119891 (120578)]119889120578 = 119866119891119891120578 + 1198661198911015840 (119891120578)1015840 + sdot sdot sdot

+ 119866119891(119899minus1) (119891120578)(119899minus1) (24)

11986610158401015840 [119891 (120578)] = 1198892119866 [119891 (120578)]1198891205782 = 119866119891119891 (119891120578)2 + 1198661198911198911015840 (119891120578)1015840


(25)

119866101584010158401015840 [119891 (120578)] = 1198893119866 [119891 (120578)]1198891205783 = 119866119891119891119891 (119891120578)3 + 1198661198911198911198911015840



(26)



119891120578 = 1198911 = 119871minus1119866 [1198910 (120578)] 119891120578120578 = 1198912 = 119871minus11198661015840 [1198910 (120578)]



(27)


119866 [1198910 (120578)]= 119867 (1198910 (120578) 11989110158400 (120578) 119891101584010158400 (120578) 119891(119899minus1)0 (120578)) (28)


(29)




(30)









+ 119866119891(119899minus1)0 (1198913)(119899minus1)

(31)


4 Application




119891 (120578) = 119891 (0) + 1198911015840 (0) 120578 + 11989110158401015840 (0) 12057822 + 119891101584010158401015840 (0) 1205783

3+ 119871minus1 [ 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578)]

(32)


119891 (120578) = 1198601 + 1198602120578 + 1198603 12057822 + 1198604 1205783

3 + 119871minus1119866 [119891 (120578)] (33)

where

1198601 = 119891 (0) 1198602 = 1198911015840 (0) 1198603 = 11989110158401015840 (0) 1198604 = 119891101584010158401015840 (0) 119866 [119891] = 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)

minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578) 119886119899119889 119871minus1 () = int120578

0int1205780int1205780int1205780(119889120578)4

(34)


119891 (120578) = 1198602120578 + 1198604 12057833 + 119871minus1 [119866 [119891 (120578)] (35)



(36)

Step 3 yields

119866 [119891 (120578])] = 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578)

+ 1198911015840 (120578) 11989110158401015840 (120578) minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578)] (37)

1198661015840 [119891 (120578)] = 119889119866 [119891 (120578)]119889120578 = 119866119891119891120578 + 1198661198911015840 (119891120578)1015840 + 11986611989110158401015840

(119891120578)10158401015840 + 119866119891101584010158401015840 (119891120578)101584010158401015840 (38)

11986610158401015840 [119891 (120578)] = 1198892119866 [119891 (120578)]1198891205782 = 119866119891119891 (119891120578)2 + 21198661198911198911015840

119891120578 (119891120578)1015840 + 211986611989111989110158401015840 119891120578 (119891120578)10158401015840 + 2119866119891119891101584010158401015840 119891120578 (119891120578)101584010158401015840+ 11986611989110158401198911015840 (119891120578)10158402 + 2119866119891101584011989110158401015840 (119891120578)1015840 (119891120578)10158401015840 + 21198661198911015840119891101584010158401015840 (119891120578)1015840 (119891120578)101584010158401015840 + 1198661198911015840101584011989110158401015840 (119891120578)101584010158402 + 211986611989110158401015840119891101584010158401015840 (119891120578)10158401015840sdot (119891120578)101584010158401015840 + 119866119891101584010158401015840119891101584010158401015840 (119891120578)1015840101584010158402 + 119866119891119891120578120578 + 1198661198911015840 (119891120578120578)1015840+ 11986611989110158401015840 (119891120578120578)10158401015840 + 119866119891101584010158401015840 (119891120578120578)101584010158401015840

(39)

119866101584010158401015840 [119891 (120578)] = 1198893119866 [119891 (120578)]1198891205783 = 119866119891119891119891 (119891120578)3 + 3

1198661198911198911198911015840 (119891120578)2 (119891120578)1015840 + 311986611989111989111989110158401015840 (119891120578)2 (119891120578)10158401015840 + 3119866119891119891119891101584010158401015840 (119891120578)2 (119891120578)101584010158401015840 + 311986611989111989110158401198911015840 (119891120578) (119891120578)10158402 + 4119866119891119891101584011989110158401015840 (119891120578) (119891120578)1015840 (119891120578)10158401015840 + 41198661198911198911015840119891101584010158401015840 (119891120578) (119891120578)1015840 (119891120578)101584010158401015840 + 31198661198911198911015840101584011989110158401015840 (119891120578) (119891120578)101584010158402 + 411986611989111989110158401015840119891101584010158401015840 (119891120578) (119891120578)10158401015840 (119891120578)101584010158401015840 + 3119866119891119891101584010158401015840119891101584010158401015840 (119891120578) (119891120578)1015840101584010158402 + 119866119891101584011989110158401198911015840 (119891120578)10158403 + 31198661198911015840119891101584011989110158401015840 (119891120578)10158402 (119891120578)10158401015840 + 11986611989110158401198911015840119891101584010158401015840 (119891120578)10158402 (119891120578)101584010158401015840 + 4119866119891101584011989110158401015840119891 (119891120578)1015840 119891120578 (119891120578)10158401015840 + 311986611989110158401198911015840101584011989110158401015840 (119891120578)1015840 (119891120578)101584010158402 + 4119866119891101584011989110158401015840119891101584010158401015840 (119891120578)1015840 (119891120578)10158401015840 (119891120578)101584010158401015840+ 41198661198911015840119891101584010158401015840119891 (119891120578)1015840 119891120578 (119891120578)101584010158401015840 + 4119866119891101584011989110158401015840101584011989110158401015840 (119891120578)1015840 (119891120578)10158401015840 (119891120578)101584010158401015840 + 31198661198911015840119891101584010158401015840119891101584010158401015840 (119891120578)1015840 (119891120578)1015840101584010158402+ 119866119891101584010158401198911015840101584011989110158401015840 (119891120578)101584010158403 + 31198661198911015840101584011989110158401015840101584011989110158401015840 (119891120578)101584010158402 (119891120578)101584010158401015840+ 311986611989110158401015840101584011989110158401015840101584011989110158401015840 (119891120578)1015840101584010158402 (119891120578)10158401015840 + 119866119891101584010158401015840119891101584010158401015840119891101584010158401015840 (119891120578)1015840101584010158403+ 3119866119891119891119891120578120578119891120578 + 31198661198911198911015840 119891120578120578 (119891120578)1015840 + 311986611989111989110158401015840 119891120578120578(119891120578)10158401015840 + 3119866119891119891101584010158401015840 119891120578120578 (119891120578)101584010158401015840 + 119866119891119891120578120578120578 + 31198661198911015840119891


(119891120578120578)1015840 (119891120578) + 311986611989110158401198911015840 (119891120578120578)1015840 (119891120578)1015840 + 3119866119891101584011989110158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 31198661198911015840119891101584010158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 1198661198911015840 (119891120578120578120578)1015840 + 311986611989110158401015840119891 (119891120578120578)10158401015840 (119891120578) + 3119866119891101584010158401198911015840 (119891120578120578)10158401015840 (119891120578)1015840 + 31198661198911015840101584011989110158401015840 (119891120578120578)10158401015840 (119891120578)10158401015840 + 311986611989110158401015840119891101584010158401015840 (119891120578120578)10158401015840 (119891120578)101584010158401015840 + 11986611989110158401015840 (119891120578120578120578)10158401015840 + 3119866119891101584010158401015840119891 (119891120578120578)101584010158401015840 (119891120578)+ 31198661198911015840101584010158401198911015840 (119891120578120578)101584010158401015840 (119891120578)1015840 + 311986611989110158401015840101584011989110158401015840 (119891120578120578)101584010158401015840 (119891120578)10158401015840+ 3119866119891101584010158401015840119891101584010158401015840 (119891120578120578)101584010158401015840 (119891120578)101584010158401015840 + 119866119891101584010158401015840 (119891120578120578120578)101584010158401015840

(40)


119866 [1198910 (120578)] = 1205741198781 + 120574 ((1205781198911015840101584010158400 (120578) + 311989110158400 (120578)+ 11989110158400 (120578) 119891101584010158400 (120578) minus 1198910 (120578) 1198911015840101584010158400 (120578)) +1198722119891101584010158400 (120578))

(41)

1198661015840 [1198910 (120578)] = 1198661198910 1198911 + 11986611989101015840 (1198911)1015840 + 119866119891101584010158400 (1198911)10158401015840+ 1198661198911015840101584010158400 (1198911)101584010158401015840

(42)

11986610158401015840 [1198910 (120578)] = 11986611989101198910 (1198911)2 + 2119866119891011989110158400 1198911 (1198911)1015840 + 21198661198910119891101584010158400 1198911 (1198911)10158401015840 + 211986611989101198911015840101584010158400 1198911 (1198911)101584010158401015840 + 1198661198911015840011989110158400 (1198911)10158402+ 211986611989110158400119891101584010158400 (1198911)1015840 (1198911)10158401015840 + 2119866119891101584001198911015840101584010158400 (1198911)1015840 (1198911)101584010158401015840+ 119866119891101584010158400 119891101584010158400 (1198911)101584010158402 + 2119866119891101584010158400 1198911015840101584010158400 (1198911)10158401015840 (1198911)101584010158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158402 + 1198661198910 1198912 + 11986611989110158400 (1198912)1015840 + 119866119891101584010158400 (1198912)10158401015840 + 1198661198911015840101584010158400 (1198912)101584010158401015840

(43)

119866101584010158401015840 [1198910 (120578)] = 119866119891011989101198910 (1198911)3 + 31198661198910119891011989110158400 (1198911)2 (1198911)1015840+ 311986611989101198910119891101584010158400 (1198911)2 (1198911)10158401015840 + 3119866119891011989101198911015840101584010158400 (1198911)2 (1198911)101584010158401015840+ 311986611989101198911015840011989110158400 (1198911) (1198911)10158402 + 4119866119891011989110158400119891101584010158400 (1198911) (1198911)1015840 (1198911)10158401015840 + 41198661198910119891101584001198911015840101584010158400 (1198911) (1198911)1015840 (1198911)101584010158401015840 + 31198661198910119891101584010158400 119891101584010158400 (1198911) (1198911)101584010158402 + 41198661198910119891101584010158400 1198911015840101584010158400 (1198911) (1198911)10158401015840 (1198911)101584010158401015840 + 311986611989101198911015840101584010158400 1198911015840101584010158400 (1198911) (1198911)1015840101584010158402 + 119866119891101584001198911015840011989110158400 (1198911)10158403 + 31198661198911015840011989110158400119891101584010158400 (1198911)10158402 (1198911)10158401015840 + 11986611989110158400119891101584001198911015840101584010158400 (1198911)10158402 (1198911)101584010158401015840 + 411986611989110158400119891101584010158400 1198910 (1198911)1015840 1198911 (1198911)10158401015840 + 311986611989110158400119891101584010158400 119891101584010158400 (1198911)1015840 (1198911)101584010158402

+ 411986611989110158400119891101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 1198910 (1198911)1015840 1198911 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 119891101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 3119866119891101584001198911015840101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)1015840101584010158402 + 119866119891101584010158400 119891101584010158400 119891101584010158400 (1198911)101584010158403 + 3119866119891101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)101584010158402 (1198911)101584010158401015840 + 31198661198911015840101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)1015840101584010158402 (1198911)10158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158403 + 311986611989101198910 11989121198911 + 3119866119891011989110158400 1198912(1198911)1015840 + 31198661198910119891101584010158400 1198912 (1198911)10158401015840 + 311986611989101198911015840101584010158400 1198912 (1198911)101584010158401015840+ 1198661198910 1198913 + 3119866119891101584001198910 (1198912)1015840 (1198911) + 31198661198911015840011989110158400 (1198912)1015840 (1198911)1015840+ 311986611989110158400119891101584010158400 (1198912)1015840 (1198911)10158401015840 + 3119866119891101584001198911015840101584010158400 (1198912)1015840 (1198911)10158401015840+ 11986611989110158400 (1198913)1015840 + 3119866119891101584010158400 1198910 (1198912)10158401015840 (1198911) + 3119866119891101584010158400 11989110158400 (1198912)10158401015840 (1198911)1015840 + 3119866119891101584010158400 119891101584010158400 (1198912)10158401015840 (1198911)10158401015840 + 3119866119891101584010158400 1198911015840101584010158400 (1198912)10158401015840 (1198911)101584010158401015840 + 119866119891101584010158400 (1198913)10158401015840 + 31198661198911015840101584010158400 1198910 (1198912)101584010158401015840 (1198911)+ 31198661198911015840101584010158400 11989110158400 (1198912)101584010158401015840 (1198911)1015840 + 31198661198911015840101584010158400 119891101584010158400 (1198912)101584010158401015840 (1198911)10158401015840+ 31198661198911015840101584010158400 1198911015840101584010158400 (1198912)101584010158401015840 (1198911)101584010158401015840 + 1198661198911015840101584010158400 (1198913)101584010158401015840

(44)


1198661198910 = minus 1205741198781 + 1205741198911015840101584010158400 (120578) 11986611989101198910 = 119866119891011989110158400 = 1198661198910119891101584010158400 = 011986611989101198911015840101584010158400 = minus 1205741198781 + 120574 119866119891011989101198910 = 1198661198910119891101584001198910 = 11986611989101198911015840011989110158400 = 1198661198910119891101584010158400 11989110158400 = 1198661198910119891011989110158400

= 11986611989101198911015840101584010158400 1198911015840101584010158400 = 1198661198910119891101584010158400 119891101584010158400 = 011986611989110158400 = 1205741198781 + 120574 = (3 + 119891101584010158400 (120578))

119866119891101584001198910 = 1198661198911015840011989110158400 = 119866119891101584001198911015840101584010158400 = 011986611989110158400119891101584010158400 = 1205741198781 + 120574 1198661198911015840011989101198910 = 11986611989110158400119891101584001198910 = 119866119891101584001198911015840011989110158400 = 11986611989110158400119891101584010158400 11989110158400 = 119866119891101584001198911015840101584010158400 11989110158400

= 119866119891101584001198911015840101584010158400 1198911015840101584010158400 = 11986611989110158400119891101584010158400 119891101584010158400 = 0119866119891101584010158400 = 1205741198781 + 1205741198911015840 (120578) + 12057411987221 + 120574

119866119891101584010158400 1198910 = 119866119891101584010158400 119891101584010158400 = 119866119891101584010158400 1198911015840101584010158400 = 0


119866119891101584010158400 11989110158400 = 1205741198781 + 120574 119866119891101584010158400 11989101198910 = 119866119891101584010158400 119891101584001198910 = 119866119891101584010158400 1198911015840011989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400

= 119866119891101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 119866119891101584010158400 119891101584010158400 119891101584010158400 = 01198661198911015840101584010158400 = 1205741198781205781 + 120574 minus 1205741198781 + 1205741198910 (120578)

1198661198911015840101584010158400 11989110158400 = 1198661198911015840101584010158400 119891101584010158400 = 1198661198911015840101584010158400 1198911015840101584010158400 = 01198661198911015840101584010158400 1198910 = minus 1205741198781 + 120574

1198661198911015840101584010158400 11989101198910 = 1198661198911015840101584010158400 119891101584001198910 = 1198661198911015840101584010158400 1198911015840011989110158400 = 1198661198911015840101584010158400 119891101584010158400 11989110158400 = 1198661198911015840101584010158400 1198911015840101584010158400 11989110158400= 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 0

(45)


1198910 = 1198602120578 + 1611986041205783 (46)

1198911 = 12520119903119878119860241205787 + 1120119903 (41198781198604 +11987221198604) 1205785 (47)


(48)


+ 11389960011990331198783119860431198602 + minus19729720011990331198722119878211986043)sdot 12057813 + 8340864824000119903311987831198604412057815

(49)



(50)




119865119899+1 = 119866 [119865119899] 119865119899 = 119899sum119896=0

119891119896119899 = 0 1 2 3

(51)






1003817100381710038171003817119865119899 minus 1198651198981003817100381710038171003817 =1003817100381710038171003817100381710038171003817100381710038171003817119899sum119896=0

119891119896 minus 119898sum119896=0

1198911198961003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817[1198910 + 119871minus1119899sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]

minus [1198910 + 119871minus1 119898sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]1003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817119871minus1119866[119899minus1sum

119896=0


119891119896]1003817100381710038171003817100381710038171003817100381710038171003817

(119904119894119899119888119890 119865119899 = 119866 [119865119899minus1])⩽ 10038161003816100381610038161003816Łminus110038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119866[119899minus1sum119896=0

119891119896] minus 119866[119898minus1sum119896=0

119891119896]1003817100381710038171003817100381710038171003817100381710038171003817


(54)


and hence


From (56) we get



(57)



(58)




119871minus1 [119866 [119865119899]] (59)


119869 (119865) = 1198650 + 119871minus1119866 [119865] (60)



(61)



119891119896]


119899sum119896=0

119891119896] = 119871minus1119866[ lim

119899997888rarrinfin119865119899]


119871minus1119866 [119865119899]

(62)






120572119896 =

1003817100381710038171003817119865119896+1 minus 119865119896100381710038171003817100381710038171003817100381710038171198651 minus 11986501003817100381710038171003817 = 1003817100381710038171003817119891119896+11003817100381710038171003817100381710038171003817100381711989111003817100381710038171003817 100381710038171003817100381711989111003817100381710038171003817 = 00 100381710038171003817100381711989111003817100381710038171003817 = 0 (63)









1205723 = 00000001976 lt 1

(64)








1205723 = 00000006301 lt 1

(65)













01 0147656 0147656 1467313 146731302 0292534 0292534 1425575 142557503 0431831 0431832 1355629 135562904 0562701 0562701 1256898 125689905 0682225 0682226 1128583 112858306 0787397 0787398 0969661 096966107 0875096 0875097 0778889 077888108 0942065 0942067 0554821 055482109 0984895 0984897 0295804 0295804


01 0150638 0150638 1485907 149590702 0298136 0298136 1448847 144884703 0439360 0439361 1370673 137067304 0571246 0571246 1261767 126176705 0690716 0690716 1122650 112265006 0794791 0794790 0953971 095397107 0880550 0880541 0756487 075648708 0945156 0945156 0531047 053104709 0985858 0985876 0278565 0278566









1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10


08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10






02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









where



(22)


119891(119899minus1) (120578)) (23)

1198661015840 [119891 (120578)] = 119889119866 [119891 (120578)]119889120578 = 119866119891119891120578 + 1198661198911015840 (119891120578)1015840 + sdot sdot sdot

+ 119866119891(119899minus1) (119891120578)(119899minus1) (24)

11986610158401015840 [119891 (120578)] = 1198892119866 [119891 (120578)]1198891205782 = 119866119891119891 (119891120578)2 + 1198661198911198911015840 (119891120578)1015840


(25)

119866101584010158401015840 [119891 (120578)] = 1198893119866 [119891 (120578)]1198891205783 = 119866119891119891119891 (119891120578)3 + 1198661198911198911198911015840



(26)



119891120578 = 1198911 = 119871minus1119866 [1198910 (120578)] 119891120578120578 = 1198912 = 119871minus11198661015840 [1198910 (120578)]



(27)


119866 [1198910 (120578)]= 119867 (1198910 (120578) 11989110158400 (120578) 119891101584010158400 (120578) 119891(119899minus1)0 (120578)) (28)


(29)




(30)









+ 119866119891(119899minus1)0 (1198913)(119899minus1)

(31)


4 Application




119891 (120578) = 119891 (0) + 1198911015840 (0) 120578 + 11989110158401015840 (0) 12057822 + 119891101584010158401015840 (0) 1205783

3+ 119871minus1 [ 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578)]

(32)


119891 (120578) = 1198601 + 1198602120578 + 1198603 12057822 + 1198604 1205783

3 + 119871minus1119866 [119891 (120578)] (33)

where

1198601 = 119891 (0) 1198602 = 1198911015840 (0) 1198603 = 11989110158401015840 (0) 1198604 = 119891101584010158401015840 (0) 119866 [119891] = 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)

minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578) 119886119899119889 119871minus1 () = int120578

0int1205780int1205780int1205780(119889120578)4

(34)


119891 (120578) = 1198602120578 + 1198604 12057833 + 119871minus1 [119866 [119891 (120578)] (35)



(36)

Step 3 yields

119866 [119891 (120578])] = 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578)

+ 1198911015840 (120578) 11989110158401015840 (120578) minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578)] (37)

1198661015840 [119891 (120578)] = 119889119866 [119891 (120578)]119889120578 = 119866119891119891120578 + 1198661198911015840 (119891120578)1015840 + 11986611989110158401015840

(119891120578)10158401015840 + 119866119891101584010158401015840 (119891120578)101584010158401015840 (38)

11986610158401015840 [119891 (120578)] = 1198892119866 [119891 (120578)]1198891205782 = 119866119891119891 (119891120578)2 + 21198661198911198911015840

119891120578 (119891120578)1015840 + 211986611989111989110158401015840 119891120578 (119891120578)10158401015840 + 2119866119891119891101584010158401015840 119891120578 (119891120578)101584010158401015840+ 11986611989110158401198911015840 (119891120578)10158402 + 2119866119891101584011989110158401015840 (119891120578)1015840 (119891120578)10158401015840 + 21198661198911015840119891101584010158401015840 (119891120578)1015840 (119891120578)101584010158401015840 + 1198661198911015840101584011989110158401015840 (119891120578)101584010158402 + 211986611989110158401015840119891101584010158401015840 (119891120578)10158401015840sdot (119891120578)101584010158401015840 + 119866119891101584010158401015840119891101584010158401015840 (119891120578)1015840101584010158402 + 119866119891119891120578120578 + 1198661198911015840 (119891120578120578)1015840+ 11986611989110158401015840 (119891120578120578)10158401015840 + 119866119891101584010158401015840 (119891120578120578)101584010158401015840

(39)

119866101584010158401015840 [119891 (120578)] = 1198893119866 [119891 (120578)]1198891205783 = 119866119891119891119891 (119891120578)3 + 3

1198661198911198911198911015840 (119891120578)2 (119891120578)1015840 + 311986611989111989111989110158401015840 (119891120578)2 (119891120578)10158401015840 + 3119866119891119891119891101584010158401015840 (119891120578)2 (119891120578)101584010158401015840 + 311986611989111989110158401198911015840 (119891120578) (119891120578)10158402 + 4119866119891119891101584011989110158401015840 (119891120578) (119891120578)1015840 (119891120578)10158401015840 + 41198661198911198911015840119891101584010158401015840 (119891120578) (119891120578)1015840 (119891120578)101584010158401015840 + 31198661198911198911015840101584011989110158401015840 (119891120578) (119891120578)101584010158402 + 411986611989111989110158401015840119891101584010158401015840 (119891120578) (119891120578)10158401015840 (119891120578)101584010158401015840 + 3119866119891119891101584010158401015840119891101584010158401015840 (119891120578) (119891120578)1015840101584010158402 + 119866119891101584011989110158401198911015840 (119891120578)10158403 + 31198661198911015840119891101584011989110158401015840 (119891120578)10158402 (119891120578)10158401015840 + 11986611989110158401198911015840119891101584010158401015840 (119891120578)10158402 (119891120578)101584010158401015840 + 4119866119891101584011989110158401015840119891 (119891120578)1015840 119891120578 (119891120578)10158401015840 + 311986611989110158401198911015840101584011989110158401015840 (119891120578)1015840 (119891120578)101584010158402 + 4119866119891101584011989110158401015840119891101584010158401015840 (119891120578)1015840 (119891120578)10158401015840 (119891120578)101584010158401015840+ 41198661198911015840119891101584010158401015840119891 (119891120578)1015840 119891120578 (119891120578)101584010158401015840 + 4119866119891101584011989110158401015840101584011989110158401015840 (119891120578)1015840 (119891120578)10158401015840 (119891120578)101584010158401015840 + 31198661198911015840119891101584010158401015840119891101584010158401015840 (119891120578)1015840 (119891120578)1015840101584010158402+ 119866119891101584010158401198911015840101584011989110158401015840 (119891120578)101584010158403 + 31198661198911015840101584011989110158401015840101584011989110158401015840 (119891120578)101584010158402 (119891120578)101584010158401015840+ 311986611989110158401015840101584011989110158401015840101584011989110158401015840 (119891120578)1015840101584010158402 (119891120578)10158401015840 + 119866119891101584010158401015840119891101584010158401015840119891101584010158401015840 (119891120578)1015840101584010158403+ 3119866119891119891119891120578120578119891120578 + 31198661198911198911015840 119891120578120578 (119891120578)1015840 + 311986611989111989110158401015840 119891120578120578(119891120578)10158401015840 + 3119866119891119891101584010158401015840 119891120578120578 (119891120578)101584010158401015840 + 119866119891119891120578120578120578 + 31198661198911015840119891


(119891120578120578)1015840 (119891120578) + 311986611989110158401198911015840 (119891120578120578)1015840 (119891120578)1015840 + 3119866119891101584011989110158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 31198661198911015840119891101584010158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 1198661198911015840 (119891120578120578120578)1015840 + 311986611989110158401015840119891 (119891120578120578)10158401015840 (119891120578) + 3119866119891101584010158401198911015840 (119891120578120578)10158401015840 (119891120578)1015840 + 31198661198911015840101584011989110158401015840 (119891120578120578)10158401015840 (119891120578)10158401015840 + 311986611989110158401015840119891101584010158401015840 (119891120578120578)10158401015840 (119891120578)101584010158401015840 + 11986611989110158401015840 (119891120578120578120578)10158401015840 + 3119866119891101584010158401015840119891 (119891120578120578)101584010158401015840 (119891120578)+ 31198661198911015840101584010158401198911015840 (119891120578120578)101584010158401015840 (119891120578)1015840 + 311986611989110158401015840101584011989110158401015840 (119891120578120578)101584010158401015840 (119891120578)10158401015840+ 3119866119891101584010158401015840119891101584010158401015840 (119891120578120578)101584010158401015840 (119891120578)101584010158401015840 + 119866119891101584010158401015840 (119891120578120578120578)101584010158401015840

(40)


119866 [1198910 (120578)] = 1205741198781 + 120574 ((1205781198911015840101584010158400 (120578) + 311989110158400 (120578)+ 11989110158400 (120578) 119891101584010158400 (120578) minus 1198910 (120578) 1198911015840101584010158400 (120578)) +1198722119891101584010158400 (120578))

(41)

1198661015840 [1198910 (120578)] = 1198661198910 1198911 + 11986611989101015840 (1198911)1015840 + 119866119891101584010158400 (1198911)10158401015840+ 1198661198911015840101584010158400 (1198911)101584010158401015840

(42)

11986610158401015840 [1198910 (120578)] = 11986611989101198910 (1198911)2 + 2119866119891011989110158400 1198911 (1198911)1015840 + 21198661198910119891101584010158400 1198911 (1198911)10158401015840 + 211986611989101198911015840101584010158400 1198911 (1198911)101584010158401015840 + 1198661198911015840011989110158400 (1198911)10158402+ 211986611989110158400119891101584010158400 (1198911)1015840 (1198911)10158401015840 + 2119866119891101584001198911015840101584010158400 (1198911)1015840 (1198911)101584010158401015840+ 119866119891101584010158400 119891101584010158400 (1198911)101584010158402 + 2119866119891101584010158400 1198911015840101584010158400 (1198911)10158401015840 (1198911)101584010158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158402 + 1198661198910 1198912 + 11986611989110158400 (1198912)1015840 + 119866119891101584010158400 (1198912)10158401015840 + 1198661198911015840101584010158400 (1198912)101584010158401015840

(43)

119866101584010158401015840 [1198910 (120578)] = 119866119891011989101198910 (1198911)3 + 31198661198910119891011989110158400 (1198911)2 (1198911)1015840+ 311986611989101198910119891101584010158400 (1198911)2 (1198911)10158401015840 + 3119866119891011989101198911015840101584010158400 (1198911)2 (1198911)101584010158401015840+ 311986611989101198911015840011989110158400 (1198911) (1198911)10158402 + 4119866119891011989110158400119891101584010158400 (1198911) (1198911)1015840 (1198911)10158401015840 + 41198661198910119891101584001198911015840101584010158400 (1198911) (1198911)1015840 (1198911)101584010158401015840 + 31198661198910119891101584010158400 119891101584010158400 (1198911) (1198911)101584010158402 + 41198661198910119891101584010158400 1198911015840101584010158400 (1198911) (1198911)10158401015840 (1198911)101584010158401015840 + 311986611989101198911015840101584010158400 1198911015840101584010158400 (1198911) (1198911)1015840101584010158402 + 119866119891101584001198911015840011989110158400 (1198911)10158403 + 31198661198911015840011989110158400119891101584010158400 (1198911)10158402 (1198911)10158401015840 + 11986611989110158400119891101584001198911015840101584010158400 (1198911)10158402 (1198911)101584010158401015840 + 411986611989110158400119891101584010158400 1198910 (1198911)1015840 1198911 (1198911)10158401015840 + 311986611989110158400119891101584010158400 119891101584010158400 (1198911)1015840 (1198911)101584010158402

+ 411986611989110158400119891101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 1198910 (1198911)1015840 1198911 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 119891101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 3119866119891101584001198911015840101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)1015840101584010158402 + 119866119891101584010158400 119891101584010158400 119891101584010158400 (1198911)101584010158403 + 3119866119891101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)101584010158402 (1198911)101584010158401015840 + 31198661198911015840101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)1015840101584010158402 (1198911)10158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158403 + 311986611989101198910 11989121198911 + 3119866119891011989110158400 1198912(1198911)1015840 + 31198661198910119891101584010158400 1198912 (1198911)10158401015840 + 311986611989101198911015840101584010158400 1198912 (1198911)101584010158401015840+ 1198661198910 1198913 + 3119866119891101584001198910 (1198912)1015840 (1198911) + 31198661198911015840011989110158400 (1198912)1015840 (1198911)1015840+ 311986611989110158400119891101584010158400 (1198912)1015840 (1198911)10158401015840 + 3119866119891101584001198911015840101584010158400 (1198912)1015840 (1198911)10158401015840+ 11986611989110158400 (1198913)1015840 + 3119866119891101584010158400 1198910 (1198912)10158401015840 (1198911) + 3119866119891101584010158400 11989110158400 (1198912)10158401015840 (1198911)1015840 + 3119866119891101584010158400 119891101584010158400 (1198912)10158401015840 (1198911)10158401015840 + 3119866119891101584010158400 1198911015840101584010158400 (1198912)10158401015840 (1198911)101584010158401015840 + 119866119891101584010158400 (1198913)10158401015840 + 31198661198911015840101584010158400 1198910 (1198912)101584010158401015840 (1198911)+ 31198661198911015840101584010158400 11989110158400 (1198912)101584010158401015840 (1198911)1015840 + 31198661198911015840101584010158400 119891101584010158400 (1198912)101584010158401015840 (1198911)10158401015840+ 31198661198911015840101584010158400 1198911015840101584010158400 (1198912)101584010158401015840 (1198911)101584010158401015840 + 1198661198911015840101584010158400 (1198913)101584010158401015840

(44)


1198661198910 = minus 1205741198781 + 1205741198911015840101584010158400 (120578) 11986611989101198910 = 119866119891011989110158400 = 1198661198910119891101584010158400 = 011986611989101198911015840101584010158400 = minus 1205741198781 + 120574 119866119891011989101198910 = 1198661198910119891101584001198910 = 11986611989101198911015840011989110158400 = 1198661198910119891101584010158400 11989110158400 = 1198661198910119891011989110158400

= 11986611989101198911015840101584010158400 1198911015840101584010158400 = 1198661198910119891101584010158400 119891101584010158400 = 011986611989110158400 = 1205741198781 + 120574 = (3 + 119891101584010158400 (120578))

119866119891101584001198910 = 1198661198911015840011989110158400 = 119866119891101584001198911015840101584010158400 = 011986611989110158400119891101584010158400 = 1205741198781 + 120574 1198661198911015840011989101198910 = 11986611989110158400119891101584001198910 = 119866119891101584001198911015840011989110158400 = 11986611989110158400119891101584010158400 11989110158400 = 119866119891101584001198911015840101584010158400 11989110158400

= 119866119891101584001198911015840101584010158400 1198911015840101584010158400 = 11986611989110158400119891101584010158400 119891101584010158400 = 0119866119891101584010158400 = 1205741198781 + 1205741198911015840 (120578) + 12057411987221 + 120574

119866119891101584010158400 1198910 = 119866119891101584010158400 119891101584010158400 = 119866119891101584010158400 1198911015840101584010158400 = 0


119866119891101584010158400 11989110158400 = 1205741198781 + 120574 119866119891101584010158400 11989101198910 = 119866119891101584010158400 119891101584001198910 = 119866119891101584010158400 1198911015840011989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400

= 119866119891101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 119866119891101584010158400 119891101584010158400 119891101584010158400 = 01198661198911015840101584010158400 = 1205741198781205781 + 120574 minus 1205741198781 + 1205741198910 (120578)

1198661198911015840101584010158400 11989110158400 = 1198661198911015840101584010158400 119891101584010158400 = 1198661198911015840101584010158400 1198911015840101584010158400 = 01198661198911015840101584010158400 1198910 = minus 1205741198781 + 120574

1198661198911015840101584010158400 11989101198910 = 1198661198911015840101584010158400 119891101584001198910 = 1198661198911015840101584010158400 1198911015840011989110158400 = 1198661198911015840101584010158400 119891101584010158400 11989110158400 = 1198661198911015840101584010158400 1198911015840101584010158400 11989110158400= 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 0

(45)


1198910 = 1198602120578 + 1611986041205783 (46)

1198911 = 12520119903119878119860241205787 + 1120119903 (41198781198604 +11987221198604) 1205785 (47)


(48)


+ 11389960011990331198783119860431198602 + minus19729720011990331198722119878211986043)sdot 12057813 + 8340864824000119903311987831198604412057815

(49)



(50)




119865119899+1 = 119866 [119865119899] 119865119899 = 119899sum119896=0

119891119896119899 = 0 1 2 3

(51)






1003817100381710038171003817119865119899 minus 1198651198981003817100381710038171003817 =1003817100381710038171003817100381710038171003817100381710038171003817119899sum119896=0

119891119896 minus 119898sum119896=0

1198911198961003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817[1198910 + 119871minus1119899sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]

minus [1198910 + 119871minus1 119898sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]1003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817119871minus1119866[119899minus1sum

119896=0


119891119896]1003817100381710038171003817100381710038171003817100381710038171003817

(119904119894119899119888119890 119865119899 = 119866 [119865119899minus1])⩽ 10038161003816100381610038161003816Łminus110038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119866[119899minus1sum119896=0

119891119896] minus 119866[119898minus1sum119896=0

119891119896]1003817100381710038171003817100381710038171003817100381710038171003817


(54)


and hence


From (56) we get



(57)



(58)




119871minus1 [119866 [119865119899]] (59)


119869 (119865) = 1198650 + 119871minus1119866 [119865] (60)



(61)



119891119896]


119899sum119896=0

119891119896] = 119871minus1119866[ lim

119899997888rarrinfin119865119899]


119871minus1119866 [119865119899]

(62)






120572119896 =

1003817100381710038171003817119865119896+1 minus 119865119896100381710038171003817100381710038171003817100381710038171198651 minus 11986501003817100381710038171003817 = 1003817100381710038171003817119891119896+11003817100381710038171003817100381710038171003817100381711989111003817100381710038171003817 100381710038171003817100381711989111003817100381710038171003817 = 00 100381710038171003817100381711989111003817100381710038171003817 = 0 (63)









1205723 = 00000001976 lt 1

(64)








1205723 = 00000006301 lt 1

(65)













01 0147656 0147656 1467313 146731302 0292534 0292534 1425575 142557503 0431831 0431832 1355629 135562904 0562701 0562701 1256898 125689905 0682225 0682226 1128583 112858306 0787397 0787398 0969661 096966107 0875096 0875097 0778889 077888108 0942065 0942067 0554821 055482109 0984895 0984897 0295804 0295804


01 0150638 0150638 1485907 149590702 0298136 0298136 1448847 144884703 0439360 0439361 1370673 137067304 0571246 0571246 1261767 126176705 0690716 0690716 1122650 112265006 0794791 0794790 0953971 095397107 0880550 0880541 0756487 075648708 0945156 0945156 0531047 053104709 0985858 0985876 0278565 0278566









1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10


08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10






02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018










(27)


119866 [1198910 (120578)]= 119867 (1198910 (120578) 11989110158400 (120578) 119891101584010158400 (120578) 119891(119899minus1)0 (120578)) (28)


(29)




(30)









+ 119866119891(119899minus1)0 (1198913)(119899minus1)

(31)


4 Application




119891 (120578) = 119891 (0) + 1198911015840 (0) 120578 + 11989110158401015840 (0) 12057822 + 119891101584010158401015840 (0) 1205783

3+ 119871minus1 [ 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578)]

(32)


119891 (120578) = 1198601 + 1198602120578 + 1198603 12057822 + 1198604 1205783

3 + 119871minus1119866 [119891 (120578)] (33)

where

1198601 = 119891 (0) 1198602 = 1198911015840 (0) 1198603 = 11989110158401015840 (0) 1198604 = 119891101584010158401015840 (0) 119866 [119891] = 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)

minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578) 119886119899119889 119871minus1 () = int120578

0int1205780int1205780int1205780(119889120578)4

(34)


119891 (120578) = 1198602120578 + 1198604 12057833 + 119871minus1 [119866 [119891 (120578)] (35)



(36)

Step 3 yields

119866 [119891 (120578])] = 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578)

+ 1198911015840 (120578) 11989110158401015840 (120578) minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578)] (37)

1198661015840 [119891 (120578)] = 119889119866 [119891 (120578)]119889120578 = 119866119891119891120578 + 1198661198911015840 (119891120578)1015840 + 11986611989110158401015840

(119891120578)10158401015840 + 119866119891101584010158401015840 (119891120578)101584010158401015840 (38)

11986610158401015840 [119891 (120578)] = 1198892119866 [119891 (120578)]1198891205782 = 119866119891119891 (119891120578)2 + 21198661198911198911015840

119891120578 (119891120578)1015840 + 211986611989111989110158401015840 119891120578 (119891120578)10158401015840 + 2119866119891119891101584010158401015840 119891120578 (119891120578)101584010158401015840+ 11986611989110158401198911015840 (119891120578)10158402 + 2119866119891101584011989110158401015840 (119891120578)1015840 (119891120578)10158401015840 + 21198661198911015840119891101584010158401015840 (119891120578)1015840 (119891120578)101584010158401015840 + 1198661198911015840101584011989110158401015840 (119891120578)101584010158402 + 211986611989110158401015840119891101584010158401015840 (119891120578)10158401015840sdot (119891120578)101584010158401015840 + 119866119891101584010158401015840119891101584010158401015840 (119891120578)1015840101584010158402 + 119866119891119891120578120578 + 1198661198911015840 (119891120578120578)1015840+ 11986611989110158401015840 (119891120578120578)10158401015840 + 119866119891101584010158401015840 (119891120578120578)101584010158401015840

(39)

119866101584010158401015840 [119891 (120578)] = 1198893119866 [119891 (120578)]1198891205783 = 119866119891119891119891 (119891120578)3 + 3

1198661198911198911198911015840 (119891120578)2 (119891120578)1015840 + 311986611989111989111989110158401015840 (119891120578)2 (119891120578)10158401015840 + 3119866119891119891119891101584010158401015840 (119891120578)2 (119891120578)101584010158401015840 + 311986611989111989110158401198911015840 (119891120578) (119891120578)10158402 + 4119866119891119891101584011989110158401015840 (119891120578) (119891120578)1015840 (119891120578)10158401015840 + 41198661198911198911015840119891101584010158401015840 (119891120578) (119891120578)1015840 (119891120578)101584010158401015840 + 31198661198911198911015840101584011989110158401015840 (119891120578) (119891120578)101584010158402 + 411986611989111989110158401015840119891101584010158401015840 (119891120578) (119891120578)10158401015840 (119891120578)101584010158401015840 + 3119866119891119891101584010158401015840119891101584010158401015840 (119891120578) (119891120578)1015840101584010158402 + 119866119891101584011989110158401198911015840 (119891120578)10158403 + 31198661198911015840119891101584011989110158401015840 (119891120578)10158402 (119891120578)10158401015840 + 11986611989110158401198911015840119891101584010158401015840 (119891120578)10158402 (119891120578)101584010158401015840 + 4119866119891101584011989110158401015840119891 (119891120578)1015840 119891120578 (119891120578)10158401015840 + 311986611989110158401198911015840101584011989110158401015840 (119891120578)1015840 (119891120578)101584010158402 + 4119866119891101584011989110158401015840119891101584010158401015840 (119891120578)1015840 (119891120578)10158401015840 (119891120578)101584010158401015840+ 41198661198911015840119891101584010158401015840119891 (119891120578)1015840 119891120578 (119891120578)101584010158401015840 + 4119866119891101584011989110158401015840101584011989110158401015840 (119891120578)1015840 (119891120578)10158401015840 (119891120578)101584010158401015840 + 31198661198911015840119891101584010158401015840119891101584010158401015840 (119891120578)1015840 (119891120578)1015840101584010158402+ 119866119891101584010158401198911015840101584011989110158401015840 (119891120578)101584010158403 + 31198661198911015840101584011989110158401015840101584011989110158401015840 (119891120578)101584010158402 (119891120578)101584010158401015840+ 311986611989110158401015840101584011989110158401015840101584011989110158401015840 (119891120578)1015840101584010158402 (119891120578)10158401015840 + 119866119891101584010158401015840119891101584010158401015840119891101584010158401015840 (119891120578)1015840101584010158403+ 3119866119891119891119891120578120578119891120578 + 31198661198911198911015840 119891120578120578 (119891120578)1015840 + 311986611989111989110158401015840 119891120578120578(119891120578)10158401015840 + 3119866119891119891101584010158401015840 119891120578120578 (119891120578)101584010158401015840 + 119866119891119891120578120578120578 + 31198661198911015840119891


(119891120578120578)1015840 (119891120578) + 311986611989110158401198911015840 (119891120578120578)1015840 (119891120578)1015840 + 3119866119891101584011989110158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 31198661198911015840119891101584010158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 1198661198911015840 (119891120578120578120578)1015840 + 311986611989110158401015840119891 (119891120578120578)10158401015840 (119891120578) + 3119866119891101584010158401198911015840 (119891120578120578)10158401015840 (119891120578)1015840 + 31198661198911015840101584011989110158401015840 (119891120578120578)10158401015840 (119891120578)10158401015840 + 311986611989110158401015840119891101584010158401015840 (119891120578120578)10158401015840 (119891120578)101584010158401015840 + 11986611989110158401015840 (119891120578120578120578)10158401015840 + 3119866119891101584010158401015840119891 (119891120578120578)101584010158401015840 (119891120578)+ 31198661198911015840101584010158401198911015840 (119891120578120578)101584010158401015840 (119891120578)1015840 + 311986611989110158401015840101584011989110158401015840 (119891120578120578)101584010158401015840 (119891120578)10158401015840+ 3119866119891101584010158401015840119891101584010158401015840 (119891120578120578)101584010158401015840 (119891120578)101584010158401015840 + 119866119891101584010158401015840 (119891120578120578120578)101584010158401015840

(40)


119866 [1198910 (120578)] = 1205741198781 + 120574 ((1205781198911015840101584010158400 (120578) + 311989110158400 (120578)+ 11989110158400 (120578) 119891101584010158400 (120578) minus 1198910 (120578) 1198911015840101584010158400 (120578)) +1198722119891101584010158400 (120578))

(41)

1198661015840 [1198910 (120578)] = 1198661198910 1198911 + 11986611989101015840 (1198911)1015840 + 119866119891101584010158400 (1198911)10158401015840+ 1198661198911015840101584010158400 (1198911)101584010158401015840

(42)

11986610158401015840 [1198910 (120578)] = 11986611989101198910 (1198911)2 + 2119866119891011989110158400 1198911 (1198911)1015840 + 21198661198910119891101584010158400 1198911 (1198911)10158401015840 + 211986611989101198911015840101584010158400 1198911 (1198911)101584010158401015840 + 1198661198911015840011989110158400 (1198911)10158402+ 211986611989110158400119891101584010158400 (1198911)1015840 (1198911)10158401015840 + 2119866119891101584001198911015840101584010158400 (1198911)1015840 (1198911)101584010158401015840+ 119866119891101584010158400 119891101584010158400 (1198911)101584010158402 + 2119866119891101584010158400 1198911015840101584010158400 (1198911)10158401015840 (1198911)101584010158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158402 + 1198661198910 1198912 + 11986611989110158400 (1198912)1015840 + 119866119891101584010158400 (1198912)10158401015840 + 1198661198911015840101584010158400 (1198912)101584010158401015840

(43)

119866101584010158401015840 [1198910 (120578)] = 119866119891011989101198910 (1198911)3 + 31198661198910119891011989110158400 (1198911)2 (1198911)1015840+ 311986611989101198910119891101584010158400 (1198911)2 (1198911)10158401015840 + 3119866119891011989101198911015840101584010158400 (1198911)2 (1198911)101584010158401015840+ 311986611989101198911015840011989110158400 (1198911) (1198911)10158402 + 4119866119891011989110158400119891101584010158400 (1198911) (1198911)1015840 (1198911)10158401015840 + 41198661198910119891101584001198911015840101584010158400 (1198911) (1198911)1015840 (1198911)101584010158401015840 + 31198661198910119891101584010158400 119891101584010158400 (1198911) (1198911)101584010158402 + 41198661198910119891101584010158400 1198911015840101584010158400 (1198911) (1198911)10158401015840 (1198911)101584010158401015840 + 311986611989101198911015840101584010158400 1198911015840101584010158400 (1198911) (1198911)1015840101584010158402 + 119866119891101584001198911015840011989110158400 (1198911)10158403 + 31198661198911015840011989110158400119891101584010158400 (1198911)10158402 (1198911)10158401015840 + 11986611989110158400119891101584001198911015840101584010158400 (1198911)10158402 (1198911)101584010158401015840 + 411986611989110158400119891101584010158400 1198910 (1198911)1015840 1198911 (1198911)10158401015840 + 311986611989110158400119891101584010158400 119891101584010158400 (1198911)1015840 (1198911)101584010158402

+ 411986611989110158400119891101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 1198910 (1198911)1015840 1198911 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 119891101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 3119866119891101584001198911015840101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)1015840101584010158402 + 119866119891101584010158400 119891101584010158400 119891101584010158400 (1198911)101584010158403 + 3119866119891101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)101584010158402 (1198911)101584010158401015840 + 31198661198911015840101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)1015840101584010158402 (1198911)10158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158403 + 311986611989101198910 11989121198911 + 3119866119891011989110158400 1198912(1198911)1015840 + 31198661198910119891101584010158400 1198912 (1198911)10158401015840 + 311986611989101198911015840101584010158400 1198912 (1198911)101584010158401015840+ 1198661198910 1198913 + 3119866119891101584001198910 (1198912)1015840 (1198911) + 31198661198911015840011989110158400 (1198912)1015840 (1198911)1015840+ 311986611989110158400119891101584010158400 (1198912)1015840 (1198911)10158401015840 + 3119866119891101584001198911015840101584010158400 (1198912)1015840 (1198911)10158401015840+ 11986611989110158400 (1198913)1015840 + 3119866119891101584010158400 1198910 (1198912)10158401015840 (1198911) + 3119866119891101584010158400 11989110158400 (1198912)10158401015840 (1198911)1015840 + 3119866119891101584010158400 119891101584010158400 (1198912)10158401015840 (1198911)10158401015840 + 3119866119891101584010158400 1198911015840101584010158400 (1198912)10158401015840 (1198911)101584010158401015840 + 119866119891101584010158400 (1198913)10158401015840 + 31198661198911015840101584010158400 1198910 (1198912)101584010158401015840 (1198911)+ 31198661198911015840101584010158400 11989110158400 (1198912)101584010158401015840 (1198911)1015840 + 31198661198911015840101584010158400 119891101584010158400 (1198912)101584010158401015840 (1198911)10158401015840+ 31198661198911015840101584010158400 1198911015840101584010158400 (1198912)101584010158401015840 (1198911)101584010158401015840 + 1198661198911015840101584010158400 (1198913)101584010158401015840

(44)


1198661198910 = minus 1205741198781 + 1205741198911015840101584010158400 (120578) 11986611989101198910 = 119866119891011989110158400 = 1198661198910119891101584010158400 = 011986611989101198911015840101584010158400 = minus 1205741198781 + 120574 119866119891011989101198910 = 1198661198910119891101584001198910 = 11986611989101198911015840011989110158400 = 1198661198910119891101584010158400 11989110158400 = 1198661198910119891011989110158400

= 11986611989101198911015840101584010158400 1198911015840101584010158400 = 1198661198910119891101584010158400 119891101584010158400 = 011986611989110158400 = 1205741198781 + 120574 = (3 + 119891101584010158400 (120578))

119866119891101584001198910 = 1198661198911015840011989110158400 = 119866119891101584001198911015840101584010158400 = 011986611989110158400119891101584010158400 = 1205741198781 + 120574 1198661198911015840011989101198910 = 11986611989110158400119891101584001198910 = 119866119891101584001198911015840011989110158400 = 11986611989110158400119891101584010158400 11989110158400 = 119866119891101584001198911015840101584010158400 11989110158400

= 119866119891101584001198911015840101584010158400 1198911015840101584010158400 = 11986611989110158400119891101584010158400 119891101584010158400 = 0119866119891101584010158400 = 1205741198781 + 1205741198911015840 (120578) + 12057411987221 + 120574

119866119891101584010158400 1198910 = 119866119891101584010158400 119891101584010158400 = 119866119891101584010158400 1198911015840101584010158400 = 0


119866119891101584010158400 11989110158400 = 1205741198781 + 120574 119866119891101584010158400 11989101198910 = 119866119891101584010158400 119891101584001198910 = 119866119891101584010158400 1198911015840011989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400

= 119866119891101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 119866119891101584010158400 119891101584010158400 119891101584010158400 = 01198661198911015840101584010158400 = 1205741198781205781 + 120574 minus 1205741198781 + 1205741198910 (120578)

1198661198911015840101584010158400 11989110158400 = 1198661198911015840101584010158400 119891101584010158400 = 1198661198911015840101584010158400 1198911015840101584010158400 = 01198661198911015840101584010158400 1198910 = minus 1205741198781 + 120574

1198661198911015840101584010158400 11989101198910 = 1198661198911015840101584010158400 119891101584001198910 = 1198661198911015840101584010158400 1198911015840011989110158400 = 1198661198911015840101584010158400 119891101584010158400 11989110158400 = 1198661198911015840101584010158400 1198911015840101584010158400 11989110158400= 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 0

(45)


1198910 = 1198602120578 + 1611986041205783 (46)

1198911 = 12520119903119878119860241205787 + 1120119903 (41198781198604 +11987221198604) 1205785 (47)


(48)


+ 11389960011990331198783119860431198602 + minus19729720011990331198722119878211986043)sdot 12057813 + 8340864824000119903311987831198604412057815

(49)



(50)




119865119899+1 = 119866 [119865119899] 119865119899 = 119899sum119896=0

119891119896119899 = 0 1 2 3

(51)






1003817100381710038171003817119865119899 minus 1198651198981003817100381710038171003817 =1003817100381710038171003817100381710038171003817100381710038171003817119899sum119896=0

119891119896 minus 119898sum119896=0

1198911198961003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817[1198910 + 119871minus1119899sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]

minus [1198910 + 119871minus1 119898sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]1003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817119871minus1119866[119899minus1sum

119896=0


119891119896]1003817100381710038171003817100381710038171003817100381710038171003817

(119904119894119899119888119890 119865119899 = 119866 [119865119899minus1])⩽ 10038161003816100381610038161003816Łminus110038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119866[119899minus1sum119896=0

119891119896] minus 119866[119898minus1sum119896=0

119891119896]1003817100381710038171003817100381710038171003817100381710038171003817


(54)


and hence


From (56) we get



(57)



(58)




119871minus1 [119866 [119865119899]] (59)


119869 (119865) = 1198650 + 119871minus1119866 [119865] (60)



(61)



119891119896]


119899sum119896=0

119891119896] = 119871minus1119866[ lim

119899997888rarrinfin119865119899]


119871minus1119866 [119865119899]

(62)






120572119896 =

1003817100381710038171003817119865119896+1 minus 119865119896100381710038171003817100381710038171003817100381710038171198651 minus 11986501003817100381710038171003817 = 1003817100381710038171003817119891119896+11003817100381710038171003817100381710038171003817100381711989111003817100381710038171003817 100381710038171003817100381711989111003817100381710038171003817 = 00 100381710038171003817100381711989111003817100381710038171003817 = 0 (63)









1205723 = 00000001976 lt 1

(64)








1205723 = 00000006301 lt 1

(65)













01 0147656 0147656 1467313 146731302 0292534 0292534 1425575 142557503 0431831 0431832 1355629 135562904 0562701 0562701 1256898 125689905 0682225 0682226 1128583 112858306 0787397 0787398 0969661 096966107 0875096 0875097 0778889 077888108 0942065 0942067 0554821 055482109 0984895 0984897 0295804 0295804


01 0150638 0150638 1485907 149590702 0298136 0298136 1448847 144884703 0439360 0439361 1370673 137067304 0571246 0571246 1261767 126176705 0690716 0690716 1122650 112265006 0794791 0794790 0953971 095397107 0880550 0880541 0756487 075648708 0945156 0945156 0531047 053104709 0985858 0985876 0278565 0278566









1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10


08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10






02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018










119891 (120578) = 119891 (0) + 1198911015840 (0) 120578 + 11989110158401015840 (0) 12057822 + 119891101584010158401015840 (0) 1205783

3+ 119871minus1 [ 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578)]

(32)


119891 (120578) = 1198601 + 1198602120578 + 1198603 12057822 + 1198604 1205783

3 + 119871minus1119866 [119891 (120578)] (33)

where

1198601 = 119891 (0) 1198602 = 1198911015840 (0) 1198603 = 11989110158401015840 (0) 1198604 = 119891101584010158401015840 (0) 119866 [119891] = 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578) + 1198911015840 (120578) 11989110158401015840 (120578)

minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578) 119886119899119889 119871minus1 () = int120578

0int1205780int1205780int1205780(119889120578)4

(34)


119891 (120578) = 1198602120578 + 1198604 12057833 + 119871minus1 [119866 [119891 (120578)] (35)



(36)

Step 3 yields

119866 [119891 (120578])] = 1205741198781 + 120574 (120578119891101584010158401015840 (120578) + 31198911015840 (120578)

+ 1198911015840 (120578) 11989110158401015840 (120578) minus 119891 (120578) 119891101584010158401015840 (120578)) + 12057411987221 + 12057411989110158401015840 (120578)] (37)

1198661015840 [119891 (120578)] = 119889119866 [119891 (120578)]119889120578 = 119866119891119891120578 + 1198661198911015840 (119891120578)1015840 + 11986611989110158401015840

(119891120578)10158401015840 + 119866119891101584010158401015840 (119891120578)101584010158401015840 (38)

11986610158401015840 [119891 (120578)] = 1198892119866 [119891 (120578)]1198891205782 = 119866119891119891 (119891120578)2 + 21198661198911198911015840

119891120578 (119891120578)1015840 + 211986611989111989110158401015840 119891120578 (119891120578)10158401015840 + 2119866119891119891101584010158401015840 119891120578 (119891120578)101584010158401015840+ 11986611989110158401198911015840 (119891120578)10158402 + 2119866119891101584011989110158401015840 (119891120578)1015840 (119891120578)10158401015840 + 21198661198911015840119891101584010158401015840 (119891120578)1015840 (119891120578)101584010158401015840 + 1198661198911015840101584011989110158401015840 (119891120578)101584010158402 + 211986611989110158401015840119891101584010158401015840 (119891120578)10158401015840sdot (119891120578)101584010158401015840 + 119866119891101584010158401015840119891101584010158401015840 (119891120578)1015840101584010158402 + 119866119891119891120578120578 + 1198661198911015840 (119891120578120578)1015840+ 11986611989110158401015840 (119891120578120578)10158401015840 + 119866119891101584010158401015840 (119891120578120578)101584010158401015840

(39)

119866101584010158401015840 [119891 (120578)] = 1198893119866 [119891 (120578)]1198891205783 = 119866119891119891119891 (119891120578)3 + 3

1198661198911198911198911015840 (119891120578)2 (119891120578)1015840 + 311986611989111989111989110158401015840 (119891120578)2 (119891120578)10158401015840 + 3119866119891119891119891101584010158401015840 (119891120578)2 (119891120578)101584010158401015840 + 311986611989111989110158401198911015840 (119891120578) (119891120578)10158402 + 4119866119891119891101584011989110158401015840 (119891120578) (119891120578)1015840 (119891120578)10158401015840 + 41198661198911198911015840119891101584010158401015840 (119891120578) (119891120578)1015840 (119891120578)101584010158401015840 + 31198661198911198911015840101584011989110158401015840 (119891120578) (119891120578)101584010158402 + 411986611989111989110158401015840119891101584010158401015840 (119891120578) (119891120578)10158401015840 (119891120578)101584010158401015840 + 3119866119891119891101584010158401015840119891101584010158401015840 (119891120578) (119891120578)1015840101584010158402 + 119866119891101584011989110158401198911015840 (119891120578)10158403 + 31198661198911015840119891101584011989110158401015840 (119891120578)10158402 (119891120578)10158401015840 + 11986611989110158401198911015840119891101584010158401015840 (119891120578)10158402 (119891120578)101584010158401015840 + 4119866119891101584011989110158401015840119891 (119891120578)1015840 119891120578 (119891120578)10158401015840 + 311986611989110158401198911015840101584011989110158401015840 (119891120578)1015840 (119891120578)101584010158402 + 4119866119891101584011989110158401015840119891101584010158401015840 (119891120578)1015840 (119891120578)10158401015840 (119891120578)101584010158401015840+ 41198661198911015840119891101584010158401015840119891 (119891120578)1015840 119891120578 (119891120578)101584010158401015840 + 4119866119891101584011989110158401015840101584011989110158401015840 (119891120578)1015840 (119891120578)10158401015840 (119891120578)101584010158401015840 + 31198661198911015840119891101584010158401015840119891101584010158401015840 (119891120578)1015840 (119891120578)1015840101584010158402+ 119866119891101584010158401198911015840101584011989110158401015840 (119891120578)101584010158403 + 31198661198911015840101584011989110158401015840101584011989110158401015840 (119891120578)101584010158402 (119891120578)101584010158401015840+ 311986611989110158401015840101584011989110158401015840101584011989110158401015840 (119891120578)1015840101584010158402 (119891120578)10158401015840 + 119866119891101584010158401015840119891101584010158401015840119891101584010158401015840 (119891120578)1015840101584010158403+ 3119866119891119891119891120578120578119891120578 + 31198661198911198911015840 119891120578120578 (119891120578)1015840 + 311986611989111989110158401015840 119891120578120578(119891120578)10158401015840 + 3119866119891119891101584010158401015840 119891120578120578 (119891120578)101584010158401015840 + 119866119891119891120578120578120578 + 31198661198911015840119891


(119891120578120578)1015840 (119891120578) + 311986611989110158401198911015840 (119891120578120578)1015840 (119891120578)1015840 + 3119866119891101584011989110158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 31198661198911015840119891101584010158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 1198661198911015840 (119891120578120578120578)1015840 + 311986611989110158401015840119891 (119891120578120578)10158401015840 (119891120578) + 3119866119891101584010158401198911015840 (119891120578120578)10158401015840 (119891120578)1015840 + 31198661198911015840101584011989110158401015840 (119891120578120578)10158401015840 (119891120578)10158401015840 + 311986611989110158401015840119891101584010158401015840 (119891120578120578)10158401015840 (119891120578)101584010158401015840 + 11986611989110158401015840 (119891120578120578120578)10158401015840 + 3119866119891101584010158401015840119891 (119891120578120578)101584010158401015840 (119891120578)+ 31198661198911015840101584010158401198911015840 (119891120578120578)101584010158401015840 (119891120578)1015840 + 311986611989110158401015840101584011989110158401015840 (119891120578120578)101584010158401015840 (119891120578)10158401015840+ 3119866119891101584010158401015840119891101584010158401015840 (119891120578120578)101584010158401015840 (119891120578)101584010158401015840 + 119866119891101584010158401015840 (119891120578120578120578)101584010158401015840

(40)


119866 [1198910 (120578)] = 1205741198781 + 120574 ((1205781198911015840101584010158400 (120578) + 311989110158400 (120578)+ 11989110158400 (120578) 119891101584010158400 (120578) minus 1198910 (120578) 1198911015840101584010158400 (120578)) +1198722119891101584010158400 (120578))

(41)

1198661015840 [1198910 (120578)] = 1198661198910 1198911 + 11986611989101015840 (1198911)1015840 + 119866119891101584010158400 (1198911)10158401015840+ 1198661198911015840101584010158400 (1198911)101584010158401015840

(42)

11986610158401015840 [1198910 (120578)] = 11986611989101198910 (1198911)2 + 2119866119891011989110158400 1198911 (1198911)1015840 + 21198661198910119891101584010158400 1198911 (1198911)10158401015840 + 211986611989101198911015840101584010158400 1198911 (1198911)101584010158401015840 + 1198661198911015840011989110158400 (1198911)10158402+ 211986611989110158400119891101584010158400 (1198911)1015840 (1198911)10158401015840 + 2119866119891101584001198911015840101584010158400 (1198911)1015840 (1198911)101584010158401015840+ 119866119891101584010158400 119891101584010158400 (1198911)101584010158402 + 2119866119891101584010158400 1198911015840101584010158400 (1198911)10158401015840 (1198911)101584010158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158402 + 1198661198910 1198912 + 11986611989110158400 (1198912)1015840 + 119866119891101584010158400 (1198912)10158401015840 + 1198661198911015840101584010158400 (1198912)101584010158401015840

(43)

119866101584010158401015840 [1198910 (120578)] = 119866119891011989101198910 (1198911)3 + 31198661198910119891011989110158400 (1198911)2 (1198911)1015840+ 311986611989101198910119891101584010158400 (1198911)2 (1198911)10158401015840 + 3119866119891011989101198911015840101584010158400 (1198911)2 (1198911)101584010158401015840+ 311986611989101198911015840011989110158400 (1198911) (1198911)10158402 + 4119866119891011989110158400119891101584010158400 (1198911) (1198911)1015840 (1198911)10158401015840 + 41198661198910119891101584001198911015840101584010158400 (1198911) (1198911)1015840 (1198911)101584010158401015840 + 31198661198910119891101584010158400 119891101584010158400 (1198911) (1198911)101584010158402 + 41198661198910119891101584010158400 1198911015840101584010158400 (1198911) (1198911)10158401015840 (1198911)101584010158401015840 + 311986611989101198911015840101584010158400 1198911015840101584010158400 (1198911) (1198911)1015840101584010158402 + 119866119891101584001198911015840011989110158400 (1198911)10158403 + 31198661198911015840011989110158400119891101584010158400 (1198911)10158402 (1198911)10158401015840 + 11986611989110158400119891101584001198911015840101584010158400 (1198911)10158402 (1198911)101584010158401015840 + 411986611989110158400119891101584010158400 1198910 (1198911)1015840 1198911 (1198911)10158401015840 + 311986611989110158400119891101584010158400 119891101584010158400 (1198911)1015840 (1198911)101584010158402

+ 411986611989110158400119891101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 1198910 (1198911)1015840 1198911 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 119891101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 3119866119891101584001198911015840101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)1015840101584010158402 + 119866119891101584010158400 119891101584010158400 119891101584010158400 (1198911)101584010158403 + 3119866119891101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)101584010158402 (1198911)101584010158401015840 + 31198661198911015840101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)1015840101584010158402 (1198911)10158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158403 + 311986611989101198910 11989121198911 + 3119866119891011989110158400 1198912(1198911)1015840 + 31198661198910119891101584010158400 1198912 (1198911)10158401015840 + 311986611989101198911015840101584010158400 1198912 (1198911)101584010158401015840+ 1198661198910 1198913 + 3119866119891101584001198910 (1198912)1015840 (1198911) + 31198661198911015840011989110158400 (1198912)1015840 (1198911)1015840+ 311986611989110158400119891101584010158400 (1198912)1015840 (1198911)10158401015840 + 3119866119891101584001198911015840101584010158400 (1198912)1015840 (1198911)10158401015840+ 11986611989110158400 (1198913)1015840 + 3119866119891101584010158400 1198910 (1198912)10158401015840 (1198911) + 3119866119891101584010158400 11989110158400 (1198912)10158401015840 (1198911)1015840 + 3119866119891101584010158400 119891101584010158400 (1198912)10158401015840 (1198911)10158401015840 + 3119866119891101584010158400 1198911015840101584010158400 (1198912)10158401015840 (1198911)101584010158401015840 + 119866119891101584010158400 (1198913)10158401015840 + 31198661198911015840101584010158400 1198910 (1198912)101584010158401015840 (1198911)+ 31198661198911015840101584010158400 11989110158400 (1198912)101584010158401015840 (1198911)1015840 + 31198661198911015840101584010158400 119891101584010158400 (1198912)101584010158401015840 (1198911)10158401015840+ 31198661198911015840101584010158400 1198911015840101584010158400 (1198912)101584010158401015840 (1198911)101584010158401015840 + 1198661198911015840101584010158400 (1198913)101584010158401015840

(44)


1198661198910 = minus 1205741198781 + 1205741198911015840101584010158400 (120578) 11986611989101198910 = 119866119891011989110158400 = 1198661198910119891101584010158400 = 011986611989101198911015840101584010158400 = minus 1205741198781 + 120574 119866119891011989101198910 = 1198661198910119891101584001198910 = 11986611989101198911015840011989110158400 = 1198661198910119891101584010158400 11989110158400 = 1198661198910119891011989110158400

= 11986611989101198911015840101584010158400 1198911015840101584010158400 = 1198661198910119891101584010158400 119891101584010158400 = 011986611989110158400 = 1205741198781 + 120574 = (3 + 119891101584010158400 (120578))

119866119891101584001198910 = 1198661198911015840011989110158400 = 119866119891101584001198911015840101584010158400 = 011986611989110158400119891101584010158400 = 1205741198781 + 120574 1198661198911015840011989101198910 = 11986611989110158400119891101584001198910 = 119866119891101584001198911015840011989110158400 = 11986611989110158400119891101584010158400 11989110158400 = 119866119891101584001198911015840101584010158400 11989110158400

= 119866119891101584001198911015840101584010158400 1198911015840101584010158400 = 11986611989110158400119891101584010158400 119891101584010158400 = 0119866119891101584010158400 = 1205741198781 + 1205741198911015840 (120578) + 12057411987221 + 120574

119866119891101584010158400 1198910 = 119866119891101584010158400 119891101584010158400 = 119866119891101584010158400 1198911015840101584010158400 = 0


119866119891101584010158400 11989110158400 = 1205741198781 + 120574 119866119891101584010158400 11989101198910 = 119866119891101584010158400 119891101584001198910 = 119866119891101584010158400 1198911015840011989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400

= 119866119891101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 119866119891101584010158400 119891101584010158400 119891101584010158400 = 01198661198911015840101584010158400 = 1205741198781205781 + 120574 minus 1205741198781 + 1205741198910 (120578)

1198661198911015840101584010158400 11989110158400 = 1198661198911015840101584010158400 119891101584010158400 = 1198661198911015840101584010158400 1198911015840101584010158400 = 01198661198911015840101584010158400 1198910 = minus 1205741198781 + 120574

1198661198911015840101584010158400 11989101198910 = 1198661198911015840101584010158400 119891101584001198910 = 1198661198911015840101584010158400 1198911015840011989110158400 = 1198661198911015840101584010158400 119891101584010158400 11989110158400 = 1198661198911015840101584010158400 1198911015840101584010158400 11989110158400= 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 0

(45)


1198910 = 1198602120578 + 1611986041205783 (46)

1198911 = 12520119903119878119860241205787 + 1120119903 (41198781198604 +11987221198604) 1205785 (47)


(48)


+ 11389960011990331198783119860431198602 + minus19729720011990331198722119878211986043)sdot 12057813 + 8340864824000119903311987831198604412057815

(49)



(50)




119865119899+1 = 119866 [119865119899] 119865119899 = 119899sum119896=0

119891119896119899 = 0 1 2 3

(51)






1003817100381710038171003817119865119899 minus 1198651198981003817100381710038171003817 =1003817100381710038171003817100381710038171003817100381710038171003817119899sum119896=0

119891119896 minus 119898sum119896=0

1198911198961003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817[1198910 + 119871minus1119899sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]

minus [1198910 + 119871minus1 119898sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]1003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817119871minus1119866[119899minus1sum

119896=0


119891119896]1003817100381710038171003817100381710038171003817100381710038171003817

(119904119894119899119888119890 119865119899 = 119866 [119865119899minus1])⩽ 10038161003816100381610038161003816Łminus110038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119866[119899minus1sum119896=0

119891119896] minus 119866[119898minus1sum119896=0

119891119896]1003817100381710038171003817100381710038171003817100381710038171003817


(54)


and hence


From (56) we get



(57)



(58)




119871minus1 [119866 [119865119899]] (59)


119869 (119865) = 1198650 + 119871minus1119866 [119865] (60)



(61)



119891119896]


119899sum119896=0

119891119896] = 119871minus1119866[ lim

119899997888rarrinfin119865119899]


119871minus1119866 [119865119899]

(62)






120572119896 =

1003817100381710038171003817119865119896+1 minus 119865119896100381710038171003817100381710038171003817100381710038171198651 minus 11986501003817100381710038171003817 = 1003817100381710038171003817119891119896+11003817100381710038171003817100381710038171003817100381711989111003817100381710038171003817 100381710038171003817100381711989111003817100381710038171003817 = 00 100381710038171003817100381711989111003817100381710038171003817 = 0 (63)









1205723 = 00000001976 lt 1

(64)








1205723 = 00000006301 lt 1

(65)













01 0147656 0147656 1467313 146731302 0292534 0292534 1425575 142557503 0431831 0431832 1355629 135562904 0562701 0562701 1256898 125689905 0682225 0682226 1128583 112858306 0787397 0787398 0969661 096966107 0875096 0875097 0778889 077888108 0942065 0942067 0554821 055482109 0984895 0984897 0295804 0295804


01 0150638 0150638 1485907 149590702 0298136 0298136 1448847 144884703 0439360 0439361 1370673 137067304 0571246 0571246 1261767 126176705 0690716 0690716 1122650 112265006 0794791 0794790 0953971 095397107 0880550 0880541 0756487 075648708 0945156 0945156 0531047 053104709 0985858 0985876 0278565 0278566









1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10


08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10






02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









(119891120578120578)1015840 (119891120578) + 311986611989110158401198911015840 (119891120578120578)1015840 (119891120578)1015840 + 3119866119891101584011989110158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 31198661198911015840119891101584010158401015840 (119891120578120578)1015840 (119891120578)10158401015840 + 1198661198911015840 (119891120578120578120578)1015840 + 311986611989110158401015840119891 (119891120578120578)10158401015840 (119891120578) + 3119866119891101584010158401198911015840 (119891120578120578)10158401015840 (119891120578)1015840 + 31198661198911015840101584011989110158401015840 (119891120578120578)10158401015840 (119891120578)10158401015840 + 311986611989110158401015840119891101584010158401015840 (119891120578120578)10158401015840 (119891120578)101584010158401015840 + 11986611989110158401015840 (119891120578120578120578)10158401015840 + 3119866119891101584010158401015840119891 (119891120578120578)101584010158401015840 (119891120578)+ 31198661198911015840101584010158401198911015840 (119891120578120578)101584010158401015840 (119891120578)1015840 + 311986611989110158401015840101584011989110158401015840 (119891120578120578)101584010158401015840 (119891120578)10158401015840+ 3119866119891101584010158401015840119891101584010158401015840 (119891120578120578)101584010158401015840 (119891120578)101584010158401015840 + 119866119891101584010158401015840 (119891120578120578120578)101584010158401015840

(40)


119866 [1198910 (120578)] = 1205741198781 + 120574 ((1205781198911015840101584010158400 (120578) + 311989110158400 (120578)+ 11989110158400 (120578) 119891101584010158400 (120578) minus 1198910 (120578) 1198911015840101584010158400 (120578)) +1198722119891101584010158400 (120578))

(41)

1198661015840 [1198910 (120578)] = 1198661198910 1198911 + 11986611989101015840 (1198911)1015840 + 119866119891101584010158400 (1198911)10158401015840+ 1198661198911015840101584010158400 (1198911)101584010158401015840

(42)

11986610158401015840 [1198910 (120578)] = 11986611989101198910 (1198911)2 + 2119866119891011989110158400 1198911 (1198911)1015840 + 21198661198910119891101584010158400 1198911 (1198911)10158401015840 + 211986611989101198911015840101584010158400 1198911 (1198911)101584010158401015840 + 1198661198911015840011989110158400 (1198911)10158402+ 211986611989110158400119891101584010158400 (1198911)1015840 (1198911)10158401015840 + 2119866119891101584001198911015840101584010158400 (1198911)1015840 (1198911)101584010158401015840+ 119866119891101584010158400 119891101584010158400 (1198911)101584010158402 + 2119866119891101584010158400 1198911015840101584010158400 (1198911)10158401015840 (1198911)101584010158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158402 + 1198661198910 1198912 + 11986611989110158400 (1198912)1015840 + 119866119891101584010158400 (1198912)10158401015840 + 1198661198911015840101584010158400 (1198912)101584010158401015840

(43)

119866101584010158401015840 [1198910 (120578)] = 119866119891011989101198910 (1198911)3 + 31198661198910119891011989110158400 (1198911)2 (1198911)1015840+ 311986611989101198910119891101584010158400 (1198911)2 (1198911)10158401015840 + 3119866119891011989101198911015840101584010158400 (1198911)2 (1198911)101584010158401015840+ 311986611989101198911015840011989110158400 (1198911) (1198911)10158402 + 4119866119891011989110158400119891101584010158400 (1198911) (1198911)1015840 (1198911)10158401015840 + 41198661198910119891101584001198911015840101584010158400 (1198911) (1198911)1015840 (1198911)101584010158401015840 + 31198661198910119891101584010158400 119891101584010158400 (1198911) (1198911)101584010158402 + 41198661198910119891101584010158400 1198911015840101584010158400 (1198911) (1198911)10158401015840 (1198911)101584010158401015840 + 311986611989101198911015840101584010158400 1198911015840101584010158400 (1198911) (1198911)1015840101584010158402 + 119866119891101584001198911015840011989110158400 (1198911)10158403 + 31198661198911015840011989110158400119891101584010158400 (1198911)10158402 (1198911)10158401015840 + 11986611989110158400119891101584001198911015840101584010158400 (1198911)10158402 (1198911)101584010158401015840 + 411986611989110158400119891101584010158400 1198910 (1198911)1015840 1198911 (1198911)10158401015840 + 311986611989110158400119891101584010158400 119891101584010158400 (1198911)1015840 (1198911)101584010158402

+ 411986611989110158400119891101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 1198910 (1198911)1015840 1198911 (1198911)101584010158401015840 + 4119866119891101584001198911015840101584010158400 119891101584010158400 (1198911)1015840 (1198911)10158401015840 (1198911)101584010158401015840 + 3119866119891101584001198911015840101584010158400 1198911015840101584010158400 (1198911)1015840 (1198911)1015840101584010158402 + 119866119891101584010158400 119891101584010158400 119891101584010158400 (1198911)101584010158403 + 3119866119891101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)101584010158402 (1198911)101584010158401015840 + 31198661198911015840101584010158400 1198911015840101584010158400 119891101584010158400 (1198911)1015840101584010158402 (1198911)10158401015840+ 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 (1198911)1015840101584010158403 + 311986611989101198910 11989121198911 + 3119866119891011989110158400 1198912(1198911)1015840 + 31198661198910119891101584010158400 1198912 (1198911)10158401015840 + 311986611989101198911015840101584010158400 1198912 (1198911)101584010158401015840+ 1198661198910 1198913 + 3119866119891101584001198910 (1198912)1015840 (1198911) + 31198661198911015840011989110158400 (1198912)1015840 (1198911)1015840+ 311986611989110158400119891101584010158400 (1198912)1015840 (1198911)10158401015840 + 3119866119891101584001198911015840101584010158400 (1198912)1015840 (1198911)10158401015840+ 11986611989110158400 (1198913)1015840 + 3119866119891101584010158400 1198910 (1198912)10158401015840 (1198911) + 3119866119891101584010158400 11989110158400 (1198912)10158401015840 (1198911)1015840 + 3119866119891101584010158400 119891101584010158400 (1198912)10158401015840 (1198911)10158401015840 + 3119866119891101584010158400 1198911015840101584010158400 (1198912)10158401015840 (1198911)101584010158401015840 + 119866119891101584010158400 (1198913)10158401015840 + 31198661198911015840101584010158400 1198910 (1198912)101584010158401015840 (1198911)+ 31198661198911015840101584010158400 11989110158400 (1198912)101584010158401015840 (1198911)1015840 + 31198661198911015840101584010158400 119891101584010158400 (1198912)101584010158401015840 (1198911)10158401015840+ 31198661198911015840101584010158400 1198911015840101584010158400 (1198912)101584010158401015840 (1198911)101584010158401015840 + 1198661198911015840101584010158400 (1198913)101584010158401015840

(44)


1198661198910 = minus 1205741198781 + 1205741198911015840101584010158400 (120578) 11986611989101198910 = 119866119891011989110158400 = 1198661198910119891101584010158400 = 011986611989101198911015840101584010158400 = minus 1205741198781 + 120574 119866119891011989101198910 = 1198661198910119891101584001198910 = 11986611989101198911015840011989110158400 = 1198661198910119891101584010158400 11989110158400 = 1198661198910119891011989110158400

= 11986611989101198911015840101584010158400 1198911015840101584010158400 = 1198661198910119891101584010158400 119891101584010158400 = 011986611989110158400 = 1205741198781 + 120574 = (3 + 119891101584010158400 (120578))

119866119891101584001198910 = 1198661198911015840011989110158400 = 119866119891101584001198911015840101584010158400 = 011986611989110158400119891101584010158400 = 1205741198781 + 120574 1198661198911015840011989101198910 = 11986611989110158400119891101584001198910 = 119866119891101584001198911015840011989110158400 = 11986611989110158400119891101584010158400 11989110158400 = 119866119891101584001198911015840101584010158400 11989110158400

= 119866119891101584001198911015840101584010158400 1198911015840101584010158400 = 11986611989110158400119891101584010158400 119891101584010158400 = 0119866119891101584010158400 = 1205741198781 + 1205741198911015840 (120578) + 12057411987221 + 120574

119866119891101584010158400 1198910 = 119866119891101584010158400 119891101584010158400 = 119866119891101584010158400 1198911015840101584010158400 = 0


119866119891101584010158400 11989110158400 = 1205741198781 + 120574 119866119891101584010158400 11989101198910 = 119866119891101584010158400 119891101584001198910 = 119866119891101584010158400 1198911015840011989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400

= 119866119891101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 119866119891101584010158400 119891101584010158400 119891101584010158400 = 01198661198911015840101584010158400 = 1205741198781205781 + 120574 minus 1205741198781 + 1205741198910 (120578)

1198661198911015840101584010158400 11989110158400 = 1198661198911015840101584010158400 119891101584010158400 = 1198661198911015840101584010158400 1198911015840101584010158400 = 01198661198911015840101584010158400 1198910 = minus 1205741198781 + 120574

1198661198911015840101584010158400 11989101198910 = 1198661198911015840101584010158400 119891101584001198910 = 1198661198911015840101584010158400 1198911015840011989110158400 = 1198661198911015840101584010158400 119891101584010158400 11989110158400 = 1198661198911015840101584010158400 1198911015840101584010158400 11989110158400= 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 0

(45)


1198910 = 1198602120578 + 1611986041205783 (46)

1198911 = 12520119903119878119860241205787 + 1120119903 (41198781198604 +11987221198604) 1205785 (47)


(48)


+ 11389960011990331198783119860431198602 + minus19729720011990331198722119878211986043)sdot 12057813 + 8340864824000119903311987831198604412057815

(49)



(50)




119865119899+1 = 119866 [119865119899] 119865119899 = 119899sum119896=0

119891119896119899 = 0 1 2 3

(51)






1003817100381710038171003817119865119899 minus 1198651198981003817100381710038171003817 =1003817100381710038171003817100381710038171003817100381710038171003817119899sum119896=0

119891119896 minus 119898sum119896=0

1198911198961003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817[1198910 + 119871minus1119899sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]

minus [1198910 + 119871minus1 119898sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]1003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817119871minus1119866[119899minus1sum

119896=0


119891119896]1003817100381710038171003817100381710038171003817100381710038171003817

(119904119894119899119888119890 119865119899 = 119866 [119865119899minus1])⩽ 10038161003816100381610038161003816Łminus110038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119866[119899minus1sum119896=0

119891119896] minus 119866[119898minus1sum119896=0

119891119896]1003817100381710038171003817100381710038171003817100381710038171003817


(54)


and hence


From (56) we get



(57)



(58)




119871minus1 [119866 [119865119899]] (59)


119869 (119865) = 1198650 + 119871minus1119866 [119865] (60)



(61)



119891119896]


119899sum119896=0

119891119896] = 119871minus1119866[ lim

119899997888rarrinfin119865119899]


119871minus1119866 [119865119899]

(62)






120572119896 =

1003817100381710038171003817119865119896+1 minus 119865119896100381710038171003817100381710038171003817100381710038171198651 minus 11986501003817100381710038171003817 = 1003817100381710038171003817119891119896+11003817100381710038171003817100381710038171003817100381711989111003817100381710038171003817 100381710038171003817100381711989111003817100381710038171003817 = 00 100381710038171003817100381711989111003817100381710038171003817 = 0 (63)









1205723 = 00000001976 lt 1

(64)








1205723 = 00000006301 lt 1

(65)













01 0147656 0147656 1467313 146731302 0292534 0292534 1425575 142557503 0431831 0431832 1355629 135562904 0562701 0562701 1256898 125689905 0682225 0682226 1128583 112858306 0787397 0787398 0969661 096966107 0875096 0875097 0778889 077888108 0942065 0942067 0554821 055482109 0984895 0984897 0295804 0295804


01 0150638 0150638 1485907 149590702 0298136 0298136 1448847 144884703 0439360 0439361 1370673 137067304 0571246 0571246 1261767 126176705 0690716 0690716 1122650 112265006 0794791 0794790 0953971 095397107 0880550 0880541 0756487 075648708 0945156 0945156 0531047 053104709 0985858 0985876 0278565 0278566









1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10


08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10






02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









119866119891101584010158400 11989110158400 = 1205741198781 + 120574 119866119891101584010158400 11989101198910 = 119866119891101584010158400 119891101584001198910 = 119866119891101584010158400 1198911015840011989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400 = 119866119891101584010158400 119891101584010158400 11989110158400

= 119866119891101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 119866119891101584010158400 119891101584010158400 119891101584010158400 = 01198661198911015840101584010158400 = 1205741198781205781 + 120574 minus 1205741198781 + 1205741198910 (120578)

1198661198911015840101584010158400 11989110158400 = 1198661198911015840101584010158400 119891101584010158400 = 1198661198911015840101584010158400 1198911015840101584010158400 = 01198661198911015840101584010158400 1198910 = minus 1205741198781 + 120574

1198661198911015840101584010158400 11989101198910 = 1198661198911015840101584010158400 119891101584001198910 = 1198661198911015840101584010158400 1198911015840011989110158400 = 1198661198911015840101584010158400 119891101584010158400 11989110158400 = 1198661198911015840101584010158400 1198911015840101584010158400 11989110158400= 1198661198911015840101584010158400 1198911015840101584010158400 1198911015840101584010158400 = 0

(45)


1198910 = 1198602120578 + 1611986041205783 (46)

1198911 = 12520119903119878119860241205787 + 1120119903 (41198781198604 +11987221198604) 1205785 (47)


(48)


+ 11389960011990331198783119860431198602 + minus19729720011990331198722119878211986043)sdot 12057813 + 8340864824000119903311987831198604412057815

(49)



(50)




119865119899+1 = 119866 [119865119899] 119865119899 = 119899sum119896=0

119891119896119899 = 0 1 2 3

(51)






1003817100381710038171003817119865119899 minus 1198651198981003817100381710038171003817 =1003817100381710038171003817100381710038171003817100381710038171003817119899sum119896=0

119891119896 minus 119898sum119896=0

1198911198961003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817[1198910 + 119871minus1119899sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]

minus [1198910 + 119871minus1 119898sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]1003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817119871minus1119866[119899minus1sum

119896=0


119891119896]1003817100381710038171003817100381710038171003817100381710038171003817

(119904119894119899119888119890 119865119899 = 119866 [119865119899minus1])⩽ 10038161003816100381610038161003816Łminus110038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119866[119899minus1sum119896=0

119891119896] minus 119866[119898minus1sum119896=0

119891119896]1003817100381710038171003817100381710038171003817100381710038171003817


(54)


and hence


From (56) we get



(57)



(58)




119871minus1 [119866 [119865119899]] (59)


119869 (119865) = 1198650 + 119871minus1119866 [119865] (60)



(61)



119891119896]


119899sum119896=0

119891119896] = 119871minus1119866[ lim

119899997888rarrinfin119865119899]


119871minus1119866 [119865119899]

(62)






120572119896 =

1003817100381710038171003817119865119896+1 minus 119865119896100381710038171003817100381710038171003817100381710038171198651 minus 11986501003817100381710038171003817 = 1003817100381710038171003817119891119896+11003817100381710038171003817100381710038171003817100381711989111003817100381710038171003817 100381710038171003817100381711989111003817100381710038171003817 = 00 100381710038171003817100381711989111003817100381710038171003817 = 0 (63)









1205723 = 00000001976 lt 1

(64)








1205723 = 00000006301 lt 1

(65)













01 0147656 0147656 1467313 146731302 0292534 0292534 1425575 142557503 0431831 0431832 1355629 135562904 0562701 0562701 1256898 125689905 0682225 0682226 1128583 112858306 0787397 0787398 0969661 096966107 0875096 0875097 0778889 077888108 0942065 0942067 0554821 055482109 0984895 0984897 0295804 0295804


01 0150638 0150638 1485907 149590702 0298136 0298136 1448847 144884703 0439360 0439361 1370673 137067304 0571246 0571246 1261767 126176705 0690716 0690716 1122650 112265006 0794791 0794790 0953971 095397107 0880550 0880541 0756487 075648708 0945156 0945156 0531047 053104709 0985858 0985876 0278565 0278566









1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10


08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10






02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018












1003817100381710038171003817119865119899 minus 1198651198981003817100381710038171003817 =1003817100381710038171003817100381710038171003817100381710038171003817119899sum119896=0

119891119896 minus 119898sum119896=0

1198911198961003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817[1198910 + 119871minus1119899sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]

minus [1198910 + 119871minus1 119898sum119896=1

119889(119896minus1)119866 [1198910 (120578)]119889120578(119896minus1) ]1003817100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038171003817119871minus1119866[119899minus1sum

119896=0


119891119896]1003817100381710038171003817100381710038171003817100381710038171003817

(119904119894119899119888119890 119865119899 = 119866 [119865119899minus1])⩽ 10038161003816100381610038161003816Łminus110038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119866[119899minus1sum119896=0

119891119896] minus 119866[119898minus1sum119896=0

119891119896]1003817100381710038171003817100381710038171003817100381710038171003817


(54)


and hence


From (56) we get



(57)



(58)




119871minus1 [119866 [119865119899]] (59)


119869 (119865) = 1198650 + 119871minus1119866 [119865] (60)



(61)



119891119896]


119899sum119896=0

119891119896] = 119871minus1119866[ lim

119899997888rarrinfin119865119899]


119871minus1119866 [119865119899]

(62)






120572119896 =

1003817100381710038171003817119865119896+1 minus 119865119896100381710038171003817100381710038171003817100381710038171198651 minus 11986501003817100381710038171003817 = 1003817100381710038171003817119891119896+11003817100381710038171003817100381710038171003817100381711989111003817100381710038171003817 100381710038171003817100381711989111003817100381710038171003817 = 00 100381710038171003817100381711989111003817100381710038171003817 = 0 (63)









1205723 = 00000001976 lt 1

(64)








1205723 = 00000006301 lt 1

(65)













01 0147656 0147656 1467313 146731302 0292534 0292534 1425575 142557503 0431831 0431832 1355629 135562904 0562701 0562701 1256898 125689905 0682225 0682226 1128583 112858306 0787397 0787398 0969661 096966107 0875096 0875097 0778889 077888108 0942065 0942067 0554821 055482109 0984895 0984897 0295804 0295804


01 0150638 0150638 1485907 149590702 0298136 0298136 1448847 144884703 0439360 0439361 1370673 137067304 0571246 0571246 1261767 126176705 0690716 0690716 1122650 112265006 0794791 0794790 0953971 095397107 0880550 0880541 0756487 075648708 0945156 0945156 0531047 053104709 0985858 0985876 0278565 0278566









1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10


08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10






02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018












120572119896 =

1003817100381710038171003817119865119896+1 minus 119865119896100381710038171003817100381710038171003817100381710038171198651 minus 11986501003817100381710038171003817 = 1003817100381710038171003817119891119896+11003817100381710038171003817100381710038171003817100381711989111003817100381710038171003817 100381710038171003817100381711989111003817100381710038171003817 = 00 100381710038171003817100381711989111003817100381710038171003817 = 0 (63)









1205723 = 00000001976 lt 1

(64)








1205723 = 00000006301 lt 1

(65)













01 0147656 0147656 1467313 146731302 0292534 0292534 1425575 142557503 0431831 0431832 1355629 135562904 0562701 0562701 1256898 125689905 0682225 0682226 1128583 112858306 0787397 0787398 0969661 096966107 0875096 0875097 0778889 077888108 0942065 0942067 0554821 055482109 0984895 0984897 0295804 0295804


01 0150638 0150638 1485907 149590702 0298136 0298136 1448847 144884703 0439360 0439361 1370673 137067304 0571246 0571246 1261767 126176705 0690716 0690716 1122650 112265006 0794791 0794790 0953971 095397107 0880550 0880541 0756487 075648708 0945156 0945156 0531047 053104709 0985858 0985876 0278565 0278566









1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10


08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10






02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018


















01 0147656 0147656 1467313 146731302 0292534 0292534 1425575 142557503 0431831 0431832 1355629 135562904 0562701 0562701 1256898 125689905 0682225 0682226 1128583 112858306 0787397 0787398 0969661 096966107 0875096 0875097 0778889 077888108 0942065 0942067 0554821 055482109 0984895 0984897 0295804 0295804


01 0150638 0150638 1485907 149590702 0298136 0298136 1448847 144884703 0439360 0439361 1370673 137067304 0571246 0571246 1261767 126176705 0690716 0690716 1122650 112265006 0794791 0794790 0953971 095397107 0880550 0880541 0756487 075648708 0945156 0945156 0531047 053104709 0985858 0985876 0278565 0278566









1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10


08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10






02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018










01 0147656 0147656 1467313 146731302 0292534 0292534 1425575 142557503 0431831 0431832 1355629 135562904 0562701 0562701 1256898 125689905 0682225 0682226 1128583 112858306 0787397 0787398 0969661 096966107 0875096 0875097 0778889 077888108 0942065 0942067 0554821 055482109 0984895 0984897 0295804 0295804


01 0150638 0150638 1485907 149590702 0298136 0298136 1448847 144884703 0439360 0439361 1370673 137067304 0571246 0571246 1261767 126176705 0690716 0690716 1122650 112265006 0794791 0794790 0953971 095397107 0880550 0880541 0756487 075648708 0945156 0945156 0531047 053104709 0985858 0985876 0278565 0278566









1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10


08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10






02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









1

08

06

04

02

0

f(

)

S=1S=4

S=7S=10

S=1S=4

S=7S=10

14

12

1

08

06

04

02

0

f ()

10 06 0802 04

02 04 06 08 10


08

06

04

02

0

f(

)

14

12

1

08

06

04

02

0

f ()

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10






02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)14

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1


02 04 06 08 10

02 04 06 08 10

1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

S=-1S=-4

S=-7S=-10

S=-1S=-4

S=-7S=-10



7 Conclusions



1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

002

0406

081

0

0204

0608

1

t x

u v

0

05-minus1

15-minus2

25-

10

09

08

1080604020

02-04-06-08-


002

0406

081

0

0204

0608

1

t x

002

0406

08105

0403

0201

0

xt

002

0406

08105

0403

0201

0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









1

08

06

04

02

0

f(

)

f ()

0

05

1

15

2

M=1M =3

M=5M=7

M=1M =3

M=5M=7

02 04 06 08 10

02 04 06 08 10


1

08

06

04

02

0

f(

)

14

16

18

12

1

08

06

04

02

0

f ()

=01=04

=07=1

=01=04

=07=1

02 04 06 08 10

10 06 0802 04





002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

0406

081

002

0406

081

t x

002

04

0608

1

002

04

0608

1

t x

002

0406

081

002

0406

081

t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

minus1 times 108

minus2 times 108

minus3 times 108

minus4 times 108


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

minus50

minus100

minus150

minus200

minus250

40

30

20

10

1600140012001000800600400200


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

3634323

282624222

1

0

minus1

minus2

minus3

minus4



002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

0

05-

minus1

15-

minus2

28

26

24

22

20

18

002-04-06-08-minus1

12-14-16-


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

002-04-06-08-minus1

12-14-

212

19181716

0

05-

minus1


002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

002

0406

08

1

t x

u v

1614121

08060402

6

5

4

3

2

2

0

minus2

minus4

minus6


002

0406

081

002

0406

081

t x

002

0406

08

1

002

0406

08

1

t x

002

0406

081

002

0406

081

t x

u v

0

minus1

minus2

minus3

35

3

25

2

002-04-06-08-minus1

12-14-



002

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t x

002

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t x

002

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002

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t x

uv

0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

0

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minus2 times 109

minus3 times 109

minus4 times 109

minus5 times 109


002

0406

081

0

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1

t x

002

0406

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0

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002

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0

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uv

0

minus20

minus40

minus60

minus80

11

10

09

08

07

40

30

20

10


002

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0

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002

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0

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0

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t x

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0

05-minus1

15-minus2

25-

10

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02-04-06-08-


002

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0

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1

t x

002

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0

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002

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0

xt

u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









002

0406

081

002

0406

081

t x

002

0406

081

002

0406

081

t x

002

0406

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002

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0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

u v

minus2

205-210-215-220-

0

minus2 times 108

minus4 times 108

minus6 times 108

minus8 times 108

minus1 times 109


002

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002

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t x

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04

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1

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t x

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t x

uv

0

minus1 times 109

minus2 times 109

minus3 times 109

186-187-188-189-190-191-192-193-194-195-

0

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minus2 times 108

minus3 times 108

minus4 times 108


002

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1

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1

t x

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1

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40

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28

26

24

22

20

18

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212

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0

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minus1 times 1010

minus15 times 1010

minus2 times 1010

minus441-42-43-44-45-46-47-

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minus2 times 109

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002

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t x

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uv

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minus40

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11

10

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40

30

20

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0

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u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









002

0406

08

1

002

0406

08

1

t x

002

0406

08

1

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28

26

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20

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002-04-06-08-minus1

12-14-

212

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0

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08

1

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35

3

25

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002

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0

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minus1 times 1010

minus15 times 1010

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minus2 times 109

minus3 times 109

minus4 times 109

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002

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002

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0

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002

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0

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uv

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minus40

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11

10

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40

30

20

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u v

002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









002

0406

081

002

0406

081

t x

002

0406

081

002

0406

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002

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0

minus5 times 109

minus1 times 1010

minus15 times 1010

minus2 times 1010

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0

minus1 times 109

minus2 times 109

minus3 times 109

minus4 times 109

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002

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1

t x

002

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11

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40

30

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002-04-06-08-minus1

12-

170

168

166

164

162

160

001-02-03-04-05-06-




Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018










Data Availability




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

A Novel Algorithm for Studying the Effects of Squeezing