Research ArticleNumerical Solution of Piecewise Constant Delay SystemsBased on a Hybrid Framework

H R Marzban and S Hajiabdolrahmani

Department of Mathematical Sciences Isfahan University of Technology Isfahan Iran

Correspondence should be addressed to H R Marzban hmarzbancciutacir

Received 5 July 2016 Accepted 22 November 2016

Academic Editor Jaume Gine

Copyright copy 2016 H R Marzban and S HajiabdolrahmaniThis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

An efficient numerical scheme for solving delay differential equations with a piecewise constant delay function is developedin this paper The proposed approach is based on a hybrid of block-pulse functions and Taylorrsquos polynomials The operationalmatrix of delay corresponding to the proposed hybrid functions is introduced The sparsity of this matrix significantly reduces thecomputation time andmemory requirementThe operational matrices of integration delay and product are employed to transformthe problem under consideration into a system of algebraic equations It is shown that the developed approach is also applicable toa special class of nonlinear piecewise constant delay differential equations Several numerical experiments are examined to verifythe validity and applicability of the presented technique

1 Introduction

Many problems arising in diverse areas of science andengineering can be described by delay differential equations(DDEs) Typical examples are transmission lines popula-tion dynamics chemical processes mathematical biologyphysics electrodynamics quantum mechanics robotics andcommunication networks [1ndash3] Many research works havebeen devoted to the numerical treatments of DDEs involvingconstant delay [4ndash16] To our knowledge a few papers havebeen paid to the numerical investigation of delay differentialequations with a piecewise constant delay function [17ndash21] Itshould be mentioned that except for some special cases it iseither difficult or impossible to obtain a closed-form solutionto a constant delay system Obviously the situation becomesmore complicated when time-delay is a piecewise constantfunction

It is generally supposed that the delay function is eitherconstant or continuous However in some actual situationstime-delay is a piecewise constant function Most of theexisting computational techniques are capable of providingsatisfactory approximation for problems whose solutions areinfinitely smooth or well-behaved To deal with a problem

whose solution is a nonsmooth function such as time-delaysystems they encounter some challenges The computationalprocedures developed in [17 18] are based on block-pulsefunctions Among piecewise constant basis functions block-pulse functions have a simple structure Simplicity in oper-ations is a distinctive feature of these functions [22] As aresult they can be implemented without too much effortAlthough these functions possess some nice properties suchas disjointedness orthogonality and completeness they donot provide accurate solutions for DDEs The numericalresults obtained by the methods implemented in [17 18]are not in good agreement with the exact solution of thementioned systems It should be pointed out that the ana-lytical solution of these systems cannot be produced solelyeither by continuous basis functions or by piecewise constantbasis functions The main difficulty arises because of thelack of smoothness of the solution associated with DDEsTo overcome these drawbacks it motivates one to design aflexible framework to accurately model the mixed featuresof continuity and jumps that occur in the solution of theproblem under consideration

Recently an effective computational technique for solvingDDEs with a piecewise constant delay function has been

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2016 Article ID 9754906 11 pageshttpdxdoiorg10115520169754906

2 International Journal of Differential Equations

introduced by Marzban and Shahsiah [19] The developedmethod is based on a hybrid of block-pulse functions andChebyshev polynomials More recently an efficient numer-ical scheme has been presented for solving linear piecewiseconstant delay systems [20] The proposed approach is basedon a hybrid of block-pulse functions and Legendre polynomi-alsThe purpose of this work is to present a simple frameworkto numerically solve piecewise constant delay systems Theproposed procedure is based on a hybrid of block-pulsefunctions and Taylorrsquos polynomials The structure of thiskind of hybrid functions is much simpler than that ofthose available in the literature particularly the hybrid ofblock-pulse functions and Chebyshev polynomialsThis typeof hybrid functions allows one to significantly reduce thecomputation time as well as thememory usageThe proposedprocedure is also applicable to nonlinear piecewise constantdelay systems but some modifications are required

The paper is organized as follows In Section 2 weexplain the fundamental properties of the hybrid of block-pulse functions and Taylorrsquos polynomials In Section 3 theoperational matrix of delay associated with the proposedhybrid functions is constructed Section 4 is dedicated tothe problem statement and its description Finally in Sec-tion 5 various types of piecewise constant delay systems areinvestigated to evaluate the performance and computationalefficiency of the suggested numerical scheme

2 Hybrid Functions

21 Hybrid of Block-Pulse Functions and Taylorrsquos PolynomialsHybrid functions120601119899119898(119905) 119899 = 1 2 119873 119898 = 0 1 119872minus1are defined on the interval [0 119905119891) as [14]120601119899119898 (119905)

= 119879119898 (119873119905 minus (119899 minus 1) 119905119891) 119905 isin [(119899 minus 1119873 ) 119905119891 119899119873119905119891) 0 otherwise (1)

where 119899 is the order of block-pulse functions Here 119879119898(119905) =119905119898 119898 = 0 1 119872 minus 1 are the well-known Taylorrsquospolynomials of degree119898 A detailed explanation of the block-pulse functions can be found in [22] It is worth noting thatthe mentioned hybrid functions constitute a semiorthogonalset on the interval [0 119905119891)22 Function Approximation A function 119891(119905) can be repre-sented in terms of the hybrid functions as

119891 (119905) = infinsum119899=1

infinsum119898=0

119888119899119898120601119899119898 (119905) (2)

where the coefficients 119888119899119898 119899 = 1 2 119873 119898 = 0 1 119872 minus1 are calculated by the following formula

119888119899119898 = 1119873119898119898 (119889119898119891 (119905)119889119905119898 )10038161003816100381610038161003816100381610038161003816119905=((119899minus1)119873)119905119891 (3)

If the series given by (2) is truncated then it can be written inthe form

119891 (119905) ≃ 119873sum119899=1

119872minus1sum119898=0

119888119899119898120601119899119898 (119905) = 119862119879Φ (119905) (4)

where

119862 = [11988810 1198881119872minus1 11988820 1198882119872minus1 1198881198730 119888119873119872minus1]119879 (5)Φ (119905) = [12060110 (119905) 1206011119872minus1 (119905) 12060120 (119905) 1206012119872minus1 (119905) 1206011198730 (119905) 120601119873119872minus1 (119905)]119879 (6)

In order to illustrate the usefulness and advantages of the pre-sented framework over the conventional Taylorrsquos expansionwe investigate two cases as follows

Case 1 Assume that

119891 (119905) = 119905 0 le 119905 lt 11 1 le 119905 lt 23 minus 119905 2 le 119905 le 3 (7)

Clearly the function 119891(119905) has two corner points at thepoints 119905 = 1 and 119905 = 2 Accordingly Taylorrsquos expansion of119891(119905) about each of the mentioned points does not exist Thisfunction can be exactly approximated by the hybrid functionsas follows

119891 (119905) = 3sum119899=1

1sum119898=0

119888119899119898120601119899119898 (119905) = 119862119879Φ (119905) (8)

in which

119862 = [0 13 1 0 1 minus13]119879 Φ (119905)= [12060110 (119905) 12060111 (119905) 12060120 (119905) 12060121 (119905) 12060130 (119905) 12060131 (119905)]119879 (9)

The elements of the vectorΦ(119905) are defined by

12060110 (119905) = 1 119905 isin [0 1) 0 otherwise

12060111 (119905) = 3119905 119905 isin [0 1) 0 otherwise

12060120 (119905) = 1 119905 isin [1 2) 0 otherwise

International Journal of Differential Equations 3

12060121 (119905) = 3119905 minus 3 119905 isin [1 2) 0 otherwise

12060130 (119905) = 1 119905 isin [2 3) 0 otherwise

12060131 (119905) = 3119905 minus 6 119905 isin [2 3) 0 otherwise

(10)

Case 2 Suppose that

119892 (119905) = exp (119905) 0 le 119905 lt 11199052 + 119905 1 le 119905 lt 2119905 + 4 2 le 119905 le 3 (11)

Obviously this function is discontinuous at the points 119905 = 1and 119905 = 2 However it can be efficiently approximated by theproposed hybrid functions To express this function in termsof hybrid functions we choose 119873 = 3 In addition Assumethat119872 is an arbitrary positive integer number greater than 2With the use of (2) we have

119892 (119905) = 3sum119899=1

119872minus1sum119898=0

119888119899119898120601119899119898 (119905) (12)

where the hybrid functions 120601119899119898(119905) 119899 = 1 2 3 119898 = 0 1 119872 minus 1 are defined by

120601119899119898 (119905) = 119879119898 (3119905 minus 3119899 + 3) = 3119898 (119905 minus 119899 + 1)119898 (13)

Also the associated coefficients 119888119899119898 are given by

1198881119898 = 13119898119898 119898 = 0 1 119872 minus 111988820 = 211988821 = 111988822 = 19 1198882119898 = 0 119898 = 3 4 119872 minus 111988830 = 611988831 = 13 1198883119898 = 0 119898 = 2 3 119872 minus 1

(14)

The hybrid expansion of 119892(119905) over the interval [1 2] ispresented by1198882012060120 (119905) + 1198882112060121 (119905) + 1198882212060122 (119905)= 2 + (3119905 minus 3) + 19 (3119905 minus 3)2 = 1199052 + 119905 (15)

Similarly the hybrid approximation of 119892(119905) on the interval[2 3) is described by1198883012060130 (119905) + 1198883112060131 (119905) = 6 + 13 (3119905 minus 6) = 119905 + 4 (16)

Moreover the hybrid expansion of119892(119905) over the interval [0 1)is expressed by

119892 (119905) = 119872minus1sum119898=0

119905119898119898 (17)

which is precisely Taylorrsquos expansion of the function exp(119905)about 119905 = 0 up to degree119872minus 1 Define119864 = 10038171003817100381710038171003817119892 (119905) minus 119862119879Φ (119905)10038171003817100381710038171003817infin= max 10038161003816100381610038161003816119892 (119905) minus 119862119879Φ (119905)10038161003816100381610038161003816 0 le 119905 le 1 (18)

It is easily verified that 119864 = 119890119872 (19)

The obtained result indicates that the maximum error con-verges to zero as the value of 119872 increases Some importantaspects of the proposed hybrid functions are clarified in [23]

23 Operational Matrices of Integration and Product Theintegration of the vector Φ(119905) presented by (6) can also beexpanded in terms of hybrid functions as

int1199050Φ (120591) 119889120591 ≃ 119875Φ (119905) (20)

in which119875 is the119872119873times119872119873 operationalmatrix of integrationcorresponding to the hybrid functions The structure of thismatrix is given by [14]

119875 = ((((

119864lowast 119867 119867 sdot sdot sdot 1198670 119864lowast 119867 sdot sdot sdot 1198670 0 119864lowast sdot sdot sdot 119867 d0 0 0 sdot sdot sdot 119864lowast))))

(21)

where

119867 = 1119873 (((((

119905119891 0 sdot sdot sdot 011990521198912 0 sdot sdot sdot 0 d119905119872119891119872 0 sdot sdot sdot 0)))))

(22)

4 International Journal of Differential Equations

and 119864lowast is defined by

119864lowast = 1119873((((((((

0 1 0 sdot sdot sdot 00 0 12 sdot sdot sdot 0 d0 0 0 sdot sdot sdot 1119872 minus 10 0 0 sdot sdot sdot 0

))))))))

(23)

It should be noted that 119864lowast is the operational matrix ofintegration associated with the Taylorrsquos polynomials on theinterval [((119899minus1)119873)119905119891 (119899119873)119905119891) Furthermore if119862 is a vectorof order 119872119873 times 1 then the expression Φ(119905)Φ119879(119905)119862 can berepresented in terms of hybrid functions asΦ (119905)Φ119879 (119905) 119862 ≃ Φ (119905) (24)

The 119872119873 times 119872119873 matrix is called the operational matrix ofproduct [14]

3 Derivation of the OperationalMatrix of Delay

In this section we shall derive the operational matrix ofdelay associated with the hybrid of block-pulse functionsand Taylorrsquos polynomials For this purpose we employ aprocedure analogous to the one developed in [19] for thehybrid of block-pulse functions and Chebyshev polynomialsThe delayed vectorΦ(119905 minus 119886(119905)) can be represented in terms ofΦ(119905) as follows Φ (119905 minus 119886 (119905)) = 119863Φ (119905) (25)

in which 119863 is called the operational matrix of delay Inaddition 119886(119905) is a piecewise constant function defined by

119886 (119905) =

1205911 1198790 le 119905 lt 11987911205912 1198791 le 119905 lt 1198792 120591119903 119879119903minus1 le 119905 lt 119879119903(26)

where 0 = 1198790 lt 1198791 lt sdot sdot sdot lt 119879119903minus1 lt 119879119903 = 119905119891 It isfurther supposed that 120591119894 119894 = 1 2 119903 are arbitrary rationalnumbers in the interval [0 119905119891) and 119879119894 isin Q 119894 = 0 1 119903 Toconstruct119863 we first obtain the value of119873 Let119860 = 119894 120591119894 = 0 (27)

and take 120574 as the smallest positive integer number such thatthe following conditions are satisfied120574120591119894 isin Z120574119879119895 isin Z119894 isin 119860 119895 = 1 2 119903 (28)

Assume that |119860| = 119896 Next we define 120582 as the greatestcommon divisor of the integers 120574120591119894 and 120574119879119895 119894 isin 119860 and119895 = 1 2 119903 that is120582 = gcd (1205741205911 1205741205912 120574120591119896 1205741198791 1205741198792 120574119879119903) (29)

Define

119873 = 120574120582119905119891 if

120574120582119905119891 isin Z[ 120574120582119905119891] + 1 otherwise (30)

where [(120574120582)119905119891] denotes the greatest integer value less thanor equal to (120574120582)119905119891 It is worth noting that 119873 is selected insuch a way that the number of subintervals is minimized Asa consequence we obtain the following subintervals

[0 120582120574) [120582120574 2120582120574) [(119873 minus 1) 120582120574 119873120582120574) (31)

For convenience we use the notations defined by

ℎ = 120582120574 119873119895 = 120574120582 (119879119895 minus 119879119895minus1) 119897119895 = 120574120582120591119895 119895 = 1 2 119903

(32)

Clearly sum119903119895=1119873119895 = 119873 Consequently we have119886 (119905) = 119896119894ℎ (119894 minus 1) ℎ le 119905 lt 119894ℎ 119894 = 1 2 119873 (33)

in which

119896119894 =

1198971 1 le 119894 le 11987311198972 1198731 + 1 le 119894 le 1198731 + 1198732 119897119895 119895minus1sum119896=1

119873119896 + 1 le 119894 le 119895sum119896=1

119873119896 119897119903 119903minus1sum119896=1

119873119896 + 1 le 119894 le 119903sum119896=1

119873119896

(34)

As a result the problem is reduced to find the delay opera-tional matrix for the delay function defined by

Φ (119905 minus 119886 (119905)) =

Φ(119905 minus 1198961ℎ) 0 le 119905 lt 1199051Φ (119905 minus 1198962ℎ) 1199051 le 119905 lt 1199052 Φ (119905 minus 119896119873ℎ) 119905119899minus1 le 119905 lt 119905119873(35)

International Journal of Differential Equations 5

where 119905119894 = 119894ℎ 119894 = 1 2 119873 (36)

To construct the matrix 119863 we first derive the matrix 119863119894 for119894 = 1 2 119873 in such a way that the following relation issatisfied Φ(119905 minus 119896119894ℎ) = 119863119894Φ (119905) 119905119894minus1 le 119905 lt 119905119894 (37)

By the properties of the hybrid functions it is easy to verifythat for the case 119905119894minus1 le 119905 lt 119905119894 the only functions with nonzerovalues are 120601(119905minus119896119894)119898(119905minus119896119894ℎ) for119898 = 0 1 119872minus1 Notice that

120601(119905minus119896119894)119898 (119905 minus 119896119894ℎ) = 120601119894119898 (119905) 119898 = 0 1 119872 minus 1 (38)

Accordingly 120601(119905minus119896119894)119898(119905minus119896119894ℎ) can be written in terms of 120601119894119898(119905)as 119863119894 = 119878119894 otimes 119868119872 119894 = 1 2 119873 (39)

where 119868119872 is the119872-dimensional identitymatrix andotimes denotesthe Kronecker product [24] In addition 119878119894 is an119873times119873matrixin which the only nonzero entry is equal to one and locatedat the (119894 minus 119896119894)th row and 119894th column Finally if we expressΦ(119905 minus 119886(119905)) in terms of Φ(119905) we find119863 = 1198631 + 1198632 + sdot sdot sdot + 119863119873 (40)

Remark 1 If 119894 minus 119896119894 le 0 then 119878119894 is a zero matrix of order119873times119873 It is worth noting that the structure of the operationalmatrix of delay associatedwith the proposed hybrid functionsis exactly the same as the structure of the hybrid of block-pulse functions and Chebyshev polynomials

4 Problem Statement and Its Approximation

Consider the time-varying piecewise constant delay systemdescribed by

(119905) = 119864 (119905)119883 (119905) + 119865 (119905)119883 (119905 minus 119886 (119905)) + 119866 (119905) 119880 (119905) 0 le 119905 le 119905119891 (41)

119883 (0) = 1198830 (42)119883 (119905) = 0 119905 lt 0 (43)

119886 (119905) =

1205911 1198790 le 119905 lt 11987911205912 1198791 le 119905 lt 1198792 120591119903 119879119903minus1 le 119905 lt 119879119903(44)

where119883(119905) isin R119897 119880(119905) isin R119902 119864(119905) 119865(119905) and119866(119905) arematricesof appropriate dimensions 1198830 is a constant specified vectorand 120591119894 119894 = 1 2 119903 are arbitrary rational numbers in [0 119905119891)The objective is to find119883(119905) 0 le 119905 le 119905119891 satisfying (41)ndash(43)

41 Description of the Method Let119883(119905) = [1199091 (119905) 1199092 (119905) 119909119897 (119905)]119879 119880 (119905) = [1199061 (119905) 1199062 (119905) 119906119902 (119905)]119879 (45)

Assume that each 119909119894(119905) and each of 119906119895(119905) 119894 = 1 2 119902 canbe written in terms of hybrid functions as119909119894 (119905) = Φ119879 (119905) 119883119894119906119895 (119905) = Φ119879 (119905) 119880119895 (46)

Consequently 119883(119905) = (119868119897 otimes Φ119879 (119905))119883119880 (119905) = (119868119902 otimes Φ119879 (119905))119880 (47)

where 119868119897 and 119868119902 are identity matrices of order 119897 times 119897 and 119902 times119902 respectively Furthermore 119883 and 119880 are vectors of order119897119873119872 times 1 and 119902119873119872 times 1 respectively given by

119883 = [1198831198791 1198831198792 119883119879119897 ]119879 119880 = [1198801198791 1198801198792 119880119879119902 ]119879 (48)

Also 119883 (0) = (119868119897 otimes Φ119879 (119905)) 119889 (49)

where 119889 is a vector of order 119897119873119872 times 1 defined by

119889 = [1198891198791 1198891198792 119889119879119897 ]119879 (50)

Similarly 119864(119905) 119865(119905) and 119866(119905) can be represented in terms ofhybrid functions as119864 (119905) = 119864119879 (119868119897 otimes Φ (119905)) 119865 (119905) = 119865119879 (119868119897 otimes Φ (119905)) 119866 (119905) = 119866119879 (119868119902 otimes Φ (119905)) (51)

where 119864119879 119865119879 and 119866119879 are of dimensions 119897 times 119897119873119872 119897 times 119897119873119872and 119897 times 119902119873119872 respectivelyThe delayed vector119883(119905 minus 119886(119905)) canalso be expressed in terms of hybrid functions as119883 (119905 minus 119886 (119905)) = (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119863119879)119883 (52)

in which119863 is the delay operational matrix presented by (25)As a result119864 (119905)119883 (119905) = 119864119879 (119868119897 otimes Φ (119905)) (119868119897 otimes Φ119879 (119905))119883= (119868l otimes Φ119879 (119905)) 119879119883119865 (119905)119883 (119905 minus 119886 (119905))= 119865119879 (119868119897 otimes Φ (119905)) (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119863119879)119883= (119868119897 otimes Φ119879 (119905)) 119879 (119868119897 otimes 119863119879)119883

6 International Journal of Differential Equations

119866 (119905)119880 (119905) = 119866119879 (119868119902 otimes Φ (119905)) (119868119902 otimes Φ119879 (119905))119880= (119868119897 otimes Φ119879 (119905)) 119879119880

(53)

where and can be determined in a manner analogousto the derivation process of the matrix described by (24)Furthermore

int1199050(119868119897 otimes Φ119879 (119904)) 119889119904 = (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879)

int1199050119865 (119904)119883 (119904 minus 119886 (119904)) 119889119904= (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)119883

(54)

in which 119875 is the operational matrix of integration describedby (20) By integrating both sides of (41) with respect to 119905 andusing (47) (49) (51) (52) and (54) we obtain

(119868119897 otimes Φ119879 (119905))119883 minus (119868119897 otimes Φ119879 (119905)) 119889= (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879119883+ (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)119883+ (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879119880(55)

From the preceding equation we deduce that

119883 = [119868119872119873 minus (119868119897 otimes 119875119879) 119879minus (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)]minus1 [119889 + (119868119897 otimes 119875119879) 119879119880] (56)

5 Simulation Results

In this section five examples of varying complexity areinvestigated to evaluate the usefulness and effectiveness ofthe proposed computational techniqueThe correct choice of119873 would greatly improve the efficiency and accuracy of thepresented procedure

51 Example 1 Consider the following piecewise constantdelay system [17] (119905) = 119909 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (57)119909 (119905) = 0 119905 le 0 (58)119906 (119905) = 1 119905 ge 0 (59)

119886 (119905) = 01 0 le 119905 lt 03503 035 le 119905 lt 0705 07 le 119905 le 1 (60)

The exact solution to this problem is given by

119909 (119905) =

119905 0 le 119905 lt 011200 + 910119905 + 121199052 01 le 119905 lt 02113000 + 2325119905 + 251199052 + 161199053 02 le 119905 lt 03961240000 + 18312000119905 + 1694001199052 + 7601199053 + 1241199054 03 le 119905 lt 0352901613840000 + 710119905 + 121199052 035 le 119905 lt 04830671280000 + 3950119905 + 3101199052 + 161199053 04 le 119905 lt 052592013840000 + 9111200119905 + 29801199052 + 1121199053 + 1241199054 05 le 119905 lt 06641781796000000 + 2293730000119905 + 68920001199052 + 171501199053 + 1601199054 + 11201199055 06 le 119905 lt 065107046748000000 + 34965613840000119905 + 151199052 + 161199053 065 le 119905 lt 0737190303192000000 + 37376000119905 + 1294001199052 + 1201199053 + 1241199054 07 le 119905 lt 08244440112800000 + 639910000119905 + 167960001199052 + 313001199053 + 11201199054 + 11201199055 08 le 119905 lt 085710323764000000 + 32661613840000119905 + 1101199052 + 161199053 085 le 119905 lt 09885283764000000 + 27996013840000119905 + 1214001199052 + 1601199053 + 1241199054 09 le 119905 le 1

(61)

International Journal of Differential Equations 7

To apply the method developed in the current paper wefirst choose the value of119873 with the use of (30) Because 120574 =20 and 120582 = 1 hence we select 119873 = 20 We also take 119872 = 6Let

119909 (119905) = 119862119879Φ (119905) (62)

where

119862 = [11988810 1198881119872minus1 1198881198730 119888119873119872minus1]119879 Φ (119905) = [12060110 (119905) 1206011119872minus1 (119905) 1206011198730 (119905) 120601119873119872minus1 (119905)]119879 (63)

By expanding 119906(119905) in terms of hybrid functions we get

119906 (119905) = 1198891198791Φ (119905) (64)

We also have

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (65)

where 119863 is the delay operational matrix given by (25)Integrating both sides of (57) with respect to 119905 and using (62)ndash(65) yield

119862119879 = 1198891198791119875 (119868119872119873 minus 119863119875)minus1 (66)

in which 119868119872119873 is the 119872119873-dimensional identity matrix and 119875is the operational matrix of integration given by (20) Withthe use of (66) and (62) we would obtain the same value asthe exact value of 119909(119905)52 Example 2 Consider the following two-dimensionaltime-varying piecewise constant delay system

(1 (119905)2 (119905)) = ( 1 1 + 1199052119905 1199052 )(1199091 (119905 minus 119886 (119905))1199092 (119905 minus 119886 (119905)))+ (12)119906 (119905) 0 le 119905 le 1 (67)

(1199091 (119905)1199092 (119905)) = (00) 119905 lt 0 (68)

(1199091 (0)1199092 (0)) = (11) (69)

119906 (119905) = 119905 0 le 119905 lt 051 minus 119905 05 le 119905 lt 1 (70)

119886 (119905) = 02 0 le 119905 lt 0307 03 le 119905 le 1 (71)

The exact solutions to this problem are given as follows

1199091 (119905) =

1 + 121199052 0 le 119905 lt 0243197500 + 10350 119905 + 18251199052 + 11301199053 + 141199054 02 le 119905 lt 03147071120000 + 121199052 03 le 119905 lt 05117071120000 + 119905 minus 121199052 05 le 119905 lt 07minus1033112000 + 747200119905 minus 1612001199052 + 1301199053 + 141199054 07 le 119905 lt 09minus 4245584594200000000 + 1148571500000 119905 minus 35588320000001199052 minus 4681300001199053 + 176740001199054 + 717501199055 minus 71201199056 + 1351199057 09 le 119905 le 1

1199092 (119905) =

1 + 1199052 0 le 119905 lt 02179473187500 + 10150 1199052 + 16751199053 + 3201199054 + 151199055 02 le 119905 lt 0331693513000000 + 1199052 03 le 119905 lt 0516693513000000 + 2119905 minus 1199052 05 le 119905 lt 07minus 2204813000000 + 2119905 + 492001199052 + 31001199053 minus 1101199054 + 151199055 07 le 119905 lt 093646033938184000000000 + 2119905 minus 63163400001199052 + 437221730000001199053 minus 11101400001199054 + 1981150001199055 + 1919001199056 minus 111401199057 + 1401199058 09 le 119905 le 1

(72)

8 International Journal of Differential Equations

To solve this problem by the proposed approach we firstdetermine the value of119873with the use of (30) Because 120574 = 10and 120582 = 1 consequently we select 119873 = 10 Also we choose119872 = 9 Let

1199091 (119905) = 1198621198791Φ (119905) 1199092 (119905) = 1198621198792Φ (119905) (73)

Expanding 1 1 + 119905 2119905 and 1199052 in terms of hybrid functionsimplies

1 = 1198911198791 Φ (119905) 1 + 119905 = 1198911198792 Φ (119905)

2119905 = 1198911198793 Φ (119905) 1199052 = 1198911198794 Φ (119905)

(74)

Also using (70) we obtain

int1199050119906 (120591) 119889120591=

121199052 = 1198911198795 Φ (119905) 0 le 119905 lt 05minus14 + 119905 minus 121199052 = 1198911198796 Φ (119905) 05 le 119905 lt 1

(75)

Moreover

1199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905) (76)

Therefore(1 + 119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198912= 11986211987921198632Φ (119905) 21199051199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905)Φ119879 (119905) 1198913= 11986211987911198633Φ (119905) 11990521199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198914= 11986211987921198634Φ (119905) (77)

in which 2 3 and 4 can be obtained in a way similar tothe construction method of the matrix defined by (24) Byintegrating both sides of (67) with respect to 119905 and using (68)(70) (71) (73) (74) (75) (76) and (77) we get

1198621198791 (119868119872119873 minus 119863119875) minus 11986211987921198632119875 = 1198911198791 + 1198911198795 + 1198911198796 1198621198792 (119868119872119873 minus 1198634119875) minus 11986211987911198633119875 = 1198911198791 + 21198911198795 + 21198911198796 (78)

By solving the resulting system described by (78) and thenusing (73) we would obtain the same values as the exactvalues of 1199091(119905) and 1199092(119905)53 Example 3 Consider the following time-delay system[3] (119905) = minus5119909 (119905) minus 5119909 (119905 minus 119886 (119905)) + 2119906 (119905) 0 le 119905 le 2 (79)119909 (0) = 1 (80)119906 (119905) = 1 119905 ge 0 (81)

119886 (119905) = 0 0 le 119905 lt 0803 08 le 119905 lt 1406 14 le 119905 le 1709 17 le 119905 le 2

(82)

The exact solution to this problem is given by [3]

119909 (119905) =

02 + 08119890minus10119905 0 le 119905 lt 0802 + 081198903minus10119905 + 08119890minus4 (1 minus 1198903) 119890minus5119905 08 le 119905 lt 1102 + 081198905minus10119905 minus 4119890minus25 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (65119890minus25 + 119890minus4) 119890minus5119905 11 le 119905 lt 1402 + 081198909minus10119905 minus 4119890minus1 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (8119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 14 le 119905 lt 1702 + 0811989012minus10119905 minus 411989005 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (9511989005 minus 05119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 17 le 119905 le 2(83)

To employ the proposed method we first determine thesuitable value of119873 Using (30) it follows that119873 = 120574120582119905119891 = 20 (84)

Assume that119872 is an arbitrary positive integer number Let

119909 (119905) = 119862119879Φ (119905) (85)

International Journal of Differential Equations 9

Similarly

119909 (0) = 119906 (119905) = 119889119879Φ (119905) 119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (86)

By integrating both sides of (79) with respect to 119905 and using(85)-(86) we conclude that119862119879 = (119889119879 + 2119889119879119875) (119868119872119873 + 5119875 + 5119863119875)minus1 (87)

where 119868119872119873 is the119872119873-dimensional identity matrix Define119864 = max 1003816100381610038161003816119909119890 (119905) minus 119909 (119905)1003816100381610038161003816 0 le 119905 le 1 (88)

in which 119909(119905) and 119909119890(119905) denote the approximate solutiondetermined by the present approach and the exact solutionrespectively In Table 1 the numerical results of themaximumerror denoted by 119864 for 119873 = 20 and various values of 119872are reported The simulation results obtained by the currentapproach show a significant improvement in the accuracy ofthe solution compared to the method developed in [19]

54 Example 4 As a more complicated problem considerthe following problem

(119905) = 1199051199092 (119905 minus 119886 (119905)) + 1199093 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (89)

119909 (119905) = 0 119905 lt 0 (90)119909 (0) = 1 (91)

119906 (119905) = 119905 0 le 119905 lt 0404 04 le 119905 le 1 (92)

119886 (119905) = 04 0 le 119905 lt 0808 08 le 119905 le 1 (93)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 046632511640625 + 2593315625119905 minus 72962501199052 + 1172501199053 minus 31001199054 + 131001199055 minus 11201199056 + 1561199057 04 le 119905 lt 082356911640625 + 4218715625119905 minus 762362501199052 + 2532501199053 minus 431001199054 + 231001199055 minus 71201199056 + 1561199057 08 le 119905 le 1

(94)

Although the above-mentioned problem is a nonlinear delaydifferential equation the method developed in the presentpaper can easily be implemented Because 120574 = 5 and 120582 = 2we take119873 = 3 Furthermore we take119872 = 8 Let

119909 (119905) = 119862119879Φ (119905) (95)

Similarly

119909 (0) = V1198791Φ (119905) 119905 = V1198792Φ (119905) (96)

int1199050119906 (120591) 119889120591 =

121199052 = V1198793Φ (119905) 0 le 119905 lt 04minus 225 + 25119905 = V1198794Φ (119905) 04 le 119905 lt 1 (97)

where V1 V2 V3 and V4 are vectors of order119872119873times1 Also wehave

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (98)1199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) 119863119879119862= 119862119879119863Φ (119905) (99)

1199093 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905)119882= 1198621198791198632Φ (119905) (100)

in which

119882 = 119863119879119862 (101)

Using (99) gives

1199051199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) V2= 119862119879119863V2Φ (119905) (102)

Integrating both sides of (89) with respect to 119905 and using (95)(97) (98) (99) (100) and (102) imply that

119862119879 (119868119872119873 minus 119863V2119875 minus 1198632119875) minus (V1198791 + V1198793 + V1198794 )= 0 (103)

By solving (103) and using (95) the exact value of 119909(119905)will beobtained

10 International Journal of Differential Equations

Table 1 Computational results of maximum error for Example 3Present method CPU time (s) 119864119873 = 20119872 = 1 0112 1165119890 minus 02119873 = 20119872 = 2 0202 3231119890 minus 03119873 = 20119872 = 4 0398 2573119890 minus 06119873 = 20119872 = 6 0769 4836119890 minus 08119873 = 20119872 = 8 0853 3942119890 minus 11119873 = 20119872 = 10 1282 2376119890 minus 13119873 = 20119872 = 12 1841 5875119890 minus 16119873 = 20119872 = 14 2167 3624119890 minus 17119873 = 20119872 = 16 2536 5117119890 minus 2055 Example 5 As the final example we consider the time-varying multidelay system defined by (119905) = 119905119909 (119905 minus 1198861 (119905)) + 11990521199092 (119905 minus 1198862 (119905)) + 119906 (119905) 0 le 119905 le 1

119909 (0) = 1119906 (119905) =

119905 0 le 119905 lt 12 12 12 le 119905 le 1(104)

where

1198861 (119905) = 12 0 le 119905 lt 34 14 34 le 119905 le 1

1198862 (119905) = 14 0 le 119905 lt 12 34 12 le 119905 le 1

(105)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 14 2037320480 + 121199052 + 11321199053 minus 1161199054 + 1101199055 14 le 119905 lt 12 4819161440 + 12119905 + 9161199052 minus 161199053 + 181199054 12 le 119905 lt 34 12897431966080 + 12119905 + 14287409601199052 + 1873841199053 minus 12561199054 + 1241199055 + 1481199056 34 le 119905 le 1(106)

Although the above-mentioned system involves multiplepiecewise constant delays the proposed procedure can beapplied To employ our approach we select 119873 = 4 Bychoosing119872 = 7 the exact solution of 119909(119905) will be obtained6 Conclusion

An efficient and flexible framework has been successfullydeveloped for solving piecewise constant delay systems Thefoundation of the proposed method is based on a hybrid ofblock-pulse functions and Taylorrsquos polynomials The opera-tional matrix of delay has been constructed The operationalmatrices of integration delay and product associated withthe hybrid functions were utilized to transform the problemunder consideration into a system of algebraic equations Itis worth emphasizing that the exact solutions of Examples1 2 4 and 5 cannot be produced solely either by block-pulse functions or by Taylorrsquos polynomials After determiningthe suitable value of 119873 the number of subintervals smallvalues for 119872 the degree of the Taylorrsquos polynomials areneeded to achieve a specific accuracy The simulation resultsdemonstrate the effectiveness of the suggested procedure Inaddition the proposed numerical scheme can be extended

to a class of nonlinear piecewise constant delay systems butsome modifications are required

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this manuscript

References

[1] RDDriverOrdinary and delay differential equations Springer-Verlag New York NY USA 1977

[2] M Malek-Zavarei and M Jamshidi Time-delay systems vol 9of North-Holland Systems and Control Series North-HollandPublishing Co Amsterdam Netherlands 1987

[3] K B Datta and B M Mohan Orthogonal functions in systemsand control vol 9 ofAdvanced Series in Electrical and ComputerEngineering World Scientific Publishing Co 1995

[4] A Bellen ldquoOne-step collocation for delay differential equa-tionsrdquo Journal of Computational and Applied Mathematics vol10 no 3 pp 275ndash283 1984

[5] C Hwang and M Y Chen ldquoAnalysis and parameter identifi-cation of time-delay systems via shifted Legendre polynomialsrdquoInternational Journal of Control vol 41 no 2 pp 403ndash415 1985

International Journal of Differential Equations 11

[6] I R Horng and J H Chou ldquoAnalysis parameter estimation andoptimal control of time-delay systems via Chebyshev seriesrdquoInternational Journal of Control vol 41 no 5 pp 1221ndash12341985

[7] A Bellen and M Zennaro ldquoNumerical solution of delaydifferential equations by uniform corrections to an implicitRunge-Kutta methodrdquo Numerische Mathematik vol 47 no 2pp 301ndash316 1985

[8] I-R Horng and J-H Chou ldquoAnalysis and parameter identi-fication of time-delay systems via shifted Jacobi polynomialsrdquoInternational Journal of Control vol 44 no 4 pp 935ndash942 1986

[9] M Razzaghi and M Razzaghi ldquoTaylor series analysis of time-varying multi-delay systemsrdquo International Journal of Controlvol 50 no 1 pp 183ndash192 1989

[10] B M Mohan and K B Datta ldquoAnalysis of linear time-invarianttime-delay systems via orthogonal functionsrdquo InternationalJournal of Systems Science vol 26 no 1 pp 91ndash111 1995

[11] A Bellen and S Maset ldquoNumerical solution of constantcoefficient linear delay differential equations as abstract Cauchyproblemsrdquo Numerische Mathematik vol 84 no 3 pp 351ndash3742000

[12] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering vol 7 no 4 pp 337ndash353 2001

[13] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004

[14] H R Marzban and M Razzaghi ldquoAnalysis of time-delaysystems via hybrid of block-pulse functions and Taylor seriesrdquoJournal of Vibration and Control vol 11 no 12 pp 1455ndash14682005

[15] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006

[16] H R Marzban ldquoOptimal control of linear multi-delay systemsbased on a multi-interval decomposition schemerdquo OptimalControl Applications and Methods vol 37 no 1 pp 190ndash2112016

[17] W L Chen and B S Jeng ldquoAnalysis of piecewise constantdelay systems via block-pulse functionsrdquo International Journalof Systems Science vol 12 no 5 pp 625ndash633 1981

[18] W-L Chen and C-H Meng ldquoA general procedure of solvingthe linear delay system via block pulse functionsrdquo Computersand Electrical Engineering vol 9 no 3-4 pp 153ndash166 1982

[19] HRMarzban andM Shahsiah ldquoSolution of piecewise constantdelay systems using hybrid of block-pulse and Chebyshevpolynomialsrdquo Optimal Control Applications and Methods vol32 no 6 pp 647ndash659 2011

[20] H R Marzban and S M Hoseini ldquoAnalysis of linear piecewiseconstant delay systems using a hybrid numerical schemerdquoAdvances in Numerical Analysis vol 2016 Article ID 8903184 9pages 2016

[21] H RMarzban and SMHoseini ldquoNumerical treatment of non-linear optimal control problems involving piecewise constantdelayrdquo IMA Journal of Mathematical Control and Information2015

[22] Z H Jiang and W Schaufelberger Block pulse functions andtheir applications in control systems vol 179 of Lecture Notesin Control and Information Sciences Springer Berlin Germany1992

[23] H R Marzban ldquoParameter identification of linear multi-delaysystems via a hybrid of block-pulse functions and Taylorrsquospolynomialsrdquo International Journal of Control 2016

[24] P LancasterTheory of matrices Academic Press New York NYUSA 1969

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Differential Equations

introduced by Marzban and Shahsiah [19] The developedmethod is based on a hybrid of block-pulse functions andChebyshev polynomials More recently an efficient numer-ical scheme has been presented for solving linear piecewiseconstant delay systems [20] The proposed approach is basedon a hybrid of block-pulse functions and Legendre polynomi-alsThe purpose of this work is to present a simple frameworkto numerically solve piecewise constant delay systems Theproposed procedure is based on a hybrid of block-pulsefunctions and Taylorrsquos polynomials The structure of thiskind of hybrid functions is much simpler than that ofthose available in the literature particularly the hybrid ofblock-pulse functions and Chebyshev polynomialsThis typeof hybrid functions allows one to significantly reduce thecomputation time as well as thememory usageThe proposedprocedure is also applicable to nonlinear piecewise constantdelay systems but some modifications are required

The paper is organized as follows In Section 2 weexplain the fundamental properties of the hybrid of block-pulse functions and Taylorrsquos polynomials In Section 3 theoperational matrix of delay associated with the proposedhybrid functions is constructed Section 4 is dedicated tothe problem statement and its description Finally in Sec-tion 5 various types of piecewise constant delay systems areinvestigated to evaluate the performance and computationalefficiency of the suggested numerical scheme

2 Hybrid Functions

21 Hybrid of Block-Pulse Functions and Taylorrsquos PolynomialsHybrid functions120601119899119898(119905) 119899 = 1 2 119873 119898 = 0 1 119872minus1are defined on the interval [0 119905119891) as [14]120601119899119898 (119905)

= 119879119898 (119873119905 minus (119899 minus 1) 119905119891) 119905 isin [(119899 minus 1119873 ) 119905119891 119899119873119905119891) 0 otherwise (1)

where 119899 is the order of block-pulse functions Here 119879119898(119905) =119905119898 119898 = 0 1 119872 minus 1 are the well-known Taylorrsquospolynomials of degree119898 A detailed explanation of the block-pulse functions can be found in [22] It is worth noting thatthe mentioned hybrid functions constitute a semiorthogonalset on the interval [0 119905119891)22 Function Approximation A function 119891(119905) can be repre-sented in terms of the hybrid functions as

119891 (119905) = infinsum119899=1

infinsum119898=0

119888119899119898120601119899119898 (119905) (2)

where the coefficients 119888119899119898 119899 = 1 2 119873 119898 = 0 1 119872 minus1 are calculated by the following formula

119888119899119898 = 1119873119898119898 (119889119898119891 (119905)119889119905119898 )10038161003816100381610038161003816100381610038161003816119905=((119899minus1)119873)119905119891 (3)

If the series given by (2) is truncated then it can be written inthe form

119891 (119905) ≃ 119873sum119899=1

119872minus1sum119898=0

119888119899119898120601119899119898 (119905) = 119862119879Φ (119905) (4)

where

119862 = [11988810 1198881119872minus1 11988820 1198882119872minus1 1198881198730 119888119873119872minus1]119879 (5)Φ (119905) = [12060110 (119905) 1206011119872minus1 (119905) 12060120 (119905) 1206012119872minus1 (119905) 1206011198730 (119905) 120601119873119872minus1 (119905)]119879 (6)

In order to illustrate the usefulness and advantages of the pre-sented framework over the conventional Taylorrsquos expansionwe investigate two cases as follows

Case 1 Assume that

119891 (119905) = 119905 0 le 119905 lt 11 1 le 119905 lt 23 minus 119905 2 le 119905 le 3 (7)

Clearly the function 119891(119905) has two corner points at thepoints 119905 = 1 and 119905 = 2 Accordingly Taylorrsquos expansion of119891(119905) about each of the mentioned points does not exist Thisfunction can be exactly approximated by the hybrid functionsas follows

119891 (119905) = 3sum119899=1

1sum119898=0

119888119899119898120601119899119898 (119905) = 119862119879Φ (119905) (8)

in which

119862 = [0 13 1 0 1 minus13]119879 Φ (119905)= [12060110 (119905) 12060111 (119905) 12060120 (119905) 12060121 (119905) 12060130 (119905) 12060131 (119905)]119879 (9)

The elements of the vectorΦ(119905) are defined by

12060110 (119905) = 1 119905 isin [0 1) 0 otherwise

12060111 (119905) = 3119905 119905 isin [0 1) 0 otherwise

12060120 (119905) = 1 119905 isin [1 2) 0 otherwise

International Journal of Differential Equations 3

12060121 (119905) = 3119905 minus 3 119905 isin [1 2) 0 otherwise

12060130 (119905) = 1 119905 isin [2 3) 0 otherwise

12060131 (119905) = 3119905 minus 6 119905 isin [2 3) 0 otherwise

(10)

Case 2 Suppose that

119892 (119905) = exp (119905) 0 le 119905 lt 11199052 + 119905 1 le 119905 lt 2119905 + 4 2 le 119905 le 3 (11)

Obviously this function is discontinuous at the points 119905 = 1and 119905 = 2 However it can be efficiently approximated by theproposed hybrid functions To express this function in termsof hybrid functions we choose 119873 = 3 In addition Assumethat119872 is an arbitrary positive integer number greater than 2With the use of (2) we have

119892 (119905) = 3sum119899=1

119872minus1sum119898=0

119888119899119898120601119899119898 (119905) (12)

where the hybrid functions 120601119899119898(119905) 119899 = 1 2 3 119898 = 0 1 119872 minus 1 are defined by

120601119899119898 (119905) = 119879119898 (3119905 minus 3119899 + 3) = 3119898 (119905 minus 119899 + 1)119898 (13)

Also the associated coefficients 119888119899119898 are given by

1198881119898 = 13119898119898 119898 = 0 1 119872 minus 111988820 = 211988821 = 111988822 = 19 1198882119898 = 0 119898 = 3 4 119872 minus 111988830 = 611988831 = 13 1198883119898 = 0 119898 = 2 3 119872 minus 1

(14)

The hybrid expansion of 119892(119905) over the interval [1 2] ispresented by1198882012060120 (119905) + 1198882112060121 (119905) + 1198882212060122 (119905)= 2 + (3119905 minus 3) + 19 (3119905 minus 3)2 = 1199052 + 119905 (15)

Similarly the hybrid approximation of 119892(119905) on the interval[2 3) is described by1198883012060130 (119905) + 1198883112060131 (119905) = 6 + 13 (3119905 minus 6) = 119905 + 4 (16)

Moreover the hybrid expansion of119892(119905) over the interval [0 1)is expressed by

119892 (119905) = 119872minus1sum119898=0

119905119898119898 (17)

which is precisely Taylorrsquos expansion of the function exp(119905)about 119905 = 0 up to degree119872minus 1 Define119864 = 10038171003817100381710038171003817119892 (119905) minus 119862119879Φ (119905)10038171003817100381710038171003817infin= max 10038161003816100381610038161003816119892 (119905) minus 119862119879Φ (119905)10038161003816100381610038161003816 0 le 119905 le 1 (18)

It is easily verified that 119864 = 119890119872 (19)

The obtained result indicates that the maximum error con-verges to zero as the value of 119872 increases Some importantaspects of the proposed hybrid functions are clarified in [23]

23 Operational Matrices of Integration and Product Theintegration of the vector Φ(119905) presented by (6) can also beexpanded in terms of hybrid functions as

int1199050Φ (120591) 119889120591 ≃ 119875Φ (119905) (20)

in which119875 is the119872119873times119872119873 operationalmatrix of integrationcorresponding to the hybrid functions The structure of thismatrix is given by [14]

119875 = ((((

119864lowast 119867 119867 sdot sdot sdot 1198670 119864lowast 119867 sdot sdot sdot 1198670 0 119864lowast sdot sdot sdot 119867 d0 0 0 sdot sdot sdot 119864lowast))))

(21)

where

119867 = 1119873 (((((

119905119891 0 sdot sdot sdot 011990521198912 0 sdot sdot sdot 0 d119905119872119891119872 0 sdot sdot sdot 0)))))

(22)

4 International Journal of Differential Equations

and 119864lowast is defined by

119864lowast = 1119873((((((((

0 1 0 sdot sdot sdot 00 0 12 sdot sdot sdot 0 d0 0 0 sdot sdot sdot 1119872 minus 10 0 0 sdot sdot sdot 0

))))))))

(23)

It should be noted that 119864lowast is the operational matrix ofintegration associated with the Taylorrsquos polynomials on theinterval [((119899minus1)119873)119905119891 (119899119873)119905119891) Furthermore if119862 is a vectorof order 119872119873 times 1 then the expression Φ(119905)Φ119879(119905)119862 can berepresented in terms of hybrid functions asΦ (119905)Φ119879 (119905) 119862 ≃ Φ (119905) (24)

The 119872119873 times 119872119873 matrix is called the operational matrix ofproduct [14]

3 Derivation of the OperationalMatrix of Delay

In this section we shall derive the operational matrix ofdelay associated with the hybrid of block-pulse functionsand Taylorrsquos polynomials For this purpose we employ aprocedure analogous to the one developed in [19] for thehybrid of block-pulse functions and Chebyshev polynomialsThe delayed vectorΦ(119905 minus 119886(119905)) can be represented in terms ofΦ(119905) as follows Φ (119905 minus 119886 (119905)) = 119863Φ (119905) (25)

in which 119863 is called the operational matrix of delay Inaddition 119886(119905) is a piecewise constant function defined by

119886 (119905) =

1205911 1198790 le 119905 lt 11987911205912 1198791 le 119905 lt 1198792 120591119903 119879119903minus1 le 119905 lt 119879119903(26)

where 0 = 1198790 lt 1198791 lt sdot sdot sdot lt 119879119903minus1 lt 119879119903 = 119905119891 It isfurther supposed that 120591119894 119894 = 1 2 119903 are arbitrary rationalnumbers in the interval [0 119905119891) and 119879119894 isin Q 119894 = 0 1 119903 Toconstruct119863 we first obtain the value of119873 Let119860 = 119894 120591119894 = 0 (27)

and take 120574 as the smallest positive integer number such thatthe following conditions are satisfied120574120591119894 isin Z120574119879119895 isin Z119894 isin 119860 119895 = 1 2 119903 (28)

Assume that |119860| = 119896 Next we define 120582 as the greatestcommon divisor of the integers 120574120591119894 and 120574119879119895 119894 isin 119860 and119895 = 1 2 119903 that is120582 = gcd (1205741205911 1205741205912 120574120591119896 1205741198791 1205741198792 120574119879119903) (29)

Define

119873 = 120574120582119905119891 if

120574120582119905119891 isin Z[ 120574120582119905119891] + 1 otherwise (30)

where [(120574120582)119905119891] denotes the greatest integer value less thanor equal to (120574120582)119905119891 It is worth noting that 119873 is selected insuch a way that the number of subintervals is minimized Asa consequence we obtain the following subintervals

[0 120582120574) [120582120574 2120582120574) [(119873 minus 1) 120582120574 119873120582120574) (31)

For convenience we use the notations defined by

ℎ = 120582120574 119873119895 = 120574120582 (119879119895 minus 119879119895minus1) 119897119895 = 120574120582120591119895 119895 = 1 2 119903

(32)

Clearly sum119903119895=1119873119895 = 119873 Consequently we have119886 (119905) = 119896119894ℎ (119894 minus 1) ℎ le 119905 lt 119894ℎ 119894 = 1 2 119873 (33)

in which

119896119894 =

1198971 1 le 119894 le 11987311198972 1198731 + 1 le 119894 le 1198731 + 1198732 119897119895 119895minus1sum119896=1

119873119896 + 1 le 119894 le 119895sum119896=1

119873119896 119897119903 119903minus1sum119896=1

119873119896 + 1 le 119894 le 119903sum119896=1

119873119896

(34)

As a result the problem is reduced to find the delay opera-tional matrix for the delay function defined by

Φ (119905 minus 119886 (119905)) =

Φ(119905 minus 1198961ℎ) 0 le 119905 lt 1199051Φ (119905 minus 1198962ℎ) 1199051 le 119905 lt 1199052 Φ (119905 minus 119896119873ℎ) 119905119899minus1 le 119905 lt 119905119873(35)

International Journal of Differential Equations 5

where 119905119894 = 119894ℎ 119894 = 1 2 119873 (36)

To construct the matrix 119863 we first derive the matrix 119863119894 for119894 = 1 2 119873 in such a way that the following relation issatisfied Φ(119905 minus 119896119894ℎ) = 119863119894Φ (119905) 119905119894minus1 le 119905 lt 119905119894 (37)

By the properties of the hybrid functions it is easy to verifythat for the case 119905119894minus1 le 119905 lt 119905119894 the only functions with nonzerovalues are 120601(119905minus119896119894)119898(119905minus119896119894ℎ) for119898 = 0 1 119872minus1 Notice that

120601(119905minus119896119894)119898 (119905 minus 119896119894ℎ) = 120601119894119898 (119905) 119898 = 0 1 119872 minus 1 (38)

Accordingly 120601(119905minus119896119894)119898(119905minus119896119894ℎ) can be written in terms of 120601119894119898(119905)as 119863119894 = 119878119894 otimes 119868119872 119894 = 1 2 119873 (39)

where 119868119872 is the119872-dimensional identitymatrix andotimes denotesthe Kronecker product [24] In addition 119878119894 is an119873times119873matrixin which the only nonzero entry is equal to one and locatedat the (119894 minus 119896119894)th row and 119894th column Finally if we expressΦ(119905 minus 119886(119905)) in terms of Φ(119905) we find119863 = 1198631 + 1198632 + sdot sdot sdot + 119863119873 (40)

Remark 1 If 119894 minus 119896119894 le 0 then 119878119894 is a zero matrix of order119873times119873 It is worth noting that the structure of the operationalmatrix of delay associatedwith the proposed hybrid functionsis exactly the same as the structure of the hybrid of block-pulse functions and Chebyshev polynomials

4 Problem Statement and Its Approximation

Consider the time-varying piecewise constant delay systemdescribed by

(119905) = 119864 (119905)119883 (119905) + 119865 (119905)119883 (119905 minus 119886 (119905)) + 119866 (119905) 119880 (119905) 0 le 119905 le 119905119891 (41)

119883 (0) = 1198830 (42)119883 (119905) = 0 119905 lt 0 (43)

119886 (119905) =

1205911 1198790 le 119905 lt 11987911205912 1198791 le 119905 lt 1198792 120591119903 119879119903minus1 le 119905 lt 119879119903(44)

where119883(119905) isin R119897 119880(119905) isin R119902 119864(119905) 119865(119905) and119866(119905) arematricesof appropriate dimensions 1198830 is a constant specified vectorand 120591119894 119894 = 1 2 119903 are arbitrary rational numbers in [0 119905119891)The objective is to find119883(119905) 0 le 119905 le 119905119891 satisfying (41)ndash(43)

41 Description of the Method Let119883(119905) = [1199091 (119905) 1199092 (119905) 119909119897 (119905)]119879 119880 (119905) = [1199061 (119905) 1199062 (119905) 119906119902 (119905)]119879 (45)

Assume that each 119909119894(119905) and each of 119906119895(119905) 119894 = 1 2 119902 canbe written in terms of hybrid functions as119909119894 (119905) = Φ119879 (119905) 119883119894119906119895 (119905) = Φ119879 (119905) 119880119895 (46)

Consequently 119883(119905) = (119868119897 otimes Φ119879 (119905))119883119880 (119905) = (119868119902 otimes Φ119879 (119905))119880 (47)

where 119868119897 and 119868119902 are identity matrices of order 119897 times 119897 and 119902 times119902 respectively Furthermore 119883 and 119880 are vectors of order119897119873119872 times 1 and 119902119873119872 times 1 respectively given by

119883 = [1198831198791 1198831198792 119883119879119897 ]119879 119880 = [1198801198791 1198801198792 119880119879119902 ]119879 (48)

Also 119883 (0) = (119868119897 otimes Φ119879 (119905)) 119889 (49)

where 119889 is a vector of order 119897119873119872 times 1 defined by

119889 = [1198891198791 1198891198792 119889119879119897 ]119879 (50)

Similarly 119864(119905) 119865(119905) and 119866(119905) can be represented in terms ofhybrid functions as119864 (119905) = 119864119879 (119868119897 otimes Φ (119905)) 119865 (119905) = 119865119879 (119868119897 otimes Φ (119905)) 119866 (119905) = 119866119879 (119868119902 otimes Φ (119905)) (51)

where 119864119879 119865119879 and 119866119879 are of dimensions 119897 times 119897119873119872 119897 times 119897119873119872and 119897 times 119902119873119872 respectivelyThe delayed vector119883(119905 minus 119886(119905)) canalso be expressed in terms of hybrid functions as119883 (119905 minus 119886 (119905)) = (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119863119879)119883 (52)

in which119863 is the delay operational matrix presented by (25)As a result119864 (119905)119883 (119905) = 119864119879 (119868119897 otimes Φ (119905)) (119868119897 otimes Φ119879 (119905))119883= (119868l otimes Φ119879 (119905)) 119879119883119865 (119905)119883 (119905 minus 119886 (119905))= 119865119879 (119868119897 otimes Φ (119905)) (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119863119879)119883= (119868119897 otimes Φ119879 (119905)) 119879 (119868119897 otimes 119863119879)119883

6 International Journal of Differential Equations

119866 (119905)119880 (119905) = 119866119879 (119868119902 otimes Φ (119905)) (119868119902 otimes Φ119879 (119905))119880= (119868119897 otimes Φ119879 (119905)) 119879119880

(53)

where and can be determined in a manner analogousto the derivation process of the matrix described by (24)Furthermore

int1199050(119868119897 otimes Φ119879 (119904)) 119889119904 = (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879)

int1199050119865 (119904)119883 (119904 minus 119886 (119904)) 119889119904= (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)119883

(54)

in which 119875 is the operational matrix of integration describedby (20) By integrating both sides of (41) with respect to 119905 andusing (47) (49) (51) (52) and (54) we obtain

(119868119897 otimes Φ119879 (119905))119883 minus (119868119897 otimes Φ119879 (119905)) 119889= (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879119883+ (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)119883+ (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879119880(55)

From the preceding equation we deduce that

119883 = [119868119872119873 minus (119868119897 otimes 119875119879) 119879minus (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)]minus1 [119889 + (119868119897 otimes 119875119879) 119879119880] (56)

5 Simulation Results

In this section five examples of varying complexity areinvestigated to evaluate the usefulness and effectiveness ofthe proposed computational techniqueThe correct choice of119873 would greatly improve the efficiency and accuracy of thepresented procedure

51 Example 1 Consider the following piecewise constantdelay system [17] (119905) = 119909 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (57)119909 (119905) = 0 119905 le 0 (58)119906 (119905) = 1 119905 ge 0 (59)

119886 (119905) = 01 0 le 119905 lt 03503 035 le 119905 lt 0705 07 le 119905 le 1 (60)

The exact solution to this problem is given by

119909 (119905) =

119905 0 le 119905 lt 011200 + 910119905 + 121199052 01 le 119905 lt 02113000 + 2325119905 + 251199052 + 161199053 02 le 119905 lt 03961240000 + 18312000119905 + 1694001199052 + 7601199053 + 1241199054 03 le 119905 lt 0352901613840000 + 710119905 + 121199052 035 le 119905 lt 04830671280000 + 3950119905 + 3101199052 + 161199053 04 le 119905 lt 052592013840000 + 9111200119905 + 29801199052 + 1121199053 + 1241199054 05 le 119905 lt 06641781796000000 + 2293730000119905 + 68920001199052 + 171501199053 + 1601199054 + 11201199055 06 le 119905 lt 065107046748000000 + 34965613840000119905 + 151199052 + 161199053 065 le 119905 lt 0737190303192000000 + 37376000119905 + 1294001199052 + 1201199053 + 1241199054 07 le 119905 lt 08244440112800000 + 639910000119905 + 167960001199052 + 313001199053 + 11201199054 + 11201199055 08 le 119905 lt 085710323764000000 + 32661613840000119905 + 1101199052 + 161199053 085 le 119905 lt 09885283764000000 + 27996013840000119905 + 1214001199052 + 1601199053 + 1241199054 09 le 119905 le 1

(61)

International Journal of Differential Equations 7

To apply the method developed in the current paper wefirst choose the value of119873 with the use of (30) Because 120574 =20 and 120582 = 1 hence we select 119873 = 20 We also take 119872 = 6Let

119909 (119905) = 119862119879Φ (119905) (62)

where

119862 = [11988810 1198881119872minus1 1198881198730 119888119873119872minus1]119879 Φ (119905) = [12060110 (119905) 1206011119872minus1 (119905) 1206011198730 (119905) 120601119873119872minus1 (119905)]119879 (63)

By expanding 119906(119905) in terms of hybrid functions we get

119906 (119905) = 1198891198791Φ (119905) (64)

We also have

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (65)

where 119863 is the delay operational matrix given by (25)Integrating both sides of (57) with respect to 119905 and using (62)ndash(65) yield

119862119879 = 1198891198791119875 (119868119872119873 minus 119863119875)minus1 (66)

in which 119868119872119873 is the 119872119873-dimensional identity matrix and 119875is the operational matrix of integration given by (20) Withthe use of (66) and (62) we would obtain the same value asthe exact value of 119909(119905)52 Example 2 Consider the following two-dimensionaltime-varying piecewise constant delay system

(1 (119905)2 (119905)) = ( 1 1 + 1199052119905 1199052 )(1199091 (119905 minus 119886 (119905))1199092 (119905 minus 119886 (119905)))+ (12)119906 (119905) 0 le 119905 le 1 (67)

(1199091 (119905)1199092 (119905)) = (00) 119905 lt 0 (68)

(1199091 (0)1199092 (0)) = (11) (69)

119906 (119905) = 119905 0 le 119905 lt 051 minus 119905 05 le 119905 lt 1 (70)

119886 (119905) = 02 0 le 119905 lt 0307 03 le 119905 le 1 (71)

The exact solutions to this problem are given as follows

1199091 (119905) =

1 + 121199052 0 le 119905 lt 0243197500 + 10350 119905 + 18251199052 + 11301199053 + 141199054 02 le 119905 lt 03147071120000 + 121199052 03 le 119905 lt 05117071120000 + 119905 minus 121199052 05 le 119905 lt 07minus1033112000 + 747200119905 minus 1612001199052 + 1301199053 + 141199054 07 le 119905 lt 09minus 4245584594200000000 + 1148571500000 119905 minus 35588320000001199052 minus 4681300001199053 + 176740001199054 + 717501199055 minus 71201199056 + 1351199057 09 le 119905 le 1

1199092 (119905) =

1 + 1199052 0 le 119905 lt 02179473187500 + 10150 1199052 + 16751199053 + 3201199054 + 151199055 02 le 119905 lt 0331693513000000 + 1199052 03 le 119905 lt 0516693513000000 + 2119905 minus 1199052 05 le 119905 lt 07minus 2204813000000 + 2119905 + 492001199052 + 31001199053 minus 1101199054 + 151199055 07 le 119905 lt 093646033938184000000000 + 2119905 minus 63163400001199052 + 437221730000001199053 minus 11101400001199054 + 1981150001199055 + 1919001199056 minus 111401199057 + 1401199058 09 le 119905 le 1

(72)

8 International Journal of Differential Equations

To solve this problem by the proposed approach we firstdetermine the value of119873with the use of (30) Because 120574 = 10and 120582 = 1 consequently we select 119873 = 10 Also we choose119872 = 9 Let

1199091 (119905) = 1198621198791Φ (119905) 1199092 (119905) = 1198621198792Φ (119905) (73)

Expanding 1 1 + 119905 2119905 and 1199052 in terms of hybrid functionsimplies

1 = 1198911198791 Φ (119905) 1 + 119905 = 1198911198792 Φ (119905)

2119905 = 1198911198793 Φ (119905) 1199052 = 1198911198794 Φ (119905)

(74)

Also using (70) we obtain

int1199050119906 (120591) 119889120591=

121199052 = 1198911198795 Φ (119905) 0 le 119905 lt 05minus14 + 119905 minus 121199052 = 1198911198796 Φ (119905) 05 le 119905 lt 1

(75)

Moreover

1199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905) (76)

Therefore(1 + 119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198912= 11986211987921198632Φ (119905) 21199051199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905)Φ119879 (119905) 1198913= 11986211987911198633Φ (119905) 11990521199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198914= 11986211987921198634Φ (119905) (77)

in which 2 3 and 4 can be obtained in a way similar tothe construction method of the matrix defined by (24) Byintegrating both sides of (67) with respect to 119905 and using (68)(70) (71) (73) (74) (75) (76) and (77) we get

1198621198791 (119868119872119873 minus 119863119875) minus 11986211987921198632119875 = 1198911198791 + 1198911198795 + 1198911198796 1198621198792 (119868119872119873 minus 1198634119875) minus 11986211987911198633119875 = 1198911198791 + 21198911198795 + 21198911198796 (78)

By solving the resulting system described by (78) and thenusing (73) we would obtain the same values as the exactvalues of 1199091(119905) and 1199092(119905)53 Example 3 Consider the following time-delay system[3] (119905) = minus5119909 (119905) minus 5119909 (119905 minus 119886 (119905)) + 2119906 (119905) 0 le 119905 le 2 (79)119909 (0) = 1 (80)119906 (119905) = 1 119905 ge 0 (81)

119886 (119905) = 0 0 le 119905 lt 0803 08 le 119905 lt 1406 14 le 119905 le 1709 17 le 119905 le 2

(82)

The exact solution to this problem is given by [3]

119909 (119905) =

02 + 08119890minus10119905 0 le 119905 lt 0802 + 081198903minus10119905 + 08119890minus4 (1 minus 1198903) 119890minus5119905 08 le 119905 lt 1102 + 081198905minus10119905 minus 4119890minus25 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (65119890minus25 + 119890minus4) 119890minus5119905 11 le 119905 lt 1402 + 081198909minus10119905 minus 4119890minus1 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (8119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 14 le 119905 lt 1702 + 0811989012minus10119905 minus 411989005 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (9511989005 minus 05119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 17 le 119905 le 2(83)

To employ the proposed method we first determine thesuitable value of119873 Using (30) it follows that119873 = 120574120582119905119891 = 20 (84)

Assume that119872 is an arbitrary positive integer number Let

119909 (119905) = 119862119879Φ (119905) (85)

International Journal of Differential Equations 9

Similarly

119909 (0) = 119906 (119905) = 119889119879Φ (119905) 119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (86)

By integrating both sides of (79) with respect to 119905 and using(85)-(86) we conclude that119862119879 = (119889119879 + 2119889119879119875) (119868119872119873 + 5119875 + 5119863119875)minus1 (87)

where 119868119872119873 is the119872119873-dimensional identity matrix Define119864 = max 1003816100381610038161003816119909119890 (119905) minus 119909 (119905)1003816100381610038161003816 0 le 119905 le 1 (88)

in which 119909(119905) and 119909119890(119905) denote the approximate solutiondetermined by the present approach and the exact solutionrespectively In Table 1 the numerical results of themaximumerror denoted by 119864 for 119873 = 20 and various values of 119872are reported The simulation results obtained by the currentapproach show a significant improvement in the accuracy ofthe solution compared to the method developed in [19]

54 Example 4 As a more complicated problem considerthe following problem

(119905) = 1199051199092 (119905 minus 119886 (119905)) + 1199093 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (89)

119909 (119905) = 0 119905 lt 0 (90)119909 (0) = 1 (91)

119906 (119905) = 119905 0 le 119905 lt 0404 04 le 119905 le 1 (92)

119886 (119905) = 04 0 le 119905 lt 0808 08 le 119905 le 1 (93)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 046632511640625 + 2593315625119905 minus 72962501199052 + 1172501199053 minus 31001199054 + 131001199055 minus 11201199056 + 1561199057 04 le 119905 lt 082356911640625 + 4218715625119905 minus 762362501199052 + 2532501199053 minus 431001199054 + 231001199055 minus 71201199056 + 1561199057 08 le 119905 le 1

(94)

Although the above-mentioned problem is a nonlinear delaydifferential equation the method developed in the presentpaper can easily be implemented Because 120574 = 5 and 120582 = 2we take119873 = 3 Furthermore we take119872 = 8 Let

119909 (119905) = 119862119879Φ (119905) (95)

Similarly

119909 (0) = V1198791Φ (119905) 119905 = V1198792Φ (119905) (96)

int1199050119906 (120591) 119889120591 =

121199052 = V1198793Φ (119905) 0 le 119905 lt 04minus 225 + 25119905 = V1198794Φ (119905) 04 le 119905 lt 1 (97)

where V1 V2 V3 and V4 are vectors of order119872119873times1 Also wehave

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (98)1199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) 119863119879119862= 119862119879119863Φ (119905) (99)

1199093 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905)119882= 1198621198791198632Φ (119905) (100)

in which

119882 = 119863119879119862 (101)

Using (99) gives

1199051199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) V2= 119862119879119863V2Φ (119905) (102)

Integrating both sides of (89) with respect to 119905 and using (95)(97) (98) (99) (100) and (102) imply that

119862119879 (119868119872119873 minus 119863V2119875 minus 1198632119875) minus (V1198791 + V1198793 + V1198794 )= 0 (103)

By solving (103) and using (95) the exact value of 119909(119905)will beobtained

10 International Journal of Differential Equations

Table 1 Computational results of maximum error for Example 3Present method CPU time (s) 119864119873 = 20119872 = 1 0112 1165119890 minus 02119873 = 20119872 = 2 0202 3231119890 minus 03119873 = 20119872 = 4 0398 2573119890 minus 06119873 = 20119872 = 6 0769 4836119890 minus 08119873 = 20119872 = 8 0853 3942119890 minus 11119873 = 20119872 = 10 1282 2376119890 minus 13119873 = 20119872 = 12 1841 5875119890 minus 16119873 = 20119872 = 14 2167 3624119890 minus 17119873 = 20119872 = 16 2536 5117119890 minus 2055 Example 5 As the final example we consider the time-varying multidelay system defined by (119905) = 119905119909 (119905 minus 1198861 (119905)) + 11990521199092 (119905 minus 1198862 (119905)) + 119906 (119905) 0 le 119905 le 1

119909 (0) = 1119906 (119905) =

119905 0 le 119905 lt 12 12 12 le 119905 le 1(104)

where

1198861 (119905) = 12 0 le 119905 lt 34 14 34 le 119905 le 1

1198862 (119905) = 14 0 le 119905 lt 12 34 12 le 119905 le 1

(105)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 14 2037320480 + 121199052 + 11321199053 minus 1161199054 + 1101199055 14 le 119905 lt 12 4819161440 + 12119905 + 9161199052 minus 161199053 + 181199054 12 le 119905 lt 34 12897431966080 + 12119905 + 14287409601199052 + 1873841199053 minus 12561199054 + 1241199055 + 1481199056 34 le 119905 le 1(106)

Although the above-mentioned system involves multiplepiecewise constant delays the proposed procedure can beapplied To employ our approach we select 119873 = 4 Bychoosing119872 = 7 the exact solution of 119909(119905) will be obtained6 Conclusion

An efficient and flexible framework has been successfullydeveloped for solving piecewise constant delay systems Thefoundation of the proposed method is based on a hybrid ofblock-pulse functions and Taylorrsquos polynomials The opera-tional matrix of delay has been constructed The operationalmatrices of integration delay and product associated withthe hybrid functions were utilized to transform the problemunder consideration into a system of algebraic equations Itis worth emphasizing that the exact solutions of Examples1 2 4 and 5 cannot be produced solely either by block-pulse functions or by Taylorrsquos polynomials After determiningthe suitable value of 119873 the number of subintervals smallvalues for 119872 the degree of the Taylorrsquos polynomials areneeded to achieve a specific accuracy The simulation resultsdemonstrate the effectiveness of the suggested procedure Inaddition the proposed numerical scheme can be extended

to a class of nonlinear piecewise constant delay systems butsome modifications are required

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this manuscript

References

[1] RDDriverOrdinary and delay differential equations Springer-Verlag New York NY USA 1977

[2] M Malek-Zavarei and M Jamshidi Time-delay systems vol 9of North-Holland Systems and Control Series North-HollandPublishing Co Amsterdam Netherlands 1987

[3] K B Datta and B M Mohan Orthogonal functions in systemsand control vol 9 ofAdvanced Series in Electrical and ComputerEngineering World Scientific Publishing Co 1995

[4] A Bellen ldquoOne-step collocation for delay differential equa-tionsrdquo Journal of Computational and Applied Mathematics vol10 no 3 pp 275ndash283 1984

[5] C Hwang and M Y Chen ldquoAnalysis and parameter identifi-cation of time-delay systems via shifted Legendre polynomialsrdquoInternational Journal of Control vol 41 no 2 pp 403ndash415 1985

International Journal of Differential Equations 11

[6] I R Horng and J H Chou ldquoAnalysis parameter estimation andoptimal control of time-delay systems via Chebyshev seriesrdquoInternational Journal of Control vol 41 no 5 pp 1221ndash12341985

[7] A Bellen and M Zennaro ldquoNumerical solution of delaydifferential equations by uniform corrections to an implicitRunge-Kutta methodrdquo Numerische Mathematik vol 47 no 2pp 301ndash316 1985

[8] I-R Horng and J-H Chou ldquoAnalysis and parameter identi-fication of time-delay systems via shifted Jacobi polynomialsrdquoInternational Journal of Control vol 44 no 4 pp 935ndash942 1986

[9] M Razzaghi and M Razzaghi ldquoTaylor series analysis of time-varying multi-delay systemsrdquo International Journal of Controlvol 50 no 1 pp 183ndash192 1989

[10] B M Mohan and K B Datta ldquoAnalysis of linear time-invarianttime-delay systems via orthogonal functionsrdquo InternationalJournal of Systems Science vol 26 no 1 pp 91ndash111 1995

[11] A Bellen and S Maset ldquoNumerical solution of constantcoefficient linear delay differential equations as abstract Cauchyproblemsrdquo Numerische Mathematik vol 84 no 3 pp 351ndash3742000

[12] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering vol 7 no 4 pp 337ndash353 2001

[13] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004

[14] H R Marzban and M Razzaghi ldquoAnalysis of time-delaysystems via hybrid of block-pulse functions and Taylor seriesrdquoJournal of Vibration and Control vol 11 no 12 pp 1455ndash14682005

[15] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006

[16] H R Marzban ldquoOptimal control of linear multi-delay systemsbased on a multi-interval decomposition schemerdquo OptimalControl Applications and Methods vol 37 no 1 pp 190ndash2112016

[17] W L Chen and B S Jeng ldquoAnalysis of piecewise constantdelay systems via block-pulse functionsrdquo International Journalof Systems Science vol 12 no 5 pp 625ndash633 1981

[18] W-L Chen and C-H Meng ldquoA general procedure of solvingthe linear delay system via block pulse functionsrdquo Computersand Electrical Engineering vol 9 no 3-4 pp 153ndash166 1982

[19] HRMarzban andM Shahsiah ldquoSolution of piecewise constantdelay systems using hybrid of block-pulse and Chebyshevpolynomialsrdquo Optimal Control Applications and Methods vol32 no 6 pp 647ndash659 2011

[20] H R Marzban and S M Hoseini ldquoAnalysis of linear piecewiseconstant delay systems using a hybrid numerical schemerdquoAdvances in Numerical Analysis vol 2016 Article ID 8903184 9pages 2016

[21] H RMarzban and SMHoseini ldquoNumerical treatment of non-linear optimal control problems involving piecewise constantdelayrdquo IMA Journal of Mathematical Control and Information2015

[22] Z H Jiang and W Schaufelberger Block pulse functions andtheir applications in control systems vol 179 of Lecture Notesin Control and Information Sciences Springer Berlin Germany1992

[23] H R Marzban ldquoParameter identification of linear multi-delaysystems via a hybrid of block-pulse functions and Taylorrsquospolynomialsrdquo International Journal of Control 2016

[24] P LancasterTheory of matrices Academic Press New York NYUSA 1969

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Mathematical Problems in Engineering

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 3

12060121 (119905) = 3119905 minus 3 119905 isin [1 2) 0 otherwise

12060130 (119905) = 1 119905 isin [2 3) 0 otherwise

12060131 (119905) = 3119905 minus 6 119905 isin [2 3) 0 otherwise

(10)

Case 2 Suppose that

119892 (119905) = exp (119905) 0 le 119905 lt 11199052 + 119905 1 le 119905 lt 2119905 + 4 2 le 119905 le 3 (11)

Obviously this function is discontinuous at the points 119905 = 1and 119905 = 2 However it can be efficiently approximated by theproposed hybrid functions To express this function in termsof hybrid functions we choose 119873 = 3 In addition Assumethat119872 is an arbitrary positive integer number greater than 2With the use of (2) we have

119892 (119905) = 3sum119899=1

119872minus1sum119898=0

119888119899119898120601119899119898 (119905) (12)

where the hybrid functions 120601119899119898(119905) 119899 = 1 2 3 119898 = 0 1 119872 minus 1 are defined by

120601119899119898 (119905) = 119879119898 (3119905 minus 3119899 + 3) = 3119898 (119905 minus 119899 + 1)119898 (13)

Also the associated coefficients 119888119899119898 are given by

1198881119898 = 13119898119898 119898 = 0 1 119872 minus 111988820 = 211988821 = 111988822 = 19 1198882119898 = 0 119898 = 3 4 119872 minus 111988830 = 611988831 = 13 1198883119898 = 0 119898 = 2 3 119872 minus 1

(14)

The hybrid expansion of 119892(119905) over the interval [1 2] ispresented by1198882012060120 (119905) + 1198882112060121 (119905) + 1198882212060122 (119905)= 2 + (3119905 minus 3) + 19 (3119905 minus 3)2 = 1199052 + 119905 (15)

Similarly the hybrid approximation of 119892(119905) on the interval[2 3) is described by1198883012060130 (119905) + 1198883112060131 (119905) = 6 + 13 (3119905 minus 6) = 119905 + 4 (16)

Moreover the hybrid expansion of119892(119905) over the interval [0 1)is expressed by

119892 (119905) = 119872minus1sum119898=0

119905119898119898 (17)

which is precisely Taylorrsquos expansion of the function exp(119905)about 119905 = 0 up to degree119872minus 1 Define119864 = 10038171003817100381710038171003817119892 (119905) minus 119862119879Φ (119905)10038171003817100381710038171003817infin= max 10038161003816100381610038161003816119892 (119905) minus 119862119879Φ (119905)10038161003816100381610038161003816 0 le 119905 le 1 (18)

It is easily verified that 119864 = 119890119872 (19)

The obtained result indicates that the maximum error con-verges to zero as the value of 119872 increases Some importantaspects of the proposed hybrid functions are clarified in [23]

23 Operational Matrices of Integration and Product Theintegration of the vector Φ(119905) presented by (6) can also beexpanded in terms of hybrid functions as

int1199050Φ (120591) 119889120591 ≃ 119875Φ (119905) (20)

in which119875 is the119872119873times119872119873 operationalmatrix of integrationcorresponding to the hybrid functions The structure of thismatrix is given by [14]

119875 = ((((

119864lowast 119867 119867 sdot sdot sdot 1198670 119864lowast 119867 sdot sdot sdot 1198670 0 119864lowast sdot sdot sdot 119867 d0 0 0 sdot sdot sdot 119864lowast))))

(21)

where

119867 = 1119873 (((((

119905119891 0 sdot sdot sdot 011990521198912 0 sdot sdot sdot 0 d119905119872119891119872 0 sdot sdot sdot 0)))))

(22)

4 International Journal of Differential Equations

and 119864lowast is defined by

119864lowast = 1119873((((((((

0 1 0 sdot sdot sdot 00 0 12 sdot sdot sdot 0 d0 0 0 sdot sdot sdot 1119872 minus 10 0 0 sdot sdot sdot 0

))))))))

(23)

It should be noted that 119864lowast is the operational matrix ofintegration associated with the Taylorrsquos polynomials on theinterval [((119899minus1)119873)119905119891 (119899119873)119905119891) Furthermore if119862 is a vectorof order 119872119873 times 1 then the expression Φ(119905)Φ119879(119905)119862 can berepresented in terms of hybrid functions asΦ (119905)Φ119879 (119905) 119862 ≃ Φ (119905) (24)

The 119872119873 times 119872119873 matrix is called the operational matrix ofproduct [14]

3 Derivation of the OperationalMatrix of Delay

In this section we shall derive the operational matrix ofdelay associated with the hybrid of block-pulse functionsand Taylorrsquos polynomials For this purpose we employ aprocedure analogous to the one developed in [19] for thehybrid of block-pulse functions and Chebyshev polynomialsThe delayed vectorΦ(119905 minus 119886(119905)) can be represented in terms ofΦ(119905) as follows Φ (119905 minus 119886 (119905)) = 119863Φ (119905) (25)

in which 119863 is called the operational matrix of delay Inaddition 119886(119905) is a piecewise constant function defined by

119886 (119905) =

1205911 1198790 le 119905 lt 11987911205912 1198791 le 119905 lt 1198792 120591119903 119879119903minus1 le 119905 lt 119879119903(26)

where 0 = 1198790 lt 1198791 lt sdot sdot sdot lt 119879119903minus1 lt 119879119903 = 119905119891 It isfurther supposed that 120591119894 119894 = 1 2 119903 are arbitrary rationalnumbers in the interval [0 119905119891) and 119879119894 isin Q 119894 = 0 1 119903 Toconstruct119863 we first obtain the value of119873 Let119860 = 119894 120591119894 = 0 (27)

and take 120574 as the smallest positive integer number such thatthe following conditions are satisfied120574120591119894 isin Z120574119879119895 isin Z119894 isin 119860 119895 = 1 2 119903 (28)

Assume that |119860| = 119896 Next we define 120582 as the greatestcommon divisor of the integers 120574120591119894 and 120574119879119895 119894 isin 119860 and119895 = 1 2 119903 that is120582 = gcd (1205741205911 1205741205912 120574120591119896 1205741198791 1205741198792 120574119879119903) (29)

Define

119873 = 120574120582119905119891 if

120574120582119905119891 isin Z[ 120574120582119905119891] + 1 otherwise (30)

where [(120574120582)119905119891] denotes the greatest integer value less thanor equal to (120574120582)119905119891 It is worth noting that 119873 is selected insuch a way that the number of subintervals is minimized Asa consequence we obtain the following subintervals

[0 120582120574) [120582120574 2120582120574) [(119873 minus 1) 120582120574 119873120582120574) (31)

For convenience we use the notations defined by

ℎ = 120582120574 119873119895 = 120574120582 (119879119895 minus 119879119895minus1) 119897119895 = 120574120582120591119895 119895 = 1 2 119903

(32)

Clearly sum119903119895=1119873119895 = 119873 Consequently we have119886 (119905) = 119896119894ℎ (119894 minus 1) ℎ le 119905 lt 119894ℎ 119894 = 1 2 119873 (33)

in which

119896119894 =

1198971 1 le 119894 le 11987311198972 1198731 + 1 le 119894 le 1198731 + 1198732 119897119895 119895minus1sum119896=1

119873119896 + 1 le 119894 le 119895sum119896=1

119873119896 119897119903 119903minus1sum119896=1

119873119896 + 1 le 119894 le 119903sum119896=1

119873119896

(34)

As a result the problem is reduced to find the delay opera-tional matrix for the delay function defined by

Φ (119905 minus 119886 (119905)) =

Φ(119905 minus 1198961ℎ) 0 le 119905 lt 1199051Φ (119905 minus 1198962ℎ) 1199051 le 119905 lt 1199052 Φ (119905 minus 119896119873ℎ) 119905119899minus1 le 119905 lt 119905119873(35)

International Journal of Differential Equations 5

where 119905119894 = 119894ℎ 119894 = 1 2 119873 (36)

To construct the matrix 119863 we first derive the matrix 119863119894 for119894 = 1 2 119873 in such a way that the following relation issatisfied Φ(119905 minus 119896119894ℎ) = 119863119894Φ (119905) 119905119894minus1 le 119905 lt 119905119894 (37)

By the properties of the hybrid functions it is easy to verifythat for the case 119905119894minus1 le 119905 lt 119905119894 the only functions with nonzerovalues are 120601(119905minus119896119894)119898(119905minus119896119894ℎ) for119898 = 0 1 119872minus1 Notice that

120601(119905minus119896119894)119898 (119905 minus 119896119894ℎ) = 120601119894119898 (119905) 119898 = 0 1 119872 minus 1 (38)

Accordingly 120601(119905minus119896119894)119898(119905minus119896119894ℎ) can be written in terms of 120601119894119898(119905)as 119863119894 = 119878119894 otimes 119868119872 119894 = 1 2 119873 (39)

where 119868119872 is the119872-dimensional identitymatrix andotimes denotesthe Kronecker product [24] In addition 119878119894 is an119873times119873matrixin which the only nonzero entry is equal to one and locatedat the (119894 minus 119896119894)th row and 119894th column Finally if we expressΦ(119905 minus 119886(119905)) in terms of Φ(119905) we find119863 = 1198631 + 1198632 + sdot sdot sdot + 119863119873 (40)

Remark 1 If 119894 minus 119896119894 le 0 then 119878119894 is a zero matrix of order119873times119873 It is worth noting that the structure of the operationalmatrix of delay associatedwith the proposed hybrid functionsis exactly the same as the structure of the hybrid of block-pulse functions and Chebyshev polynomials

4 Problem Statement and Its Approximation

Consider the time-varying piecewise constant delay systemdescribed by

(119905) = 119864 (119905)119883 (119905) + 119865 (119905)119883 (119905 minus 119886 (119905)) + 119866 (119905) 119880 (119905) 0 le 119905 le 119905119891 (41)

119883 (0) = 1198830 (42)119883 (119905) = 0 119905 lt 0 (43)

119886 (119905) =

1205911 1198790 le 119905 lt 11987911205912 1198791 le 119905 lt 1198792 120591119903 119879119903minus1 le 119905 lt 119879119903(44)

where119883(119905) isin R119897 119880(119905) isin R119902 119864(119905) 119865(119905) and119866(119905) arematricesof appropriate dimensions 1198830 is a constant specified vectorand 120591119894 119894 = 1 2 119903 are arbitrary rational numbers in [0 119905119891)The objective is to find119883(119905) 0 le 119905 le 119905119891 satisfying (41)ndash(43)

41 Description of the Method Let119883(119905) = [1199091 (119905) 1199092 (119905) 119909119897 (119905)]119879 119880 (119905) = [1199061 (119905) 1199062 (119905) 119906119902 (119905)]119879 (45)

Assume that each 119909119894(119905) and each of 119906119895(119905) 119894 = 1 2 119902 canbe written in terms of hybrid functions as119909119894 (119905) = Φ119879 (119905) 119883119894119906119895 (119905) = Φ119879 (119905) 119880119895 (46)

Consequently 119883(119905) = (119868119897 otimes Φ119879 (119905))119883119880 (119905) = (119868119902 otimes Φ119879 (119905))119880 (47)

where 119868119897 and 119868119902 are identity matrices of order 119897 times 119897 and 119902 times119902 respectively Furthermore 119883 and 119880 are vectors of order119897119873119872 times 1 and 119902119873119872 times 1 respectively given by

119883 = [1198831198791 1198831198792 119883119879119897 ]119879 119880 = [1198801198791 1198801198792 119880119879119902 ]119879 (48)

Also 119883 (0) = (119868119897 otimes Φ119879 (119905)) 119889 (49)

where 119889 is a vector of order 119897119873119872 times 1 defined by

119889 = [1198891198791 1198891198792 119889119879119897 ]119879 (50)

Similarly 119864(119905) 119865(119905) and 119866(119905) can be represented in terms ofhybrid functions as119864 (119905) = 119864119879 (119868119897 otimes Φ (119905)) 119865 (119905) = 119865119879 (119868119897 otimes Φ (119905)) 119866 (119905) = 119866119879 (119868119902 otimes Φ (119905)) (51)

where 119864119879 119865119879 and 119866119879 are of dimensions 119897 times 119897119873119872 119897 times 119897119873119872and 119897 times 119902119873119872 respectivelyThe delayed vector119883(119905 minus 119886(119905)) canalso be expressed in terms of hybrid functions as119883 (119905 minus 119886 (119905)) = (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119863119879)119883 (52)

in which119863 is the delay operational matrix presented by (25)As a result119864 (119905)119883 (119905) = 119864119879 (119868119897 otimes Φ (119905)) (119868119897 otimes Φ119879 (119905))119883= (119868l otimes Φ119879 (119905)) 119879119883119865 (119905)119883 (119905 minus 119886 (119905))= 119865119879 (119868119897 otimes Φ (119905)) (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119863119879)119883= (119868119897 otimes Φ119879 (119905)) 119879 (119868119897 otimes 119863119879)119883

6 International Journal of Differential Equations

119866 (119905)119880 (119905) = 119866119879 (119868119902 otimes Φ (119905)) (119868119902 otimes Φ119879 (119905))119880= (119868119897 otimes Φ119879 (119905)) 119879119880

(53)

where and can be determined in a manner analogousto the derivation process of the matrix described by (24)Furthermore

int1199050(119868119897 otimes Φ119879 (119904)) 119889119904 = (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879)

int1199050119865 (119904)119883 (119904 minus 119886 (119904)) 119889119904= (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)119883

(54)

in which 119875 is the operational matrix of integration describedby (20) By integrating both sides of (41) with respect to 119905 andusing (47) (49) (51) (52) and (54) we obtain

(119868119897 otimes Φ119879 (119905))119883 minus (119868119897 otimes Φ119879 (119905)) 119889= (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879119883+ (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)119883+ (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879119880(55)

From the preceding equation we deduce that

119883 = [119868119872119873 minus (119868119897 otimes 119875119879) 119879minus (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)]minus1 [119889 + (119868119897 otimes 119875119879) 119879119880] (56)

5 Simulation Results

In this section five examples of varying complexity areinvestigated to evaluate the usefulness and effectiveness ofthe proposed computational techniqueThe correct choice of119873 would greatly improve the efficiency and accuracy of thepresented procedure

51 Example 1 Consider the following piecewise constantdelay system [17] (119905) = 119909 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (57)119909 (119905) = 0 119905 le 0 (58)119906 (119905) = 1 119905 ge 0 (59)

119886 (119905) = 01 0 le 119905 lt 03503 035 le 119905 lt 0705 07 le 119905 le 1 (60)

The exact solution to this problem is given by

119909 (119905) =

119905 0 le 119905 lt 011200 + 910119905 + 121199052 01 le 119905 lt 02113000 + 2325119905 + 251199052 + 161199053 02 le 119905 lt 03961240000 + 18312000119905 + 1694001199052 + 7601199053 + 1241199054 03 le 119905 lt 0352901613840000 + 710119905 + 121199052 035 le 119905 lt 04830671280000 + 3950119905 + 3101199052 + 161199053 04 le 119905 lt 052592013840000 + 9111200119905 + 29801199052 + 1121199053 + 1241199054 05 le 119905 lt 06641781796000000 + 2293730000119905 + 68920001199052 + 171501199053 + 1601199054 + 11201199055 06 le 119905 lt 065107046748000000 + 34965613840000119905 + 151199052 + 161199053 065 le 119905 lt 0737190303192000000 + 37376000119905 + 1294001199052 + 1201199053 + 1241199054 07 le 119905 lt 08244440112800000 + 639910000119905 + 167960001199052 + 313001199053 + 11201199054 + 11201199055 08 le 119905 lt 085710323764000000 + 32661613840000119905 + 1101199052 + 161199053 085 le 119905 lt 09885283764000000 + 27996013840000119905 + 1214001199052 + 1601199053 + 1241199054 09 le 119905 le 1

(61)

International Journal of Differential Equations 7

To apply the method developed in the current paper wefirst choose the value of119873 with the use of (30) Because 120574 =20 and 120582 = 1 hence we select 119873 = 20 We also take 119872 = 6Let

119909 (119905) = 119862119879Φ (119905) (62)

where

119862 = [11988810 1198881119872minus1 1198881198730 119888119873119872minus1]119879 Φ (119905) = [12060110 (119905) 1206011119872minus1 (119905) 1206011198730 (119905) 120601119873119872minus1 (119905)]119879 (63)

By expanding 119906(119905) in terms of hybrid functions we get

119906 (119905) = 1198891198791Φ (119905) (64)

We also have

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (65)

where 119863 is the delay operational matrix given by (25)Integrating both sides of (57) with respect to 119905 and using (62)ndash(65) yield

119862119879 = 1198891198791119875 (119868119872119873 minus 119863119875)minus1 (66)

in which 119868119872119873 is the 119872119873-dimensional identity matrix and 119875is the operational matrix of integration given by (20) Withthe use of (66) and (62) we would obtain the same value asthe exact value of 119909(119905)52 Example 2 Consider the following two-dimensionaltime-varying piecewise constant delay system

(1 (119905)2 (119905)) = ( 1 1 + 1199052119905 1199052 )(1199091 (119905 minus 119886 (119905))1199092 (119905 minus 119886 (119905)))+ (12)119906 (119905) 0 le 119905 le 1 (67)

(1199091 (119905)1199092 (119905)) = (00) 119905 lt 0 (68)

(1199091 (0)1199092 (0)) = (11) (69)

119906 (119905) = 119905 0 le 119905 lt 051 minus 119905 05 le 119905 lt 1 (70)

119886 (119905) = 02 0 le 119905 lt 0307 03 le 119905 le 1 (71)

The exact solutions to this problem are given as follows

1199091 (119905) =

1 + 121199052 0 le 119905 lt 0243197500 + 10350 119905 + 18251199052 + 11301199053 + 141199054 02 le 119905 lt 03147071120000 + 121199052 03 le 119905 lt 05117071120000 + 119905 minus 121199052 05 le 119905 lt 07minus1033112000 + 747200119905 minus 1612001199052 + 1301199053 + 141199054 07 le 119905 lt 09minus 4245584594200000000 + 1148571500000 119905 minus 35588320000001199052 minus 4681300001199053 + 176740001199054 + 717501199055 minus 71201199056 + 1351199057 09 le 119905 le 1

1199092 (119905) =

1 + 1199052 0 le 119905 lt 02179473187500 + 10150 1199052 + 16751199053 + 3201199054 + 151199055 02 le 119905 lt 0331693513000000 + 1199052 03 le 119905 lt 0516693513000000 + 2119905 minus 1199052 05 le 119905 lt 07minus 2204813000000 + 2119905 + 492001199052 + 31001199053 minus 1101199054 + 151199055 07 le 119905 lt 093646033938184000000000 + 2119905 minus 63163400001199052 + 437221730000001199053 minus 11101400001199054 + 1981150001199055 + 1919001199056 minus 111401199057 + 1401199058 09 le 119905 le 1

(72)

8 International Journal of Differential Equations

To solve this problem by the proposed approach we firstdetermine the value of119873with the use of (30) Because 120574 = 10and 120582 = 1 consequently we select 119873 = 10 Also we choose119872 = 9 Let

1199091 (119905) = 1198621198791Φ (119905) 1199092 (119905) = 1198621198792Φ (119905) (73)

Expanding 1 1 + 119905 2119905 and 1199052 in terms of hybrid functionsimplies

1 = 1198911198791 Φ (119905) 1 + 119905 = 1198911198792 Φ (119905)

2119905 = 1198911198793 Φ (119905) 1199052 = 1198911198794 Φ (119905)

(74)

Also using (70) we obtain

int1199050119906 (120591) 119889120591=

121199052 = 1198911198795 Φ (119905) 0 le 119905 lt 05minus14 + 119905 minus 121199052 = 1198911198796 Φ (119905) 05 le 119905 lt 1

(75)

Moreover

1199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905) (76)

Therefore(1 + 119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198912= 11986211987921198632Φ (119905) 21199051199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905)Φ119879 (119905) 1198913= 11986211987911198633Φ (119905) 11990521199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198914= 11986211987921198634Φ (119905) (77)

in which 2 3 and 4 can be obtained in a way similar tothe construction method of the matrix defined by (24) Byintegrating both sides of (67) with respect to 119905 and using (68)(70) (71) (73) (74) (75) (76) and (77) we get

1198621198791 (119868119872119873 minus 119863119875) minus 11986211987921198632119875 = 1198911198791 + 1198911198795 + 1198911198796 1198621198792 (119868119872119873 minus 1198634119875) minus 11986211987911198633119875 = 1198911198791 + 21198911198795 + 21198911198796 (78)

By solving the resulting system described by (78) and thenusing (73) we would obtain the same values as the exactvalues of 1199091(119905) and 1199092(119905)53 Example 3 Consider the following time-delay system[3] (119905) = minus5119909 (119905) minus 5119909 (119905 minus 119886 (119905)) + 2119906 (119905) 0 le 119905 le 2 (79)119909 (0) = 1 (80)119906 (119905) = 1 119905 ge 0 (81)

119886 (119905) = 0 0 le 119905 lt 0803 08 le 119905 lt 1406 14 le 119905 le 1709 17 le 119905 le 2

(82)

The exact solution to this problem is given by [3]

119909 (119905) =

02 + 08119890minus10119905 0 le 119905 lt 0802 + 081198903minus10119905 + 08119890minus4 (1 minus 1198903) 119890minus5119905 08 le 119905 lt 1102 + 081198905minus10119905 minus 4119890minus25 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (65119890minus25 + 119890minus4) 119890minus5119905 11 le 119905 lt 1402 + 081198909minus10119905 minus 4119890minus1 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (8119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 14 le 119905 lt 1702 + 0811989012minus10119905 minus 411989005 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (9511989005 minus 05119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 17 le 119905 le 2(83)

To employ the proposed method we first determine thesuitable value of119873 Using (30) it follows that119873 = 120574120582119905119891 = 20 (84)

Assume that119872 is an arbitrary positive integer number Let

119909 (119905) = 119862119879Φ (119905) (85)

International Journal of Differential Equations 9

Similarly

119909 (0) = 119906 (119905) = 119889119879Φ (119905) 119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (86)

By integrating both sides of (79) with respect to 119905 and using(85)-(86) we conclude that119862119879 = (119889119879 + 2119889119879119875) (119868119872119873 + 5119875 + 5119863119875)minus1 (87)

where 119868119872119873 is the119872119873-dimensional identity matrix Define119864 = max 1003816100381610038161003816119909119890 (119905) minus 119909 (119905)1003816100381610038161003816 0 le 119905 le 1 (88)

in which 119909(119905) and 119909119890(119905) denote the approximate solutiondetermined by the present approach and the exact solutionrespectively In Table 1 the numerical results of themaximumerror denoted by 119864 for 119873 = 20 and various values of 119872are reported The simulation results obtained by the currentapproach show a significant improvement in the accuracy ofthe solution compared to the method developed in [19]

54 Example 4 As a more complicated problem considerthe following problem

(119905) = 1199051199092 (119905 minus 119886 (119905)) + 1199093 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (89)

119909 (119905) = 0 119905 lt 0 (90)119909 (0) = 1 (91)

119906 (119905) = 119905 0 le 119905 lt 0404 04 le 119905 le 1 (92)

119886 (119905) = 04 0 le 119905 lt 0808 08 le 119905 le 1 (93)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 046632511640625 + 2593315625119905 minus 72962501199052 + 1172501199053 minus 31001199054 + 131001199055 minus 11201199056 + 1561199057 04 le 119905 lt 082356911640625 + 4218715625119905 minus 762362501199052 + 2532501199053 minus 431001199054 + 231001199055 minus 71201199056 + 1561199057 08 le 119905 le 1

(94)

Although the above-mentioned problem is a nonlinear delaydifferential equation the method developed in the presentpaper can easily be implemented Because 120574 = 5 and 120582 = 2we take119873 = 3 Furthermore we take119872 = 8 Let

119909 (119905) = 119862119879Φ (119905) (95)

Similarly

119909 (0) = V1198791Φ (119905) 119905 = V1198792Φ (119905) (96)

int1199050119906 (120591) 119889120591 =

121199052 = V1198793Φ (119905) 0 le 119905 lt 04minus 225 + 25119905 = V1198794Φ (119905) 04 le 119905 lt 1 (97)

where V1 V2 V3 and V4 are vectors of order119872119873times1 Also wehave

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (98)1199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) 119863119879119862= 119862119879119863Φ (119905) (99)

1199093 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905)119882= 1198621198791198632Φ (119905) (100)

in which

119882 = 119863119879119862 (101)

Using (99) gives

1199051199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) V2= 119862119879119863V2Φ (119905) (102)

Integrating both sides of (89) with respect to 119905 and using (95)(97) (98) (99) (100) and (102) imply that

119862119879 (119868119872119873 minus 119863V2119875 minus 1198632119875) minus (V1198791 + V1198793 + V1198794 )= 0 (103)

By solving (103) and using (95) the exact value of 119909(119905)will beobtained

10 International Journal of Differential Equations

Table 1 Computational results of maximum error for Example 3Present method CPU time (s) 119864119873 = 20119872 = 1 0112 1165119890 minus 02119873 = 20119872 = 2 0202 3231119890 minus 03119873 = 20119872 = 4 0398 2573119890 minus 06119873 = 20119872 = 6 0769 4836119890 minus 08119873 = 20119872 = 8 0853 3942119890 minus 11119873 = 20119872 = 10 1282 2376119890 minus 13119873 = 20119872 = 12 1841 5875119890 minus 16119873 = 20119872 = 14 2167 3624119890 minus 17119873 = 20119872 = 16 2536 5117119890 minus 2055 Example 5 As the final example we consider the time-varying multidelay system defined by (119905) = 119905119909 (119905 minus 1198861 (119905)) + 11990521199092 (119905 minus 1198862 (119905)) + 119906 (119905) 0 le 119905 le 1

119909 (0) = 1119906 (119905) =

119905 0 le 119905 lt 12 12 12 le 119905 le 1(104)

where

1198861 (119905) = 12 0 le 119905 lt 34 14 34 le 119905 le 1

1198862 (119905) = 14 0 le 119905 lt 12 34 12 le 119905 le 1

(105)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 14 2037320480 + 121199052 + 11321199053 minus 1161199054 + 1101199055 14 le 119905 lt 12 4819161440 + 12119905 + 9161199052 minus 161199053 + 181199054 12 le 119905 lt 34 12897431966080 + 12119905 + 14287409601199052 + 1873841199053 minus 12561199054 + 1241199055 + 1481199056 34 le 119905 le 1(106)

Although the above-mentioned system involves multiplepiecewise constant delays the proposed procedure can beapplied To employ our approach we select 119873 = 4 Bychoosing119872 = 7 the exact solution of 119909(119905) will be obtained6 Conclusion

An efficient and flexible framework has been successfullydeveloped for solving piecewise constant delay systems Thefoundation of the proposed method is based on a hybrid ofblock-pulse functions and Taylorrsquos polynomials The opera-tional matrix of delay has been constructed The operationalmatrices of integration delay and product associated withthe hybrid functions were utilized to transform the problemunder consideration into a system of algebraic equations Itis worth emphasizing that the exact solutions of Examples1 2 4 and 5 cannot be produced solely either by block-pulse functions or by Taylorrsquos polynomials After determiningthe suitable value of 119873 the number of subintervals smallvalues for 119872 the degree of the Taylorrsquos polynomials areneeded to achieve a specific accuracy The simulation resultsdemonstrate the effectiveness of the suggested procedure Inaddition the proposed numerical scheme can be extended

to a class of nonlinear piecewise constant delay systems butsome modifications are required

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this manuscript

References

[1] RDDriverOrdinary and delay differential equations Springer-Verlag New York NY USA 1977

[2] M Malek-Zavarei and M Jamshidi Time-delay systems vol 9of North-Holland Systems and Control Series North-HollandPublishing Co Amsterdam Netherlands 1987

[3] K B Datta and B M Mohan Orthogonal functions in systemsand control vol 9 ofAdvanced Series in Electrical and ComputerEngineering World Scientific Publishing Co 1995

[4] A Bellen ldquoOne-step collocation for delay differential equa-tionsrdquo Journal of Computational and Applied Mathematics vol10 no 3 pp 275ndash283 1984

[5] C Hwang and M Y Chen ldquoAnalysis and parameter identifi-cation of time-delay systems via shifted Legendre polynomialsrdquoInternational Journal of Control vol 41 no 2 pp 403ndash415 1985

International Journal of Differential Equations 11

[6] I R Horng and J H Chou ldquoAnalysis parameter estimation andoptimal control of time-delay systems via Chebyshev seriesrdquoInternational Journal of Control vol 41 no 5 pp 1221ndash12341985

[7] A Bellen and M Zennaro ldquoNumerical solution of delaydifferential equations by uniform corrections to an implicitRunge-Kutta methodrdquo Numerische Mathematik vol 47 no 2pp 301ndash316 1985

[8] I-R Horng and J-H Chou ldquoAnalysis and parameter identi-fication of time-delay systems via shifted Jacobi polynomialsrdquoInternational Journal of Control vol 44 no 4 pp 935ndash942 1986

[9] M Razzaghi and M Razzaghi ldquoTaylor series analysis of time-varying multi-delay systemsrdquo International Journal of Controlvol 50 no 1 pp 183ndash192 1989

[10] B M Mohan and K B Datta ldquoAnalysis of linear time-invarianttime-delay systems via orthogonal functionsrdquo InternationalJournal of Systems Science vol 26 no 1 pp 91ndash111 1995

[11] A Bellen and S Maset ldquoNumerical solution of constantcoefficient linear delay differential equations as abstract Cauchyproblemsrdquo Numerische Mathematik vol 84 no 3 pp 351ndash3742000

[12] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering vol 7 no 4 pp 337ndash353 2001

[13] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004

[14] H R Marzban and M Razzaghi ldquoAnalysis of time-delaysystems via hybrid of block-pulse functions and Taylor seriesrdquoJournal of Vibration and Control vol 11 no 12 pp 1455ndash14682005

[15] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006

[16] H R Marzban ldquoOptimal control of linear multi-delay systemsbased on a multi-interval decomposition schemerdquo OptimalControl Applications and Methods vol 37 no 1 pp 190ndash2112016

[17] W L Chen and B S Jeng ldquoAnalysis of piecewise constantdelay systems via block-pulse functionsrdquo International Journalof Systems Science vol 12 no 5 pp 625ndash633 1981

[18] W-L Chen and C-H Meng ldquoA general procedure of solvingthe linear delay system via block pulse functionsrdquo Computersand Electrical Engineering vol 9 no 3-4 pp 153ndash166 1982

[19] HRMarzban andM Shahsiah ldquoSolution of piecewise constantdelay systems using hybrid of block-pulse and Chebyshevpolynomialsrdquo Optimal Control Applications and Methods vol32 no 6 pp 647ndash659 2011

[20] H R Marzban and S M Hoseini ldquoAnalysis of linear piecewiseconstant delay systems using a hybrid numerical schemerdquoAdvances in Numerical Analysis vol 2016 Article ID 8903184 9pages 2016

[21] H RMarzban and SMHoseini ldquoNumerical treatment of non-linear optimal control problems involving piecewise constantdelayrdquo IMA Journal of Mathematical Control and Information2015

[22] Z H Jiang and W Schaufelberger Block pulse functions andtheir applications in control systems vol 179 of Lecture Notesin Control and Information Sciences Springer Berlin Germany1992

[23] H R Marzban ldquoParameter identification of linear multi-delaysystems via a hybrid of block-pulse functions and Taylorrsquospolynomialsrdquo International Journal of Control 2016

[24] P LancasterTheory of matrices Academic Press New York NYUSA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Differential Equations

and 119864lowast is defined by

119864lowast = 1119873((((((((

0 1 0 sdot sdot sdot 00 0 12 sdot sdot sdot 0 d0 0 0 sdot sdot sdot 1119872 minus 10 0 0 sdot sdot sdot 0

))))))))

(23)

It should be noted that 119864lowast is the operational matrix ofintegration associated with the Taylorrsquos polynomials on theinterval [((119899minus1)119873)119905119891 (119899119873)119905119891) Furthermore if119862 is a vectorof order 119872119873 times 1 then the expression Φ(119905)Φ119879(119905)119862 can berepresented in terms of hybrid functions asΦ (119905)Φ119879 (119905) 119862 ≃ Φ (119905) (24)

The 119872119873 times 119872119873 matrix is called the operational matrix ofproduct [14]

3 Derivation of the OperationalMatrix of Delay

In this section we shall derive the operational matrix ofdelay associated with the hybrid of block-pulse functionsand Taylorrsquos polynomials For this purpose we employ aprocedure analogous to the one developed in [19] for thehybrid of block-pulse functions and Chebyshev polynomialsThe delayed vectorΦ(119905 minus 119886(119905)) can be represented in terms ofΦ(119905) as follows Φ (119905 minus 119886 (119905)) = 119863Φ (119905) (25)

in which 119863 is called the operational matrix of delay Inaddition 119886(119905) is a piecewise constant function defined by

119886 (119905) =

1205911 1198790 le 119905 lt 11987911205912 1198791 le 119905 lt 1198792 120591119903 119879119903minus1 le 119905 lt 119879119903(26)

where 0 = 1198790 lt 1198791 lt sdot sdot sdot lt 119879119903minus1 lt 119879119903 = 119905119891 It isfurther supposed that 120591119894 119894 = 1 2 119903 are arbitrary rationalnumbers in the interval [0 119905119891) and 119879119894 isin Q 119894 = 0 1 119903 Toconstruct119863 we first obtain the value of119873 Let119860 = 119894 120591119894 = 0 (27)

and take 120574 as the smallest positive integer number such thatthe following conditions are satisfied120574120591119894 isin Z120574119879119895 isin Z119894 isin 119860 119895 = 1 2 119903 (28)

Assume that |119860| = 119896 Next we define 120582 as the greatestcommon divisor of the integers 120574120591119894 and 120574119879119895 119894 isin 119860 and119895 = 1 2 119903 that is120582 = gcd (1205741205911 1205741205912 120574120591119896 1205741198791 1205741198792 120574119879119903) (29)

Define

119873 = 120574120582119905119891 if

120574120582119905119891 isin Z[ 120574120582119905119891] + 1 otherwise (30)

where [(120574120582)119905119891] denotes the greatest integer value less thanor equal to (120574120582)119905119891 It is worth noting that 119873 is selected insuch a way that the number of subintervals is minimized Asa consequence we obtain the following subintervals

[0 120582120574) [120582120574 2120582120574) [(119873 minus 1) 120582120574 119873120582120574) (31)

For convenience we use the notations defined by

ℎ = 120582120574 119873119895 = 120574120582 (119879119895 minus 119879119895minus1) 119897119895 = 120574120582120591119895 119895 = 1 2 119903

(32)

Clearly sum119903119895=1119873119895 = 119873 Consequently we have119886 (119905) = 119896119894ℎ (119894 minus 1) ℎ le 119905 lt 119894ℎ 119894 = 1 2 119873 (33)

in which

119896119894 =

1198971 1 le 119894 le 11987311198972 1198731 + 1 le 119894 le 1198731 + 1198732 119897119895 119895minus1sum119896=1

119873119896 + 1 le 119894 le 119895sum119896=1

119873119896 119897119903 119903minus1sum119896=1

119873119896 + 1 le 119894 le 119903sum119896=1

119873119896

(34)

As a result the problem is reduced to find the delay opera-tional matrix for the delay function defined by

Φ (119905 minus 119886 (119905)) =

Φ(119905 minus 1198961ℎ) 0 le 119905 lt 1199051Φ (119905 minus 1198962ℎ) 1199051 le 119905 lt 1199052 Φ (119905 minus 119896119873ℎ) 119905119899minus1 le 119905 lt 119905119873(35)

International Journal of Differential Equations 5

where 119905119894 = 119894ℎ 119894 = 1 2 119873 (36)

To construct the matrix 119863 we first derive the matrix 119863119894 for119894 = 1 2 119873 in such a way that the following relation issatisfied Φ(119905 minus 119896119894ℎ) = 119863119894Φ (119905) 119905119894minus1 le 119905 lt 119905119894 (37)

By the properties of the hybrid functions it is easy to verifythat for the case 119905119894minus1 le 119905 lt 119905119894 the only functions with nonzerovalues are 120601(119905minus119896119894)119898(119905minus119896119894ℎ) for119898 = 0 1 119872minus1 Notice that

120601(119905minus119896119894)119898 (119905 minus 119896119894ℎ) = 120601119894119898 (119905) 119898 = 0 1 119872 minus 1 (38)

Accordingly 120601(119905minus119896119894)119898(119905minus119896119894ℎ) can be written in terms of 120601119894119898(119905)as 119863119894 = 119878119894 otimes 119868119872 119894 = 1 2 119873 (39)

where 119868119872 is the119872-dimensional identitymatrix andotimes denotesthe Kronecker product [24] In addition 119878119894 is an119873times119873matrixin which the only nonzero entry is equal to one and locatedat the (119894 minus 119896119894)th row and 119894th column Finally if we expressΦ(119905 minus 119886(119905)) in terms of Φ(119905) we find119863 = 1198631 + 1198632 + sdot sdot sdot + 119863119873 (40)

Remark 1 If 119894 minus 119896119894 le 0 then 119878119894 is a zero matrix of order119873times119873 It is worth noting that the structure of the operationalmatrix of delay associatedwith the proposed hybrid functionsis exactly the same as the structure of the hybrid of block-pulse functions and Chebyshev polynomials

4 Problem Statement and Its Approximation

Consider the time-varying piecewise constant delay systemdescribed by

(119905) = 119864 (119905)119883 (119905) + 119865 (119905)119883 (119905 minus 119886 (119905)) + 119866 (119905) 119880 (119905) 0 le 119905 le 119905119891 (41)

119883 (0) = 1198830 (42)119883 (119905) = 0 119905 lt 0 (43)

119886 (119905) =

1205911 1198790 le 119905 lt 11987911205912 1198791 le 119905 lt 1198792 120591119903 119879119903minus1 le 119905 lt 119879119903(44)

where119883(119905) isin R119897 119880(119905) isin R119902 119864(119905) 119865(119905) and119866(119905) arematricesof appropriate dimensions 1198830 is a constant specified vectorand 120591119894 119894 = 1 2 119903 are arbitrary rational numbers in [0 119905119891)The objective is to find119883(119905) 0 le 119905 le 119905119891 satisfying (41)ndash(43)

41 Description of the Method Let119883(119905) = [1199091 (119905) 1199092 (119905) 119909119897 (119905)]119879 119880 (119905) = [1199061 (119905) 1199062 (119905) 119906119902 (119905)]119879 (45)

Assume that each 119909119894(119905) and each of 119906119895(119905) 119894 = 1 2 119902 canbe written in terms of hybrid functions as119909119894 (119905) = Φ119879 (119905) 119883119894119906119895 (119905) = Φ119879 (119905) 119880119895 (46)

Consequently 119883(119905) = (119868119897 otimes Φ119879 (119905))119883119880 (119905) = (119868119902 otimes Φ119879 (119905))119880 (47)

where 119868119897 and 119868119902 are identity matrices of order 119897 times 119897 and 119902 times119902 respectively Furthermore 119883 and 119880 are vectors of order119897119873119872 times 1 and 119902119873119872 times 1 respectively given by

119883 = [1198831198791 1198831198792 119883119879119897 ]119879 119880 = [1198801198791 1198801198792 119880119879119902 ]119879 (48)

Also 119883 (0) = (119868119897 otimes Φ119879 (119905)) 119889 (49)

where 119889 is a vector of order 119897119873119872 times 1 defined by

119889 = [1198891198791 1198891198792 119889119879119897 ]119879 (50)

Similarly 119864(119905) 119865(119905) and 119866(119905) can be represented in terms ofhybrid functions as119864 (119905) = 119864119879 (119868119897 otimes Φ (119905)) 119865 (119905) = 119865119879 (119868119897 otimes Φ (119905)) 119866 (119905) = 119866119879 (119868119902 otimes Φ (119905)) (51)

where 119864119879 119865119879 and 119866119879 are of dimensions 119897 times 119897119873119872 119897 times 119897119873119872and 119897 times 119902119873119872 respectivelyThe delayed vector119883(119905 minus 119886(119905)) canalso be expressed in terms of hybrid functions as119883 (119905 minus 119886 (119905)) = (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119863119879)119883 (52)

in which119863 is the delay operational matrix presented by (25)As a result119864 (119905)119883 (119905) = 119864119879 (119868119897 otimes Φ (119905)) (119868119897 otimes Φ119879 (119905))119883= (119868l otimes Φ119879 (119905)) 119879119883119865 (119905)119883 (119905 minus 119886 (119905))= 119865119879 (119868119897 otimes Φ (119905)) (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119863119879)119883= (119868119897 otimes Φ119879 (119905)) 119879 (119868119897 otimes 119863119879)119883

6 International Journal of Differential Equations

119866 (119905)119880 (119905) = 119866119879 (119868119902 otimes Φ (119905)) (119868119902 otimes Φ119879 (119905))119880= (119868119897 otimes Φ119879 (119905)) 119879119880

(53)

where and can be determined in a manner analogousto the derivation process of the matrix described by (24)Furthermore

int1199050(119868119897 otimes Φ119879 (119904)) 119889119904 = (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879)

int1199050119865 (119904)119883 (119904 minus 119886 (119904)) 119889119904= (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)119883

(54)

in which 119875 is the operational matrix of integration describedby (20) By integrating both sides of (41) with respect to 119905 andusing (47) (49) (51) (52) and (54) we obtain

(119868119897 otimes Φ119879 (119905))119883 minus (119868119897 otimes Φ119879 (119905)) 119889= (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879119883+ (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)119883+ (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879119880(55)

From the preceding equation we deduce that

119883 = [119868119872119873 minus (119868119897 otimes 119875119879) 119879minus (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)]minus1 [119889 + (119868119897 otimes 119875119879) 119879119880] (56)

5 Simulation Results

In this section five examples of varying complexity areinvestigated to evaluate the usefulness and effectiveness ofthe proposed computational techniqueThe correct choice of119873 would greatly improve the efficiency and accuracy of thepresented procedure

51 Example 1 Consider the following piecewise constantdelay system [17] (119905) = 119909 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (57)119909 (119905) = 0 119905 le 0 (58)119906 (119905) = 1 119905 ge 0 (59)

119886 (119905) = 01 0 le 119905 lt 03503 035 le 119905 lt 0705 07 le 119905 le 1 (60)

The exact solution to this problem is given by

119909 (119905) =

119905 0 le 119905 lt 011200 + 910119905 + 121199052 01 le 119905 lt 02113000 + 2325119905 + 251199052 + 161199053 02 le 119905 lt 03961240000 + 18312000119905 + 1694001199052 + 7601199053 + 1241199054 03 le 119905 lt 0352901613840000 + 710119905 + 121199052 035 le 119905 lt 04830671280000 + 3950119905 + 3101199052 + 161199053 04 le 119905 lt 052592013840000 + 9111200119905 + 29801199052 + 1121199053 + 1241199054 05 le 119905 lt 06641781796000000 + 2293730000119905 + 68920001199052 + 171501199053 + 1601199054 + 11201199055 06 le 119905 lt 065107046748000000 + 34965613840000119905 + 151199052 + 161199053 065 le 119905 lt 0737190303192000000 + 37376000119905 + 1294001199052 + 1201199053 + 1241199054 07 le 119905 lt 08244440112800000 + 639910000119905 + 167960001199052 + 313001199053 + 11201199054 + 11201199055 08 le 119905 lt 085710323764000000 + 32661613840000119905 + 1101199052 + 161199053 085 le 119905 lt 09885283764000000 + 27996013840000119905 + 1214001199052 + 1601199053 + 1241199054 09 le 119905 le 1

(61)

International Journal of Differential Equations 7

To apply the method developed in the current paper wefirst choose the value of119873 with the use of (30) Because 120574 =20 and 120582 = 1 hence we select 119873 = 20 We also take 119872 = 6Let

119909 (119905) = 119862119879Φ (119905) (62)

where

119862 = [11988810 1198881119872minus1 1198881198730 119888119873119872minus1]119879 Φ (119905) = [12060110 (119905) 1206011119872minus1 (119905) 1206011198730 (119905) 120601119873119872minus1 (119905)]119879 (63)

By expanding 119906(119905) in terms of hybrid functions we get

119906 (119905) = 1198891198791Φ (119905) (64)

We also have

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (65)

where 119863 is the delay operational matrix given by (25)Integrating both sides of (57) with respect to 119905 and using (62)ndash(65) yield

119862119879 = 1198891198791119875 (119868119872119873 minus 119863119875)minus1 (66)

in which 119868119872119873 is the 119872119873-dimensional identity matrix and 119875is the operational matrix of integration given by (20) Withthe use of (66) and (62) we would obtain the same value asthe exact value of 119909(119905)52 Example 2 Consider the following two-dimensionaltime-varying piecewise constant delay system

(1 (119905)2 (119905)) = ( 1 1 + 1199052119905 1199052 )(1199091 (119905 minus 119886 (119905))1199092 (119905 minus 119886 (119905)))+ (12)119906 (119905) 0 le 119905 le 1 (67)

(1199091 (119905)1199092 (119905)) = (00) 119905 lt 0 (68)

(1199091 (0)1199092 (0)) = (11) (69)

119906 (119905) = 119905 0 le 119905 lt 051 minus 119905 05 le 119905 lt 1 (70)

119886 (119905) = 02 0 le 119905 lt 0307 03 le 119905 le 1 (71)

The exact solutions to this problem are given as follows

1199091 (119905) =

1 + 121199052 0 le 119905 lt 0243197500 + 10350 119905 + 18251199052 + 11301199053 + 141199054 02 le 119905 lt 03147071120000 + 121199052 03 le 119905 lt 05117071120000 + 119905 minus 121199052 05 le 119905 lt 07minus1033112000 + 747200119905 minus 1612001199052 + 1301199053 + 141199054 07 le 119905 lt 09minus 4245584594200000000 + 1148571500000 119905 minus 35588320000001199052 minus 4681300001199053 + 176740001199054 + 717501199055 minus 71201199056 + 1351199057 09 le 119905 le 1

1199092 (119905) =

1 + 1199052 0 le 119905 lt 02179473187500 + 10150 1199052 + 16751199053 + 3201199054 + 151199055 02 le 119905 lt 0331693513000000 + 1199052 03 le 119905 lt 0516693513000000 + 2119905 minus 1199052 05 le 119905 lt 07minus 2204813000000 + 2119905 + 492001199052 + 31001199053 minus 1101199054 + 151199055 07 le 119905 lt 093646033938184000000000 + 2119905 minus 63163400001199052 + 437221730000001199053 minus 11101400001199054 + 1981150001199055 + 1919001199056 minus 111401199057 + 1401199058 09 le 119905 le 1

(72)

8 International Journal of Differential Equations

To solve this problem by the proposed approach we firstdetermine the value of119873with the use of (30) Because 120574 = 10and 120582 = 1 consequently we select 119873 = 10 Also we choose119872 = 9 Let

1199091 (119905) = 1198621198791Φ (119905) 1199092 (119905) = 1198621198792Φ (119905) (73)

Expanding 1 1 + 119905 2119905 and 1199052 in terms of hybrid functionsimplies

1 = 1198911198791 Φ (119905) 1 + 119905 = 1198911198792 Φ (119905)

2119905 = 1198911198793 Φ (119905) 1199052 = 1198911198794 Φ (119905)

(74)

Also using (70) we obtain

int1199050119906 (120591) 119889120591=

121199052 = 1198911198795 Φ (119905) 0 le 119905 lt 05minus14 + 119905 minus 121199052 = 1198911198796 Φ (119905) 05 le 119905 lt 1

(75)

Moreover

1199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905) (76)

Therefore(1 + 119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198912= 11986211987921198632Φ (119905) 21199051199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905)Φ119879 (119905) 1198913= 11986211987911198633Φ (119905) 11990521199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198914= 11986211987921198634Φ (119905) (77)

in which 2 3 and 4 can be obtained in a way similar tothe construction method of the matrix defined by (24) Byintegrating both sides of (67) with respect to 119905 and using (68)(70) (71) (73) (74) (75) (76) and (77) we get

1198621198791 (119868119872119873 minus 119863119875) minus 11986211987921198632119875 = 1198911198791 + 1198911198795 + 1198911198796 1198621198792 (119868119872119873 minus 1198634119875) minus 11986211987911198633119875 = 1198911198791 + 21198911198795 + 21198911198796 (78)

By solving the resulting system described by (78) and thenusing (73) we would obtain the same values as the exactvalues of 1199091(119905) and 1199092(119905)53 Example 3 Consider the following time-delay system[3] (119905) = minus5119909 (119905) minus 5119909 (119905 minus 119886 (119905)) + 2119906 (119905) 0 le 119905 le 2 (79)119909 (0) = 1 (80)119906 (119905) = 1 119905 ge 0 (81)

119886 (119905) = 0 0 le 119905 lt 0803 08 le 119905 lt 1406 14 le 119905 le 1709 17 le 119905 le 2

(82)

The exact solution to this problem is given by [3]

119909 (119905) =

02 + 08119890minus10119905 0 le 119905 lt 0802 + 081198903minus10119905 + 08119890minus4 (1 minus 1198903) 119890minus5119905 08 le 119905 lt 1102 + 081198905minus10119905 minus 4119890minus25 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (65119890minus25 + 119890minus4) 119890minus5119905 11 le 119905 lt 1402 + 081198909minus10119905 minus 4119890minus1 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (8119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 14 le 119905 lt 1702 + 0811989012minus10119905 minus 411989005 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (9511989005 minus 05119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 17 le 119905 le 2(83)

To employ the proposed method we first determine thesuitable value of119873 Using (30) it follows that119873 = 120574120582119905119891 = 20 (84)

Assume that119872 is an arbitrary positive integer number Let

119909 (119905) = 119862119879Φ (119905) (85)

International Journal of Differential Equations 9

Similarly

119909 (0) = 119906 (119905) = 119889119879Φ (119905) 119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (86)

By integrating both sides of (79) with respect to 119905 and using(85)-(86) we conclude that119862119879 = (119889119879 + 2119889119879119875) (119868119872119873 + 5119875 + 5119863119875)minus1 (87)

where 119868119872119873 is the119872119873-dimensional identity matrix Define119864 = max 1003816100381610038161003816119909119890 (119905) minus 119909 (119905)1003816100381610038161003816 0 le 119905 le 1 (88)

in which 119909(119905) and 119909119890(119905) denote the approximate solutiondetermined by the present approach and the exact solutionrespectively In Table 1 the numerical results of themaximumerror denoted by 119864 for 119873 = 20 and various values of 119872are reported The simulation results obtained by the currentapproach show a significant improvement in the accuracy ofthe solution compared to the method developed in [19]

54 Example 4 As a more complicated problem considerthe following problem

(119905) = 1199051199092 (119905 minus 119886 (119905)) + 1199093 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (89)

119909 (119905) = 0 119905 lt 0 (90)119909 (0) = 1 (91)

119906 (119905) = 119905 0 le 119905 lt 0404 04 le 119905 le 1 (92)

119886 (119905) = 04 0 le 119905 lt 0808 08 le 119905 le 1 (93)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 046632511640625 + 2593315625119905 minus 72962501199052 + 1172501199053 minus 31001199054 + 131001199055 minus 11201199056 + 1561199057 04 le 119905 lt 082356911640625 + 4218715625119905 minus 762362501199052 + 2532501199053 minus 431001199054 + 231001199055 minus 71201199056 + 1561199057 08 le 119905 le 1

(94)

Although the above-mentioned problem is a nonlinear delaydifferential equation the method developed in the presentpaper can easily be implemented Because 120574 = 5 and 120582 = 2we take119873 = 3 Furthermore we take119872 = 8 Let

119909 (119905) = 119862119879Φ (119905) (95)

Similarly

119909 (0) = V1198791Φ (119905) 119905 = V1198792Φ (119905) (96)

int1199050119906 (120591) 119889120591 =

121199052 = V1198793Φ (119905) 0 le 119905 lt 04minus 225 + 25119905 = V1198794Φ (119905) 04 le 119905 lt 1 (97)

where V1 V2 V3 and V4 are vectors of order119872119873times1 Also wehave

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (98)1199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) 119863119879119862= 119862119879119863Φ (119905) (99)

1199093 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905)119882= 1198621198791198632Φ (119905) (100)

in which

119882 = 119863119879119862 (101)

Using (99) gives

1199051199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) V2= 119862119879119863V2Φ (119905) (102)

Integrating both sides of (89) with respect to 119905 and using (95)(97) (98) (99) (100) and (102) imply that

119862119879 (119868119872119873 minus 119863V2119875 minus 1198632119875) minus (V1198791 + V1198793 + V1198794 )= 0 (103)

By solving (103) and using (95) the exact value of 119909(119905)will beobtained

10 International Journal of Differential Equations

Table 1 Computational results of maximum error for Example 3Present method CPU time (s) 119864119873 = 20119872 = 1 0112 1165119890 minus 02119873 = 20119872 = 2 0202 3231119890 minus 03119873 = 20119872 = 4 0398 2573119890 minus 06119873 = 20119872 = 6 0769 4836119890 minus 08119873 = 20119872 = 8 0853 3942119890 minus 11119873 = 20119872 = 10 1282 2376119890 minus 13119873 = 20119872 = 12 1841 5875119890 minus 16119873 = 20119872 = 14 2167 3624119890 minus 17119873 = 20119872 = 16 2536 5117119890 minus 2055 Example 5 As the final example we consider the time-varying multidelay system defined by (119905) = 119905119909 (119905 minus 1198861 (119905)) + 11990521199092 (119905 minus 1198862 (119905)) + 119906 (119905) 0 le 119905 le 1

119909 (0) = 1119906 (119905) =

119905 0 le 119905 lt 12 12 12 le 119905 le 1(104)

where

1198861 (119905) = 12 0 le 119905 lt 34 14 34 le 119905 le 1

1198862 (119905) = 14 0 le 119905 lt 12 34 12 le 119905 le 1

(105)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 14 2037320480 + 121199052 + 11321199053 minus 1161199054 + 1101199055 14 le 119905 lt 12 4819161440 + 12119905 + 9161199052 minus 161199053 + 181199054 12 le 119905 lt 34 12897431966080 + 12119905 + 14287409601199052 + 1873841199053 minus 12561199054 + 1241199055 + 1481199056 34 le 119905 le 1(106)

Although the above-mentioned system involves multiplepiecewise constant delays the proposed procedure can beapplied To employ our approach we select 119873 = 4 Bychoosing119872 = 7 the exact solution of 119909(119905) will be obtained6 Conclusion

An efficient and flexible framework has been successfullydeveloped for solving piecewise constant delay systems Thefoundation of the proposed method is based on a hybrid ofblock-pulse functions and Taylorrsquos polynomials The opera-tional matrix of delay has been constructed The operationalmatrices of integration delay and product associated withthe hybrid functions were utilized to transform the problemunder consideration into a system of algebraic equations Itis worth emphasizing that the exact solutions of Examples1 2 4 and 5 cannot be produced solely either by block-pulse functions or by Taylorrsquos polynomials After determiningthe suitable value of 119873 the number of subintervals smallvalues for 119872 the degree of the Taylorrsquos polynomials areneeded to achieve a specific accuracy The simulation resultsdemonstrate the effectiveness of the suggested procedure Inaddition the proposed numerical scheme can be extended

to a class of nonlinear piecewise constant delay systems butsome modifications are required

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this manuscript

References

[1] RDDriverOrdinary and delay differential equations Springer-Verlag New York NY USA 1977

[2] M Malek-Zavarei and M Jamshidi Time-delay systems vol 9of North-Holland Systems and Control Series North-HollandPublishing Co Amsterdam Netherlands 1987

[3] K B Datta and B M Mohan Orthogonal functions in systemsand control vol 9 ofAdvanced Series in Electrical and ComputerEngineering World Scientific Publishing Co 1995

[4] A Bellen ldquoOne-step collocation for delay differential equa-tionsrdquo Journal of Computational and Applied Mathematics vol10 no 3 pp 275ndash283 1984

[5] C Hwang and M Y Chen ldquoAnalysis and parameter identifi-cation of time-delay systems via shifted Legendre polynomialsrdquoInternational Journal of Control vol 41 no 2 pp 403ndash415 1985

International Journal of Differential Equations 11

[6] I R Horng and J H Chou ldquoAnalysis parameter estimation andoptimal control of time-delay systems via Chebyshev seriesrdquoInternational Journal of Control vol 41 no 5 pp 1221ndash12341985

[7] A Bellen and M Zennaro ldquoNumerical solution of delaydifferential equations by uniform corrections to an implicitRunge-Kutta methodrdquo Numerische Mathematik vol 47 no 2pp 301ndash316 1985

[8] I-R Horng and J-H Chou ldquoAnalysis and parameter identi-fication of time-delay systems via shifted Jacobi polynomialsrdquoInternational Journal of Control vol 44 no 4 pp 935ndash942 1986

[9] M Razzaghi and M Razzaghi ldquoTaylor series analysis of time-varying multi-delay systemsrdquo International Journal of Controlvol 50 no 1 pp 183ndash192 1989

[10] B M Mohan and K B Datta ldquoAnalysis of linear time-invarianttime-delay systems via orthogonal functionsrdquo InternationalJournal of Systems Science vol 26 no 1 pp 91ndash111 1995

[11] A Bellen and S Maset ldquoNumerical solution of constantcoefficient linear delay differential equations as abstract Cauchyproblemsrdquo Numerische Mathematik vol 84 no 3 pp 351ndash3742000

[12] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering vol 7 no 4 pp 337ndash353 2001

[13] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004

[14] H R Marzban and M Razzaghi ldquoAnalysis of time-delaysystems via hybrid of block-pulse functions and Taylor seriesrdquoJournal of Vibration and Control vol 11 no 12 pp 1455ndash14682005

[15] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006

[16] H R Marzban ldquoOptimal control of linear multi-delay systemsbased on a multi-interval decomposition schemerdquo OptimalControl Applications and Methods vol 37 no 1 pp 190ndash2112016

[17] W L Chen and B S Jeng ldquoAnalysis of piecewise constantdelay systems via block-pulse functionsrdquo International Journalof Systems Science vol 12 no 5 pp 625ndash633 1981

[18] W-L Chen and C-H Meng ldquoA general procedure of solvingthe linear delay system via block pulse functionsrdquo Computersand Electrical Engineering vol 9 no 3-4 pp 153ndash166 1982

[19] HRMarzban andM Shahsiah ldquoSolution of piecewise constantdelay systems using hybrid of block-pulse and Chebyshevpolynomialsrdquo Optimal Control Applications and Methods vol32 no 6 pp 647ndash659 2011

[20] H R Marzban and S M Hoseini ldquoAnalysis of linear piecewiseconstant delay systems using a hybrid numerical schemerdquoAdvances in Numerical Analysis vol 2016 Article ID 8903184 9pages 2016

[21] H RMarzban and SMHoseini ldquoNumerical treatment of non-linear optimal control problems involving piecewise constantdelayrdquo IMA Journal of Mathematical Control and Information2015

[22] Z H Jiang and W Schaufelberger Block pulse functions andtheir applications in control systems vol 179 of Lecture Notesin Control and Information Sciences Springer Berlin Germany1992

[23] H R Marzban ldquoParameter identification of linear multi-delaysystems via a hybrid of block-pulse functions and Taylorrsquospolynomialsrdquo International Journal of Control 2016

[24] P LancasterTheory of matrices Academic Press New York NYUSA 1969

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 5

where 119905119894 = 119894ℎ 119894 = 1 2 119873 (36)

To construct the matrix 119863 we first derive the matrix 119863119894 for119894 = 1 2 119873 in such a way that the following relation issatisfied Φ(119905 minus 119896119894ℎ) = 119863119894Φ (119905) 119905119894minus1 le 119905 lt 119905119894 (37)

By the properties of the hybrid functions it is easy to verifythat for the case 119905119894minus1 le 119905 lt 119905119894 the only functions with nonzerovalues are 120601(119905minus119896119894)119898(119905minus119896119894ℎ) for119898 = 0 1 119872minus1 Notice that

120601(119905minus119896119894)119898 (119905 minus 119896119894ℎ) = 120601119894119898 (119905) 119898 = 0 1 119872 minus 1 (38)

Accordingly 120601(119905minus119896119894)119898(119905minus119896119894ℎ) can be written in terms of 120601119894119898(119905)as 119863119894 = 119878119894 otimes 119868119872 119894 = 1 2 119873 (39)

where 119868119872 is the119872-dimensional identitymatrix andotimes denotesthe Kronecker product [24] In addition 119878119894 is an119873times119873matrixin which the only nonzero entry is equal to one and locatedat the (119894 minus 119896119894)th row and 119894th column Finally if we expressΦ(119905 minus 119886(119905)) in terms of Φ(119905) we find119863 = 1198631 + 1198632 + sdot sdot sdot + 119863119873 (40)

Remark 1 If 119894 minus 119896119894 le 0 then 119878119894 is a zero matrix of order119873times119873 It is worth noting that the structure of the operationalmatrix of delay associatedwith the proposed hybrid functionsis exactly the same as the structure of the hybrid of block-pulse functions and Chebyshev polynomials

4 Problem Statement and Its Approximation

Consider the time-varying piecewise constant delay systemdescribed by

(119905) = 119864 (119905)119883 (119905) + 119865 (119905)119883 (119905 minus 119886 (119905)) + 119866 (119905) 119880 (119905) 0 le 119905 le 119905119891 (41)

119883 (0) = 1198830 (42)119883 (119905) = 0 119905 lt 0 (43)

119886 (119905) =

1205911 1198790 le 119905 lt 11987911205912 1198791 le 119905 lt 1198792 120591119903 119879119903minus1 le 119905 lt 119879119903(44)

where119883(119905) isin R119897 119880(119905) isin R119902 119864(119905) 119865(119905) and119866(119905) arematricesof appropriate dimensions 1198830 is a constant specified vectorand 120591119894 119894 = 1 2 119903 are arbitrary rational numbers in [0 119905119891)The objective is to find119883(119905) 0 le 119905 le 119905119891 satisfying (41)ndash(43)

41 Description of the Method Let119883(119905) = [1199091 (119905) 1199092 (119905) 119909119897 (119905)]119879 119880 (119905) = [1199061 (119905) 1199062 (119905) 119906119902 (119905)]119879 (45)

Assume that each 119909119894(119905) and each of 119906119895(119905) 119894 = 1 2 119902 canbe written in terms of hybrid functions as119909119894 (119905) = Φ119879 (119905) 119883119894119906119895 (119905) = Φ119879 (119905) 119880119895 (46)

Consequently 119883(119905) = (119868119897 otimes Φ119879 (119905))119883119880 (119905) = (119868119902 otimes Φ119879 (119905))119880 (47)

where 119868119897 and 119868119902 are identity matrices of order 119897 times 119897 and 119902 times119902 respectively Furthermore 119883 and 119880 are vectors of order119897119873119872 times 1 and 119902119873119872 times 1 respectively given by

119883 = [1198831198791 1198831198792 119883119879119897 ]119879 119880 = [1198801198791 1198801198792 119880119879119902 ]119879 (48)

Also 119883 (0) = (119868119897 otimes Φ119879 (119905)) 119889 (49)

where 119889 is a vector of order 119897119873119872 times 1 defined by

119889 = [1198891198791 1198891198792 119889119879119897 ]119879 (50)

Similarly 119864(119905) 119865(119905) and 119866(119905) can be represented in terms ofhybrid functions as119864 (119905) = 119864119879 (119868119897 otimes Φ (119905)) 119865 (119905) = 119865119879 (119868119897 otimes Φ (119905)) 119866 (119905) = 119866119879 (119868119902 otimes Φ (119905)) (51)

where 119864119879 119865119879 and 119866119879 are of dimensions 119897 times 119897119873119872 119897 times 119897119873119872and 119897 times 119902119873119872 respectivelyThe delayed vector119883(119905 minus 119886(119905)) canalso be expressed in terms of hybrid functions as119883 (119905 minus 119886 (119905)) = (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119863119879)119883 (52)

in which119863 is the delay operational matrix presented by (25)As a result119864 (119905)119883 (119905) = 119864119879 (119868119897 otimes Φ (119905)) (119868119897 otimes Φ119879 (119905))119883= (119868l otimes Φ119879 (119905)) 119879119883119865 (119905)119883 (119905 minus 119886 (119905))= 119865119879 (119868119897 otimes Φ (119905)) (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119863119879)119883= (119868119897 otimes Φ119879 (119905)) 119879 (119868119897 otimes 119863119879)119883

6 International Journal of Differential Equations

119866 (119905)119880 (119905) = 119866119879 (119868119902 otimes Φ (119905)) (119868119902 otimes Φ119879 (119905))119880= (119868119897 otimes Φ119879 (119905)) 119879119880

(53)

where and can be determined in a manner analogousto the derivation process of the matrix described by (24)Furthermore

int1199050(119868119897 otimes Φ119879 (119904)) 119889119904 = (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879)

int1199050119865 (119904)119883 (119904 minus 119886 (119904)) 119889119904= (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)119883

(54)

in which 119875 is the operational matrix of integration describedby (20) By integrating both sides of (41) with respect to 119905 andusing (47) (49) (51) (52) and (54) we obtain

(119868119897 otimes Φ119879 (119905))119883 minus (119868119897 otimes Φ119879 (119905)) 119889= (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879119883+ (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)119883+ (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879119880(55)

From the preceding equation we deduce that

119883 = [119868119872119873 minus (119868119897 otimes 119875119879) 119879minus (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)]minus1 [119889 + (119868119897 otimes 119875119879) 119879119880] (56)

5 Simulation Results

In this section five examples of varying complexity areinvestigated to evaluate the usefulness and effectiveness ofthe proposed computational techniqueThe correct choice of119873 would greatly improve the efficiency and accuracy of thepresented procedure

51 Example 1 Consider the following piecewise constantdelay system [17] (119905) = 119909 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (57)119909 (119905) = 0 119905 le 0 (58)119906 (119905) = 1 119905 ge 0 (59)

119886 (119905) = 01 0 le 119905 lt 03503 035 le 119905 lt 0705 07 le 119905 le 1 (60)

The exact solution to this problem is given by

119909 (119905) =

119905 0 le 119905 lt 011200 + 910119905 + 121199052 01 le 119905 lt 02113000 + 2325119905 + 251199052 + 161199053 02 le 119905 lt 03961240000 + 18312000119905 + 1694001199052 + 7601199053 + 1241199054 03 le 119905 lt 0352901613840000 + 710119905 + 121199052 035 le 119905 lt 04830671280000 + 3950119905 + 3101199052 + 161199053 04 le 119905 lt 052592013840000 + 9111200119905 + 29801199052 + 1121199053 + 1241199054 05 le 119905 lt 06641781796000000 + 2293730000119905 + 68920001199052 + 171501199053 + 1601199054 + 11201199055 06 le 119905 lt 065107046748000000 + 34965613840000119905 + 151199052 + 161199053 065 le 119905 lt 0737190303192000000 + 37376000119905 + 1294001199052 + 1201199053 + 1241199054 07 le 119905 lt 08244440112800000 + 639910000119905 + 167960001199052 + 313001199053 + 11201199054 + 11201199055 08 le 119905 lt 085710323764000000 + 32661613840000119905 + 1101199052 + 161199053 085 le 119905 lt 09885283764000000 + 27996013840000119905 + 1214001199052 + 1601199053 + 1241199054 09 le 119905 le 1

(61)

International Journal of Differential Equations 7

To apply the method developed in the current paper wefirst choose the value of119873 with the use of (30) Because 120574 =20 and 120582 = 1 hence we select 119873 = 20 We also take 119872 = 6Let

119909 (119905) = 119862119879Φ (119905) (62)

where

119862 = [11988810 1198881119872minus1 1198881198730 119888119873119872minus1]119879 Φ (119905) = [12060110 (119905) 1206011119872minus1 (119905) 1206011198730 (119905) 120601119873119872minus1 (119905)]119879 (63)

By expanding 119906(119905) in terms of hybrid functions we get

119906 (119905) = 1198891198791Φ (119905) (64)

We also have

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (65)

where 119863 is the delay operational matrix given by (25)Integrating both sides of (57) with respect to 119905 and using (62)ndash(65) yield

119862119879 = 1198891198791119875 (119868119872119873 minus 119863119875)minus1 (66)

in which 119868119872119873 is the 119872119873-dimensional identity matrix and 119875is the operational matrix of integration given by (20) Withthe use of (66) and (62) we would obtain the same value asthe exact value of 119909(119905)52 Example 2 Consider the following two-dimensionaltime-varying piecewise constant delay system

(1 (119905)2 (119905)) = ( 1 1 + 1199052119905 1199052 )(1199091 (119905 minus 119886 (119905))1199092 (119905 minus 119886 (119905)))+ (12)119906 (119905) 0 le 119905 le 1 (67)

(1199091 (119905)1199092 (119905)) = (00) 119905 lt 0 (68)

(1199091 (0)1199092 (0)) = (11) (69)

119906 (119905) = 119905 0 le 119905 lt 051 minus 119905 05 le 119905 lt 1 (70)

119886 (119905) = 02 0 le 119905 lt 0307 03 le 119905 le 1 (71)

The exact solutions to this problem are given as follows

1199091 (119905) =

1 + 121199052 0 le 119905 lt 0243197500 + 10350 119905 + 18251199052 + 11301199053 + 141199054 02 le 119905 lt 03147071120000 + 121199052 03 le 119905 lt 05117071120000 + 119905 minus 121199052 05 le 119905 lt 07minus1033112000 + 747200119905 minus 1612001199052 + 1301199053 + 141199054 07 le 119905 lt 09minus 4245584594200000000 + 1148571500000 119905 minus 35588320000001199052 minus 4681300001199053 + 176740001199054 + 717501199055 minus 71201199056 + 1351199057 09 le 119905 le 1

1199092 (119905) =

1 + 1199052 0 le 119905 lt 02179473187500 + 10150 1199052 + 16751199053 + 3201199054 + 151199055 02 le 119905 lt 0331693513000000 + 1199052 03 le 119905 lt 0516693513000000 + 2119905 minus 1199052 05 le 119905 lt 07minus 2204813000000 + 2119905 + 492001199052 + 31001199053 minus 1101199054 + 151199055 07 le 119905 lt 093646033938184000000000 + 2119905 minus 63163400001199052 + 437221730000001199053 minus 11101400001199054 + 1981150001199055 + 1919001199056 minus 111401199057 + 1401199058 09 le 119905 le 1

(72)

8 International Journal of Differential Equations

To solve this problem by the proposed approach we firstdetermine the value of119873with the use of (30) Because 120574 = 10and 120582 = 1 consequently we select 119873 = 10 Also we choose119872 = 9 Let

1199091 (119905) = 1198621198791Φ (119905) 1199092 (119905) = 1198621198792Φ (119905) (73)

Expanding 1 1 + 119905 2119905 and 1199052 in terms of hybrid functionsimplies

1 = 1198911198791 Φ (119905) 1 + 119905 = 1198911198792 Φ (119905)

2119905 = 1198911198793 Φ (119905) 1199052 = 1198911198794 Φ (119905)

(74)

Also using (70) we obtain

int1199050119906 (120591) 119889120591=

121199052 = 1198911198795 Φ (119905) 0 le 119905 lt 05minus14 + 119905 minus 121199052 = 1198911198796 Φ (119905) 05 le 119905 lt 1

(75)

Moreover

1199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905) (76)

Therefore(1 + 119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198912= 11986211987921198632Φ (119905) 21199051199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905)Φ119879 (119905) 1198913= 11986211987911198633Φ (119905) 11990521199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198914= 11986211987921198634Φ (119905) (77)

in which 2 3 and 4 can be obtained in a way similar tothe construction method of the matrix defined by (24) Byintegrating both sides of (67) with respect to 119905 and using (68)(70) (71) (73) (74) (75) (76) and (77) we get

1198621198791 (119868119872119873 minus 119863119875) minus 11986211987921198632119875 = 1198911198791 + 1198911198795 + 1198911198796 1198621198792 (119868119872119873 minus 1198634119875) minus 11986211987911198633119875 = 1198911198791 + 21198911198795 + 21198911198796 (78)

By solving the resulting system described by (78) and thenusing (73) we would obtain the same values as the exactvalues of 1199091(119905) and 1199092(119905)53 Example 3 Consider the following time-delay system[3] (119905) = minus5119909 (119905) minus 5119909 (119905 minus 119886 (119905)) + 2119906 (119905) 0 le 119905 le 2 (79)119909 (0) = 1 (80)119906 (119905) = 1 119905 ge 0 (81)

119886 (119905) = 0 0 le 119905 lt 0803 08 le 119905 lt 1406 14 le 119905 le 1709 17 le 119905 le 2

(82)

The exact solution to this problem is given by [3]

119909 (119905) =

02 + 08119890minus10119905 0 le 119905 lt 0802 + 081198903minus10119905 + 08119890minus4 (1 minus 1198903) 119890minus5119905 08 le 119905 lt 1102 + 081198905minus10119905 minus 4119890minus25 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (65119890minus25 + 119890minus4) 119890minus5119905 11 le 119905 lt 1402 + 081198909minus10119905 minus 4119890minus1 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (8119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 14 le 119905 lt 1702 + 0811989012minus10119905 minus 411989005 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (9511989005 minus 05119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 17 le 119905 le 2(83)

To employ the proposed method we first determine thesuitable value of119873 Using (30) it follows that119873 = 120574120582119905119891 = 20 (84)

Assume that119872 is an arbitrary positive integer number Let

119909 (119905) = 119862119879Φ (119905) (85)

International Journal of Differential Equations 9

Similarly

119909 (0) = 119906 (119905) = 119889119879Φ (119905) 119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (86)

By integrating both sides of (79) with respect to 119905 and using(85)-(86) we conclude that119862119879 = (119889119879 + 2119889119879119875) (119868119872119873 + 5119875 + 5119863119875)minus1 (87)

where 119868119872119873 is the119872119873-dimensional identity matrix Define119864 = max 1003816100381610038161003816119909119890 (119905) minus 119909 (119905)1003816100381610038161003816 0 le 119905 le 1 (88)

in which 119909(119905) and 119909119890(119905) denote the approximate solutiondetermined by the present approach and the exact solutionrespectively In Table 1 the numerical results of themaximumerror denoted by 119864 for 119873 = 20 and various values of 119872are reported The simulation results obtained by the currentapproach show a significant improvement in the accuracy ofthe solution compared to the method developed in [19]

54 Example 4 As a more complicated problem considerthe following problem

(119905) = 1199051199092 (119905 minus 119886 (119905)) + 1199093 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (89)

119909 (119905) = 0 119905 lt 0 (90)119909 (0) = 1 (91)

119906 (119905) = 119905 0 le 119905 lt 0404 04 le 119905 le 1 (92)

119886 (119905) = 04 0 le 119905 lt 0808 08 le 119905 le 1 (93)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 046632511640625 + 2593315625119905 minus 72962501199052 + 1172501199053 minus 31001199054 + 131001199055 minus 11201199056 + 1561199057 04 le 119905 lt 082356911640625 + 4218715625119905 minus 762362501199052 + 2532501199053 minus 431001199054 + 231001199055 minus 71201199056 + 1561199057 08 le 119905 le 1

(94)

Although the above-mentioned problem is a nonlinear delaydifferential equation the method developed in the presentpaper can easily be implemented Because 120574 = 5 and 120582 = 2we take119873 = 3 Furthermore we take119872 = 8 Let

119909 (119905) = 119862119879Φ (119905) (95)

Similarly

119909 (0) = V1198791Φ (119905) 119905 = V1198792Φ (119905) (96)

int1199050119906 (120591) 119889120591 =

121199052 = V1198793Φ (119905) 0 le 119905 lt 04minus 225 + 25119905 = V1198794Φ (119905) 04 le 119905 lt 1 (97)

where V1 V2 V3 and V4 are vectors of order119872119873times1 Also wehave

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (98)1199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) 119863119879119862= 119862119879119863Φ (119905) (99)

1199093 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905)119882= 1198621198791198632Φ (119905) (100)

in which

119882 = 119863119879119862 (101)

Using (99) gives

1199051199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) V2= 119862119879119863V2Φ (119905) (102)

Integrating both sides of (89) with respect to 119905 and using (95)(97) (98) (99) (100) and (102) imply that

119862119879 (119868119872119873 minus 119863V2119875 minus 1198632119875) minus (V1198791 + V1198793 + V1198794 )= 0 (103)

By solving (103) and using (95) the exact value of 119909(119905)will beobtained

10 International Journal of Differential Equations

Table 1 Computational results of maximum error for Example 3Present method CPU time (s) 119864119873 = 20119872 = 1 0112 1165119890 minus 02119873 = 20119872 = 2 0202 3231119890 minus 03119873 = 20119872 = 4 0398 2573119890 minus 06119873 = 20119872 = 6 0769 4836119890 minus 08119873 = 20119872 = 8 0853 3942119890 minus 11119873 = 20119872 = 10 1282 2376119890 minus 13119873 = 20119872 = 12 1841 5875119890 minus 16119873 = 20119872 = 14 2167 3624119890 minus 17119873 = 20119872 = 16 2536 5117119890 minus 2055 Example 5 As the final example we consider the time-varying multidelay system defined by (119905) = 119905119909 (119905 minus 1198861 (119905)) + 11990521199092 (119905 minus 1198862 (119905)) + 119906 (119905) 0 le 119905 le 1

119909 (0) = 1119906 (119905) =

119905 0 le 119905 lt 12 12 12 le 119905 le 1(104)

where

1198861 (119905) = 12 0 le 119905 lt 34 14 34 le 119905 le 1

1198862 (119905) = 14 0 le 119905 lt 12 34 12 le 119905 le 1

(105)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 14 2037320480 + 121199052 + 11321199053 minus 1161199054 + 1101199055 14 le 119905 lt 12 4819161440 + 12119905 + 9161199052 minus 161199053 + 181199054 12 le 119905 lt 34 12897431966080 + 12119905 + 14287409601199052 + 1873841199053 minus 12561199054 + 1241199055 + 1481199056 34 le 119905 le 1(106)

Although the above-mentioned system involves multiplepiecewise constant delays the proposed procedure can beapplied To employ our approach we select 119873 = 4 Bychoosing119872 = 7 the exact solution of 119909(119905) will be obtained6 Conclusion

An efficient and flexible framework has been successfullydeveloped for solving piecewise constant delay systems Thefoundation of the proposed method is based on a hybrid ofblock-pulse functions and Taylorrsquos polynomials The opera-tional matrix of delay has been constructed The operationalmatrices of integration delay and product associated withthe hybrid functions were utilized to transform the problemunder consideration into a system of algebraic equations Itis worth emphasizing that the exact solutions of Examples1 2 4 and 5 cannot be produced solely either by block-pulse functions or by Taylorrsquos polynomials After determiningthe suitable value of 119873 the number of subintervals smallvalues for 119872 the degree of the Taylorrsquos polynomials areneeded to achieve a specific accuracy The simulation resultsdemonstrate the effectiveness of the suggested procedure Inaddition the proposed numerical scheme can be extended

to a class of nonlinear piecewise constant delay systems butsome modifications are required

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this manuscript

References

[1] RDDriverOrdinary and delay differential equations Springer-Verlag New York NY USA 1977

[2] M Malek-Zavarei and M Jamshidi Time-delay systems vol 9of North-Holland Systems and Control Series North-HollandPublishing Co Amsterdam Netherlands 1987

[3] K B Datta and B M Mohan Orthogonal functions in systemsand control vol 9 ofAdvanced Series in Electrical and ComputerEngineering World Scientific Publishing Co 1995

[4] A Bellen ldquoOne-step collocation for delay differential equa-tionsrdquo Journal of Computational and Applied Mathematics vol10 no 3 pp 275ndash283 1984

[5] C Hwang and M Y Chen ldquoAnalysis and parameter identifi-cation of time-delay systems via shifted Legendre polynomialsrdquoInternational Journal of Control vol 41 no 2 pp 403ndash415 1985

International Journal of Differential Equations 11

[6] I R Horng and J H Chou ldquoAnalysis parameter estimation andoptimal control of time-delay systems via Chebyshev seriesrdquoInternational Journal of Control vol 41 no 5 pp 1221ndash12341985

[7] A Bellen and M Zennaro ldquoNumerical solution of delaydifferential equations by uniform corrections to an implicitRunge-Kutta methodrdquo Numerische Mathematik vol 47 no 2pp 301ndash316 1985

[8] I-R Horng and J-H Chou ldquoAnalysis and parameter identi-fication of time-delay systems via shifted Jacobi polynomialsrdquoInternational Journal of Control vol 44 no 4 pp 935ndash942 1986

[9] M Razzaghi and M Razzaghi ldquoTaylor series analysis of time-varying multi-delay systemsrdquo International Journal of Controlvol 50 no 1 pp 183ndash192 1989

[10] B M Mohan and K B Datta ldquoAnalysis of linear time-invarianttime-delay systems via orthogonal functionsrdquo InternationalJournal of Systems Science vol 26 no 1 pp 91ndash111 1995

[11] A Bellen and S Maset ldquoNumerical solution of constantcoefficient linear delay differential equations as abstract Cauchyproblemsrdquo Numerische Mathematik vol 84 no 3 pp 351ndash3742000

[12] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering vol 7 no 4 pp 337ndash353 2001

[13] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004

[14] H R Marzban and M Razzaghi ldquoAnalysis of time-delaysystems via hybrid of block-pulse functions and Taylor seriesrdquoJournal of Vibration and Control vol 11 no 12 pp 1455ndash14682005

[15] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006

[16] H R Marzban ldquoOptimal control of linear multi-delay systemsbased on a multi-interval decomposition schemerdquo OptimalControl Applications and Methods vol 37 no 1 pp 190ndash2112016

[17] W L Chen and B S Jeng ldquoAnalysis of piecewise constantdelay systems via block-pulse functionsrdquo International Journalof Systems Science vol 12 no 5 pp 625ndash633 1981

[18] W-L Chen and C-H Meng ldquoA general procedure of solvingthe linear delay system via block pulse functionsrdquo Computersand Electrical Engineering vol 9 no 3-4 pp 153ndash166 1982

[19] HRMarzban andM Shahsiah ldquoSolution of piecewise constantdelay systems using hybrid of block-pulse and Chebyshevpolynomialsrdquo Optimal Control Applications and Methods vol32 no 6 pp 647ndash659 2011

[20] H R Marzban and S M Hoseini ldquoAnalysis of linear piecewiseconstant delay systems using a hybrid numerical schemerdquoAdvances in Numerical Analysis vol 2016 Article ID 8903184 9pages 2016

[21] H RMarzban and SMHoseini ldquoNumerical treatment of non-linear optimal control problems involving piecewise constantdelayrdquo IMA Journal of Mathematical Control and Information2015

[22] Z H Jiang and W Schaufelberger Block pulse functions andtheir applications in control systems vol 179 of Lecture Notesin Control and Information Sciences Springer Berlin Germany1992

[23] H R Marzban ldquoParameter identification of linear multi-delaysystems via a hybrid of block-pulse functions and Taylorrsquospolynomialsrdquo International Journal of Control 2016

[24] P LancasterTheory of matrices Academic Press New York NYUSA 1969

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Stochastic AnalysisInternational Journal of

6 International Journal of Differential Equations

119866 (119905)119880 (119905) = 119866119879 (119868119902 otimes Φ (119905)) (119868119902 otimes Φ119879 (119905))119880= (119868119897 otimes Φ119879 (119905)) 119879119880

(53)

where and can be determined in a manner analogousto the derivation process of the matrix described by (24)Furthermore

int1199050(119868119897 otimes Φ119879 (119904)) 119889119904 = (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879)

int1199050119865 (119904)119883 (119904 minus 119886 (119904)) 119889119904= (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)119883

(54)

in which 119875 is the operational matrix of integration describedby (20) By integrating both sides of (41) with respect to 119905 andusing (47) (49) (51) (52) and (54) we obtain

(119868119897 otimes Φ119879 (119905))119883 minus (119868119897 otimes Φ119879 (119905)) 119889= (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879119883+ (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)119883+ (119868119897 otimes Φ119879 (119905)) (119868119897 otimes 119875119879) 119879119880(55)

From the preceding equation we deduce that

119883 = [119868119872119873 minus (119868119897 otimes 119875119879) 119879minus (119868119897 otimes 119875119879) 119879 (119868119897 otimes 119863119879)]minus1 [119889 + (119868119897 otimes 119875119879) 119879119880] (56)

5 Simulation Results

In this section five examples of varying complexity areinvestigated to evaluate the usefulness and effectiveness ofthe proposed computational techniqueThe correct choice of119873 would greatly improve the efficiency and accuracy of thepresented procedure

51 Example 1 Consider the following piecewise constantdelay system [17] (119905) = 119909 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (57)119909 (119905) = 0 119905 le 0 (58)119906 (119905) = 1 119905 ge 0 (59)

119886 (119905) = 01 0 le 119905 lt 03503 035 le 119905 lt 0705 07 le 119905 le 1 (60)

The exact solution to this problem is given by

119909 (119905) =

119905 0 le 119905 lt 011200 + 910119905 + 121199052 01 le 119905 lt 02113000 + 2325119905 + 251199052 + 161199053 02 le 119905 lt 03961240000 + 18312000119905 + 1694001199052 + 7601199053 + 1241199054 03 le 119905 lt 0352901613840000 + 710119905 + 121199052 035 le 119905 lt 04830671280000 + 3950119905 + 3101199052 + 161199053 04 le 119905 lt 052592013840000 + 9111200119905 + 29801199052 + 1121199053 + 1241199054 05 le 119905 lt 06641781796000000 + 2293730000119905 + 68920001199052 + 171501199053 + 1601199054 + 11201199055 06 le 119905 lt 065107046748000000 + 34965613840000119905 + 151199052 + 161199053 065 le 119905 lt 0737190303192000000 + 37376000119905 + 1294001199052 + 1201199053 + 1241199054 07 le 119905 lt 08244440112800000 + 639910000119905 + 167960001199052 + 313001199053 + 11201199054 + 11201199055 08 le 119905 lt 085710323764000000 + 32661613840000119905 + 1101199052 + 161199053 085 le 119905 lt 09885283764000000 + 27996013840000119905 + 1214001199052 + 1601199053 + 1241199054 09 le 119905 le 1

(61)

International Journal of Differential Equations 7

To apply the method developed in the current paper wefirst choose the value of119873 with the use of (30) Because 120574 =20 and 120582 = 1 hence we select 119873 = 20 We also take 119872 = 6Let

119909 (119905) = 119862119879Φ (119905) (62)

where

119862 = [11988810 1198881119872minus1 1198881198730 119888119873119872minus1]119879 Φ (119905) = [12060110 (119905) 1206011119872minus1 (119905) 1206011198730 (119905) 120601119873119872minus1 (119905)]119879 (63)

By expanding 119906(119905) in terms of hybrid functions we get

119906 (119905) = 1198891198791Φ (119905) (64)

We also have

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (65)

where 119863 is the delay operational matrix given by (25)Integrating both sides of (57) with respect to 119905 and using (62)ndash(65) yield

119862119879 = 1198891198791119875 (119868119872119873 minus 119863119875)minus1 (66)

in which 119868119872119873 is the 119872119873-dimensional identity matrix and 119875is the operational matrix of integration given by (20) Withthe use of (66) and (62) we would obtain the same value asthe exact value of 119909(119905)52 Example 2 Consider the following two-dimensionaltime-varying piecewise constant delay system

(1 (119905)2 (119905)) = ( 1 1 + 1199052119905 1199052 )(1199091 (119905 minus 119886 (119905))1199092 (119905 minus 119886 (119905)))+ (12)119906 (119905) 0 le 119905 le 1 (67)

(1199091 (119905)1199092 (119905)) = (00) 119905 lt 0 (68)

(1199091 (0)1199092 (0)) = (11) (69)

119906 (119905) = 119905 0 le 119905 lt 051 minus 119905 05 le 119905 lt 1 (70)

119886 (119905) = 02 0 le 119905 lt 0307 03 le 119905 le 1 (71)

The exact solutions to this problem are given as follows

1199091 (119905) =

1 + 121199052 0 le 119905 lt 0243197500 + 10350 119905 + 18251199052 + 11301199053 + 141199054 02 le 119905 lt 03147071120000 + 121199052 03 le 119905 lt 05117071120000 + 119905 minus 121199052 05 le 119905 lt 07minus1033112000 + 747200119905 minus 1612001199052 + 1301199053 + 141199054 07 le 119905 lt 09minus 4245584594200000000 + 1148571500000 119905 minus 35588320000001199052 minus 4681300001199053 + 176740001199054 + 717501199055 minus 71201199056 + 1351199057 09 le 119905 le 1

1199092 (119905) =

1 + 1199052 0 le 119905 lt 02179473187500 + 10150 1199052 + 16751199053 + 3201199054 + 151199055 02 le 119905 lt 0331693513000000 + 1199052 03 le 119905 lt 0516693513000000 + 2119905 minus 1199052 05 le 119905 lt 07minus 2204813000000 + 2119905 + 492001199052 + 31001199053 minus 1101199054 + 151199055 07 le 119905 lt 093646033938184000000000 + 2119905 minus 63163400001199052 + 437221730000001199053 minus 11101400001199054 + 1981150001199055 + 1919001199056 minus 111401199057 + 1401199058 09 le 119905 le 1

(72)

8 International Journal of Differential Equations

To solve this problem by the proposed approach we firstdetermine the value of119873with the use of (30) Because 120574 = 10and 120582 = 1 consequently we select 119873 = 10 Also we choose119872 = 9 Let

1199091 (119905) = 1198621198791Φ (119905) 1199092 (119905) = 1198621198792Φ (119905) (73)

Expanding 1 1 + 119905 2119905 and 1199052 in terms of hybrid functionsimplies

1 = 1198911198791 Φ (119905) 1 + 119905 = 1198911198792 Φ (119905)

2119905 = 1198911198793 Φ (119905) 1199052 = 1198911198794 Φ (119905)

(74)

Also using (70) we obtain

int1199050119906 (120591) 119889120591=

121199052 = 1198911198795 Φ (119905) 0 le 119905 lt 05minus14 + 119905 minus 121199052 = 1198911198796 Φ (119905) 05 le 119905 lt 1

(75)

Moreover

1199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905) (76)

Therefore(1 + 119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198912= 11986211987921198632Φ (119905) 21199051199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905)Φ119879 (119905) 1198913= 11986211987911198633Φ (119905) 11990521199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198914= 11986211987921198634Φ (119905) (77)

in which 2 3 and 4 can be obtained in a way similar tothe construction method of the matrix defined by (24) Byintegrating both sides of (67) with respect to 119905 and using (68)(70) (71) (73) (74) (75) (76) and (77) we get

1198621198791 (119868119872119873 minus 119863119875) minus 11986211987921198632119875 = 1198911198791 + 1198911198795 + 1198911198796 1198621198792 (119868119872119873 minus 1198634119875) minus 11986211987911198633119875 = 1198911198791 + 21198911198795 + 21198911198796 (78)

By solving the resulting system described by (78) and thenusing (73) we would obtain the same values as the exactvalues of 1199091(119905) and 1199092(119905)53 Example 3 Consider the following time-delay system[3] (119905) = minus5119909 (119905) minus 5119909 (119905 minus 119886 (119905)) + 2119906 (119905) 0 le 119905 le 2 (79)119909 (0) = 1 (80)119906 (119905) = 1 119905 ge 0 (81)

119886 (119905) = 0 0 le 119905 lt 0803 08 le 119905 lt 1406 14 le 119905 le 1709 17 le 119905 le 2

(82)

The exact solution to this problem is given by [3]

119909 (119905) =

02 + 08119890minus10119905 0 le 119905 lt 0802 + 081198903minus10119905 + 08119890minus4 (1 minus 1198903) 119890minus5119905 08 le 119905 lt 1102 + 081198905minus10119905 minus 4119890minus25 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (65119890minus25 + 119890minus4) 119890minus5119905 11 le 119905 lt 1402 + 081198909minus10119905 minus 4119890minus1 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (8119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 14 le 119905 lt 1702 + 0811989012minus10119905 minus 411989005 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (9511989005 minus 05119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 17 le 119905 le 2(83)

To employ the proposed method we first determine thesuitable value of119873 Using (30) it follows that119873 = 120574120582119905119891 = 20 (84)

Assume that119872 is an arbitrary positive integer number Let

119909 (119905) = 119862119879Φ (119905) (85)

International Journal of Differential Equations 9

Similarly

119909 (0) = 119906 (119905) = 119889119879Φ (119905) 119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (86)

By integrating both sides of (79) with respect to 119905 and using(85)-(86) we conclude that119862119879 = (119889119879 + 2119889119879119875) (119868119872119873 + 5119875 + 5119863119875)minus1 (87)

where 119868119872119873 is the119872119873-dimensional identity matrix Define119864 = max 1003816100381610038161003816119909119890 (119905) minus 119909 (119905)1003816100381610038161003816 0 le 119905 le 1 (88)

in which 119909(119905) and 119909119890(119905) denote the approximate solutiondetermined by the present approach and the exact solutionrespectively In Table 1 the numerical results of themaximumerror denoted by 119864 for 119873 = 20 and various values of 119872are reported The simulation results obtained by the currentapproach show a significant improvement in the accuracy ofthe solution compared to the method developed in [19]

54 Example 4 As a more complicated problem considerthe following problem

(119905) = 1199051199092 (119905 minus 119886 (119905)) + 1199093 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (89)

119909 (119905) = 0 119905 lt 0 (90)119909 (0) = 1 (91)

119906 (119905) = 119905 0 le 119905 lt 0404 04 le 119905 le 1 (92)

119886 (119905) = 04 0 le 119905 lt 0808 08 le 119905 le 1 (93)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 046632511640625 + 2593315625119905 minus 72962501199052 + 1172501199053 minus 31001199054 + 131001199055 minus 11201199056 + 1561199057 04 le 119905 lt 082356911640625 + 4218715625119905 minus 762362501199052 + 2532501199053 minus 431001199054 + 231001199055 minus 71201199056 + 1561199057 08 le 119905 le 1

(94)

Although the above-mentioned problem is a nonlinear delaydifferential equation the method developed in the presentpaper can easily be implemented Because 120574 = 5 and 120582 = 2we take119873 = 3 Furthermore we take119872 = 8 Let

119909 (119905) = 119862119879Φ (119905) (95)

Similarly

119909 (0) = V1198791Φ (119905) 119905 = V1198792Φ (119905) (96)

int1199050119906 (120591) 119889120591 =

121199052 = V1198793Φ (119905) 0 le 119905 lt 04minus 225 + 25119905 = V1198794Φ (119905) 04 le 119905 lt 1 (97)

where V1 V2 V3 and V4 are vectors of order119872119873times1 Also wehave

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (98)1199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) 119863119879119862= 119862119879119863Φ (119905) (99)

1199093 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905)119882= 1198621198791198632Φ (119905) (100)

in which

119882 = 119863119879119862 (101)

Using (99) gives

1199051199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) V2= 119862119879119863V2Φ (119905) (102)

Integrating both sides of (89) with respect to 119905 and using (95)(97) (98) (99) (100) and (102) imply that

119862119879 (119868119872119873 minus 119863V2119875 minus 1198632119875) minus (V1198791 + V1198793 + V1198794 )= 0 (103)

By solving (103) and using (95) the exact value of 119909(119905)will beobtained

10 International Journal of Differential Equations

Table 1 Computational results of maximum error for Example 3Present method CPU time (s) 119864119873 = 20119872 = 1 0112 1165119890 minus 02119873 = 20119872 = 2 0202 3231119890 minus 03119873 = 20119872 = 4 0398 2573119890 minus 06119873 = 20119872 = 6 0769 4836119890 minus 08119873 = 20119872 = 8 0853 3942119890 minus 11119873 = 20119872 = 10 1282 2376119890 minus 13119873 = 20119872 = 12 1841 5875119890 minus 16119873 = 20119872 = 14 2167 3624119890 minus 17119873 = 20119872 = 16 2536 5117119890 minus 2055 Example 5 As the final example we consider the time-varying multidelay system defined by (119905) = 119905119909 (119905 minus 1198861 (119905)) + 11990521199092 (119905 minus 1198862 (119905)) + 119906 (119905) 0 le 119905 le 1

119909 (0) = 1119906 (119905) =

119905 0 le 119905 lt 12 12 12 le 119905 le 1(104)

where

1198861 (119905) = 12 0 le 119905 lt 34 14 34 le 119905 le 1

1198862 (119905) = 14 0 le 119905 lt 12 34 12 le 119905 le 1

(105)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 14 2037320480 + 121199052 + 11321199053 minus 1161199054 + 1101199055 14 le 119905 lt 12 4819161440 + 12119905 + 9161199052 minus 161199053 + 181199054 12 le 119905 lt 34 12897431966080 + 12119905 + 14287409601199052 + 1873841199053 minus 12561199054 + 1241199055 + 1481199056 34 le 119905 le 1(106)

Although the above-mentioned system involves multiplepiecewise constant delays the proposed procedure can beapplied To employ our approach we select 119873 = 4 Bychoosing119872 = 7 the exact solution of 119909(119905) will be obtained6 Conclusion

An efficient and flexible framework has been successfullydeveloped for solving piecewise constant delay systems Thefoundation of the proposed method is based on a hybrid ofblock-pulse functions and Taylorrsquos polynomials The opera-tional matrix of delay has been constructed The operationalmatrices of integration delay and product associated withthe hybrid functions were utilized to transform the problemunder consideration into a system of algebraic equations Itis worth emphasizing that the exact solutions of Examples1 2 4 and 5 cannot be produced solely either by block-pulse functions or by Taylorrsquos polynomials After determiningthe suitable value of 119873 the number of subintervals smallvalues for 119872 the degree of the Taylorrsquos polynomials areneeded to achieve a specific accuracy The simulation resultsdemonstrate the effectiveness of the suggested procedure Inaddition the proposed numerical scheme can be extended

to a class of nonlinear piecewise constant delay systems butsome modifications are required

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this manuscript

References

[1] RDDriverOrdinary and delay differential equations Springer-Verlag New York NY USA 1977

[2] M Malek-Zavarei and M Jamshidi Time-delay systems vol 9of North-Holland Systems and Control Series North-HollandPublishing Co Amsterdam Netherlands 1987

[3] K B Datta and B M Mohan Orthogonal functions in systemsand control vol 9 ofAdvanced Series in Electrical and ComputerEngineering World Scientific Publishing Co 1995

[4] A Bellen ldquoOne-step collocation for delay differential equa-tionsrdquo Journal of Computational and Applied Mathematics vol10 no 3 pp 275ndash283 1984

[5] C Hwang and M Y Chen ldquoAnalysis and parameter identifi-cation of time-delay systems via shifted Legendre polynomialsrdquoInternational Journal of Control vol 41 no 2 pp 403ndash415 1985

International Journal of Differential Equations 11

[6] I R Horng and J H Chou ldquoAnalysis parameter estimation andoptimal control of time-delay systems via Chebyshev seriesrdquoInternational Journal of Control vol 41 no 5 pp 1221ndash12341985

[7] A Bellen and M Zennaro ldquoNumerical solution of delaydifferential equations by uniform corrections to an implicitRunge-Kutta methodrdquo Numerische Mathematik vol 47 no 2pp 301ndash316 1985

[8] I-R Horng and J-H Chou ldquoAnalysis and parameter identi-fication of time-delay systems via shifted Jacobi polynomialsrdquoInternational Journal of Control vol 44 no 4 pp 935ndash942 1986

[9] M Razzaghi and M Razzaghi ldquoTaylor series analysis of time-varying multi-delay systemsrdquo International Journal of Controlvol 50 no 1 pp 183ndash192 1989

[10] B M Mohan and K B Datta ldquoAnalysis of linear time-invarianttime-delay systems via orthogonal functionsrdquo InternationalJournal of Systems Science vol 26 no 1 pp 91ndash111 1995

[11] A Bellen and S Maset ldquoNumerical solution of constantcoefficient linear delay differential equations as abstract Cauchyproblemsrdquo Numerische Mathematik vol 84 no 3 pp 351ndash3742000

[12] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering vol 7 no 4 pp 337ndash353 2001

[13] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004

[14] H R Marzban and M Razzaghi ldquoAnalysis of time-delaysystems via hybrid of block-pulse functions and Taylor seriesrdquoJournal of Vibration and Control vol 11 no 12 pp 1455ndash14682005

[15] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006

[16] H R Marzban ldquoOptimal control of linear multi-delay systemsbased on a multi-interval decomposition schemerdquo OptimalControl Applications and Methods vol 37 no 1 pp 190ndash2112016

[17] W L Chen and B S Jeng ldquoAnalysis of piecewise constantdelay systems via block-pulse functionsrdquo International Journalof Systems Science vol 12 no 5 pp 625ndash633 1981

[18] W-L Chen and C-H Meng ldquoA general procedure of solvingthe linear delay system via block pulse functionsrdquo Computersand Electrical Engineering vol 9 no 3-4 pp 153ndash166 1982

[19] HRMarzban andM Shahsiah ldquoSolution of piecewise constantdelay systems using hybrid of block-pulse and Chebyshevpolynomialsrdquo Optimal Control Applications and Methods vol32 no 6 pp 647ndash659 2011

[20] H R Marzban and S M Hoseini ldquoAnalysis of linear piecewiseconstant delay systems using a hybrid numerical schemerdquoAdvances in Numerical Analysis vol 2016 Article ID 8903184 9pages 2016

[21] H RMarzban and SMHoseini ldquoNumerical treatment of non-linear optimal control problems involving piecewise constantdelayrdquo IMA Journal of Mathematical Control and Information2015

[22] Z H Jiang and W Schaufelberger Block pulse functions andtheir applications in control systems vol 179 of Lecture Notesin Control and Information Sciences Springer Berlin Germany1992

[23] H R Marzban ldquoParameter identification of linear multi-delaysystems via a hybrid of block-pulse functions and Taylorrsquospolynomialsrdquo International Journal of Control 2016

[24] P LancasterTheory of matrices Academic Press New York NYUSA 1969

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 7

To apply the method developed in the current paper wefirst choose the value of119873 with the use of (30) Because 120574 =20 and 120582 = 1 hence we select 119873 = 20 We also take 119872 = 6Let

119909 (119905) = 119862119879Φ (119905) (62)

where

119862 = [11988810 1198881119872minus1 1198881198730 119888119873119872minus1]119879 Φ (119905) = [12060110 (119905) 1206011119872minus1 (119905) 1206011198730 (119905) 120601119873119872minus1 (119905)]119879 (63)

By expanding 119906(119905) in terms of hybrid functions we get

119906 (119905) = 1198891198791Φ (119905) (64)

We also have

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (65)

where 119863 is the delay operational matrix given by (25)Integrating both sides of (57) with respect to 119905 and using (62)ndash(65) yield

119862119879 = 1198891198791119875 (119868119872119873 minus 119863119875)minus1 (66)

in which 119868119872119873 is the 119872119873-dimensional identity matrix and 119875is the operational matrix of integration given by (20) Withthe use of (66) and (62) we would obtain the same value asthe exact value of 119909(119905)52 Example 2 Consider the following two-dimensionaltime-varying piecewise constant delay system

(1 (119905)2 (119905)) = ( 1 1 + 1199052119905 1199052 )(1199091 (119905 minus 119886 (119905))1199092 (119905 minus 119886 (119905)))+ (12)119906 (119905) 0 le 119905 le 1 (67)

(1199091 (119905)1199092 (119905)) = (00) 119905 lt 0 (68)

(1199091 (0)1199092 (0)) = (11) (69)

119906 (119905) = 119905 0 le 119905 lt 051 minus 119905 05 le 119905 lt 1 (70)

119886 (119905) = 02 0 le 119905 lt 0307 03 le 119905 le 1 (71)

The exact solutions to this problem are given as follows

1199091 (119905) =

1 + 121199052 0 le 119905 lt 0243197500 + 10350 119905 + 18251199052 + 11301199053 + 141199054 02 le 119905 lt 03147071120000 + 121199052 03 le 119905 lt 05117071120000 + 119905 minus 121199052 05 le 119905 lt 07minus1033112000 + 747200119905 minus 1612001199052 + 1301199053 + 141199054 07 le 119905 lt 09minus 4245584594200000000 + 1148571500000 119905 minus 35588320000001199052 minus 4681300001199053 + 176740001199054 + 717501199055 minus 71201199056 + 1351199057 09 le 119905 le 1

1199092 (119905) =

1 + 1199052 0 le 119905 lt 02179473187500 + 10150 1199052 + 16751199053 + 3201199054 + 151199055 02 le 119905 lt 0331693513000000 + 1199052 03 le 119905 lt 0516693513000000 + 2119905 minus 1199052 05 le 119905 lt 07minus 2204813000000 + 2119905 + 492001199052 + 31001199053 minus 1101199054 + 151199055 07 le 119905 lt 093646033938184000000000 + 2119905 minus 63163400001199052 + 437221730000001199053 minus 11101400001199054 + 1981150001199055 + 1919001199056 minus 111401199057 + 1401199058 09 le 119905 le 1

(72)

8 International Journal of Differential Equations

To solve this problem by the proposed approach we firstdetermine the value of119873with the use of (30) Because 120574 = 10and 120582 = 1 consequently we select 119873 = 10 Also we choose119872 = 9 Let

1199091 (119905) = 1198621198791Φ (119905) 1199092 (119905) = 1198621198792Φ (119905) (73)

Expanding 1 1 + 119905 2119905 and 1199052 in terms of hybrid functionsimplies

1 = 1198911198791 Φ (119905) 1 + 119905 = 1198911198792 Φ (119905)

2119905 = 1198911198793 Φ (119905) 1199052 = 1198911198794 Φ (119905)

(74)

Also using (70) we obtain

int1199050119906 (120591) 119889120591=

121199052 = 1198911198795 Φ (119905) 0 le 119905 lt 05minus14 + 119905 minus 121199052 = 1198911198796 Φ (119905) 05 le 119905 lt 1

(75)

Moreover

1199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905) (76)

Therefore(1 + 119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198912= 11986211987921198632Φ (119905) 21199051199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905)Φ119879 (119905) 1198913= 11986211987911198633Φ (119905) 11990521199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198914= 11986211987921198634Φ (119905) (77)

in which 2 3 and 4 can be obtained in a way similar tothe construction method of the matrix defined by (24) Byintegrating both sides of (67) with respect to 119905 and using (68)(70) (71) (73) (74) (75) (76) and (77) we get

1198621198791 (119868119872119873 minus 119863119875) minus 11986211987921198632119875 = 1198911198791 + 1198911198795 + 1198911198796 1198621198792 (119868119872119873 minus 1198634119875) minus 11986211987911198633119875 = 1198911198791 + 21198911198795 + 21198911198796 (78)

By solving the resulting system described by (78) and thenusing (73) we would obtain the same values as the exactvalues of 1199091(119905) and 1199092(119905)53 Example 3 Consider the following time-delay system[3] (119905) = minus5119909 (119905) minus 5119909 (119905 minus 119886 (119905)) + 2119906 (119905) 0 le 119905 le 2 (79)119909 (0) = 1 (80)119906 (119905) = 1 119905 ge 0 (81)

119886 (119905) = 0 0 le 119905 lt 0803 08 le 119905 lt 1406 14 le 119905 le 1709 17 le 119905 le 2

(82)

The exact solution to this problem is given by [3]

119909 (119905) =

02 + 08119890minus10119905 0 le 119905 lt 0802 + 081198903minus10119905 + 08119890minus4 (1 minus 1198903) 119890minus5119905 08 le 119905 lt 1102 + 081198905minus10119905 minus 4119890minus25 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (65119890minus25 + 119890minus4) 119890minus5119905 11 le 119905 lt 1402 + 081198909minus10119905 minus 4119890minus1 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (8119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 14 le 119905 lt 1702 + 0811989012minus10119905 minus 411989005 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (9511989005 minus 05119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 17 le 119905 le 2(83)

To employ the proposed method we first determine thesuitable value of119873 Using (30) it follows that119873 = 120574120582119905119891 = 20 (84)

Assume that119872 is an arbitrary positive integer number Let

119909 (119905) = 119862119879Φ (119905) (85)

International Journal of Differential Equations 9

Similarly

119909 (0) = 119906 (119905) = 119889119879Φ (119905) 119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (86)

By integrating both sides of (79) with respect to 119905 and using(85)-(86) we conclude that119862119879 = (119889119879 + 2119889119879119875) (119868119872119873 + 5119875 + 5119863119875)minus1 (87)

where 119868119872119873 is the119872119873-dimensional identity matrix Define119864 = max 1003816100381610038161003816119909119890 (119905) minus 119909 (119905)1003816100381610038161003816 0 le 119905 le 1 (88)

in which 119909(119905) and 119909119890(119905) denote the approximate solutiondetermined by the present approach and the exact solutionrespectively In Table 1 the numerical results of themaximumerror denoted by 119864 for 119873 = 20 and various values of 119872are reported The simulation results obtained by the currentapproach show a significant improvement in the accuracy ofthe solution compared to the method developed in [19]

54 Example 4 As a more complicated problem considerthe following problem

(119905) = 1199051199092 (119905 minus 119886 (119905)) + 1199093 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (89)

119909 (119905) = 0 119905 lt 0 (90)119909 (0) = 1 (91)

119906 (119905) = 119905 0 le 119905 lt 0404 04 le 119905 le 1 (92)

119886 (119905) = 04 0 le 119905 lt 0808 08 le 119905 le 1 (93)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 046632511640625 + 2593315625119905 minus 72962501199052 + 1172501199053 minus 31001199054 + 131001199055 minus 11201199056 + 1561199057 04 le 119905 lt 082356911640625 + 4218715625119905 minus 762362501199052 + 2532501199053 minus 431001199054 + 231001199055 minus 71201199056 + 1561199057 08 le 119905 le 1

(94)

Although the above-mentioned problem is a nonlinear delaydifferential equation the method developed in the presentpaper can easily be implemented Because 120574 = 5 and 120582 = 2we take119873 = 3 Furthermore we take119872 = 8 Let

119909 (119905) = 119862119879Φ (119905) (95)

Similarly

119909 (0) = V1198791Φ (119905) 119905 = V1198792Φ (119905) (96)

int1199050119906 (120591) 119889120591 =

121199052 = V1198793Φ (119905) 0 le 119905 lt 04minus 225 + 25119905 = V1198794Φ (119905) 04 le 119905 lt 1 (97)

where V1 V2 V3 and V4 are vectors of order119872119873times1 Also wehave

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (98)1199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) 119863119879119862= 119862119879119863Φ (119905) (99)

1199093 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905)119882= 1198621198791198632Φ (119905) (100)

in which

119882 = 119863119879119862 (101)

Using (99) gives

1199051199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) V2= 119862119879119863V2Φ (119905) (102)

Integrating both sides of (89) with respect to 119905 and using (95)(97) (98) (99) (100) and (102) imply that

119862119879 (119868119872119873 minus 119863V2119875 minus 1198632119875) minus (V1198791 + V1198793 + V1198794 )= 0 (103)

By solving (103) and using (95) the exact value of 119909(119905)will beobtained

10 International Journal of Differential Equations

Table 1 Computational results of maximum error for Example 3Present method CPU time (s) 119864119873 = 20119872 = 1 0112 1165119890 minus 02119873 = 20119872 = 2 0202 3231119890 minus 03119873 = 20119872 = 4 0398 2573119890 minus 06119873 = 20119872 = 6 0769 4836119890 minus 08119873 = 20119872 = 8 0853 3942119890 minus 11119873 = 20119872 = 10 1282 2376119890 minus 13119873 = 20119872 = 12 1841 5875119890 minus 16119873 = 20119872 = 14 2167 3624119890 minus 17119873 = 20119872 = 16 2536 5117119890 minus 2055 Example 5 As the final example we consider the time-varying multidelay system defined by (119905) = 119905119909 (119905 minus 1198861 (119905)) + 11990521199092 (119905 minus 1198862 (119905)) + 119906 (119905) 0 le 119905 le 1

119909 (0) = 1119906 (119905) =

119905 0 le 119905 lt 12 12 12 le 119905 le 1(104)

where

1198861 (119905) = 12 0 le 119905 lt 34 14 34 le 119905 le 1

1198862 (119905) = 14 0 le 119905 lt 12 34 12 le 119905 le 1

(105)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 14 2037320480 + 121199052 + 11321199053 minus 1161199054 + 1101199055 14 le 119905 lt 12 4819161440 + 12119905 + 9161199052 minus 161199053 + 181199054 12 le 119905 lt 34 12897431966080 + 12119905 + 14287409601199052 + 1873841199053 minus 12561199054 + 1241199055 + 1481199056 34 le 119905 le 1(106)

Although the above-mentioned system involves multiplepiecewise constant delays the proposed procedure can beapplied To employ our approach we select 119873 = 4 Bychoosing119872 = 7 the exact solution of 119909(119905) will be obtained6 Conclusion

An efficient and flexible framework has been successfullydeveloped for solving piecewise constant delay systems Thefoundation of the proposed method is based on a hybrid ofblock-pulse functions and Taylorrsquos polynomials The opera-tional matrix of delay has been constructed The operationalmatrices of integration delay and product associated withthe hybrid functions were utilized to transform the problemunder consideration into a system of algebraic equations Itis worth emphasizing that the exact solutions of Examples1 2 4 and 5 cannot be produced solely either by block-pulse functions or by Taylorrsquos polynomials After determiningthe suitable value of 119873 the number of subintervals smallvalues for 119872 the degree of the Taylorrsquos polynomials areneeded to achieve a specific accuracy The simulation resultsdemonstrate the effectiveness of the suggested procedure Inaddition the proposed numerical scheme can be extended

to a class of nonlinear piecewise constant delay systems butsome modifications are required

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this manuscript

References

[1] RDDriverOrdinary and delay differential equations Springer-Verlag New York NY USA 1977

[2] M Malek-Zavarei and M Jamshidi Time-delay systems vol 9of North-Holland Systems and Control Series North-HollandPublishing Co Amsterdam Netherlands 1987

[3] K B Datta and B M Mohan Orthogonal functions in systemsand control vol 9 ofAdvanced Series in Electrical and ComputerEngineering World Scientific Publishing Co 1995

[4] A Bellen ldquoOne-step collocation for delay differential equa-tionsrdquo Journal of Computational and Applied Mathematics vol10 no 3 pp 275ndash283 1984

[5] C Hwang and M Y Chen ldquoAnalysis and parameter identifi-cation of time-delay systems via shifted Legendre polynomialsrdquoInternational Journal of Control vol 41 no 2 pp 403ndash415 1985

International Journal of Differential Equations 11

[6] I R Horng and J H Chou ldquoAnalysis parameter estimation andoptimal control of time-delay systems via Chebyshev seriesrdquoInternational Journal of Control vol 41 no 5 pp 1221ndash12341985

[7] A Bellen and M Zennaro ldquoNumerical solution of delaydifferential equations by uniform corrections to an implicitRunge-Kutta methodrdquo Numerische Mathematik vol 47 no 2pp 301ndash316 1985

[8] I-R Horng and J-H Chou ldquoAnalysis and parameter identi-fication of time-delay systems via shifted Jacobi polynomialsrdquoInternational Journal of Control vol 44 no 4 pp 935ndash942 1986

[9] M Razzaghi and M Razzaghi ldquoTaylor series analysis of time-varying multi-delay systemsrdquo International Journal of Controlvol 50 no 1 pp 183ndash192 1989

[10] B M Mohan and K B Datta ldquoAnalysis of linear time-invarianttime-delay systems via orthogonal functionsrdquo InternationalJournal of Systems Science vol 26 no 1 pp 91ndash111 1995

[11] A Bellen and S Maset ldquoNumerical solution of constantcoefficient linear delay differential equations as abstract Cauchyproblemsrdquo Numerische Mathematik vol 84 no 3 pp 351ndash3742000

[12] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering vol 7 no 4 pp 337ndash353 2001

[13] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004

[14] H R Marzban and M Razzaghi ldquoAnalysis of time-delaysystems via hybrid of block-pulse functions and Taylor seriesrdquoJournal of Vibration and Control vol 11 no 12 pp 1455ndash14682005

[15] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006

[16] H R Marzban ldquoOptimal control of linear multi-delay systemsbased on a multi-interval decomposition schemerdquo OptimalControl Applications and Methods vol 37 no 1 pp 190ndash2112016

[17] W L Chen and B S Jeng ldquoAnalysis of piecewise constantdelay systems via block-pulse functionsrdquo International Journalof Systems Science vol 12 no 5 pp 625ndash633 1981

[18] W-L Chen and C-H Meng ldquoA general procedure of solvingthe linear delay system via block pulse functionsrdquo Computersand Electrical Engineering vol 9 no 3-4 pp 153ndash166 1982

[19] HRMarzban andM Shahsiah ldquoSolution of piecewise constantdelay systems using hybrid of block-pulse and Chebyshevpolynomialsrdquo Optimal Control Applications and Methods vol32 no 6 pp 647ndash659 2011

[20] H R Marzban and S M Hoseini ldquoAnalysis of linear piecewiseconstant delay systems using a hybrid numerical schemerdquoAdvances in Numerical Analysis vol 2016 Article ID 8903184 9pages 2016

[21] H RMarzban and SMHoseini ldquoNumerical treatment of non-linear optimal control problems involving piecewise constantdelayrdquo IMA Journal of Mathematical Control and Information2015

[22] Z H Jiang and W Schaufelberger Block pulse functions andtheir applications in control systems vol 179 of Lecture Notesin Control and Information Sciences Springer Berlin Germany1992

[23] H R Marzban ldquoParameter identification of linear multi-delaysystems via a hybrid of block-pulse functions and Taylorrsquospolynomialsrdquo International Journal of Control 2016

[24] P LancasterTheory of matrices Academic Press New York NYUSA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Differential Equations

To solve this problem by the proposed approach we firstdetermine the value of119873with the use of (30) Because 120574 = 10and 120582 = 1 consequently we select 119873 = 10 Also we choose119872 = 9 Let

1199091 (119905) = 1198621198791Φ (119905) 1199092 (119905) = 1198621198792Φ (119905) (73)

Expanding 1 1 + 119905 2119905 and 1199052 in terms of hybrid functionsimplies

1 = 1198911198791 Φ (119905) 1 + 119905 = 1198911198792 Φ (119905)

2119905 = 1198911198793 Φ (119905) 1199052 = 1198911198794 Φ (119905)

(74)

Also using (70) we obtain

int1199050119906 (120591) 119889120591=

121199052 = 1198911198795 Φ (119905) 0 le 119905 lt 05minus14 + 119905 minus 121199052 = 1198911198796 Φ (119905) 05 le 119905 lt 1

(75)

Moreover

1199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905) (76)

Therefore(1 + 119905) 1199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198912= 11986211987921198632Φ (119905) 21199051199091 (119905 minus 119886 (119905)) = 1198621198791119863Φ (119905)Φ119879 (119905) 1198913= 11986211987911198633Φ (119905) 11990521199092 (119905 minus 119886 (119905)) = 1198621198792119863Φ (119905)Φ119879 (119905) 1198914= 11986211987921198634Φ (119905) (77)

in which 2 3 and 4 can be obtained in a way similar tothe construction method of the matrix defined by (24) Byintegrating both sides of (67) with respect to 119905 and using (68)(70) (71) (73) (74) (75) (76) and (77) we get

1198621198791 (119868119872119873 minus 119863119875) minus 11986211987921198632119875 = 1198911198791 + 1198911198795 + 1198911198796 1198621198792 (119868119872119873 minus 1198634119875) minus 11986211987911198633119875 = 1198911198791 + 21198911198795 + 21198911198796 (78)

By solving the resulting system described by (78) and thenusing (73) we would obtain the same values as the exactvalues of 1199091(119905) and 1199092(119905)53 Example 3 Consider the following time-delay system[3] (119905) = minus5119909 (119905) minus 5119909 (119905 minus 119886 (119905)) + 2119906 (119905) 0 le 119905 le 2 (79)119909 (0) = 1 (80)119906 (119905) = 1 119905 ge 0 (81)

119886 (119905) = 0 0 le 119905 lt 0803 08 le 119905 lt 1406 14 le 119905 le 1709 17 le 119905 le 2

(82)

The exact solution to this problem is given by [3]

119909 (119905) =

02 + 08119890minus10119905 0 le 119905 lt 0802 + 081198903minus10119905 + 08119890minus4 (1 minus 1198903) 119890minus5119905 08 le 119905 lt 1102 + 081198905minus10119905 minus 4119890minus25 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (65119890minus25 + 119890minus4) 119890minus5119905 11 le 119905 lt 1402 + 081198909minus10119905 minus 4119890minus1 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (8119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 14 le 119905 lt 1702 + 0811989012minus10119905 minus 411989005 (1 minus 1198903) 119905119890minus5119905 + 08 (1 minus 1198903) (9511989005 minus 05119890minus1 minus 05119890minus25 + 119890minus4) 119890minus5119905 17 le 119905 le 2(83)

To employ the proposed method we first determine thesuitable value of119873 Using (30) it follows that119873 = 120574120582119905119891 = 20 (84)

Assume that119872 is an arbitrary positive integer number Let

119909 (119905) = 119862119879Φ (119905) (85)

International Journal of Differential Equations 9

Similarly

119909 (0) = 119906 (119905) = 119889119879Φ (119905) 119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (86)

By integrating both sides of (79) with respect to 119905 and using(85)-(86) we conclude that119862119879 = (119889119879 + 2119889119879119875) (119868119872119873 + 5119875 + 5119863119875)minus1 (87)

where 119868119872119873 is the119872119873-dimensional identity matrix Define119864 = max 1003816100381610038161003816119909119890 (119905) minus 119909 (119905)1003816100381610038161003816 0 le 119905 le 1 (88)

in which 119909(119905) and 119909119890(119905) denote the approximate solutiondetermined by the present approach and the exact solutionrespectively In Table 1 the numerical results of themaximumerror denoted by 119864 for 119873 = 20 and various values of 119872are reported The simulation results obtained by the currentapproach show a significant improvement in the accuracy ofthe solution compared to the method developed in [19]

54 Example 4 As a more complicated problem considerthe following problem

(119905) = 1199051199092 (119905 minus 119886 (119905)) + 1199093 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (89)

119909 (119905) = 0 119905 lt 0 (90)119909 (0) = 1 (91)

119906 (119905) = 119905 0 le 119905 lt 0404 04 le 119905 le 1 (92)

119886 (119905) = 04 0 le 119905 lt 0808 08 le 119905 le 1 (93)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 046632511640625 + 2593315625119905 minus 72962501199052 + 1172501199053 minus 31001199054 + 131001199055 minus 11201199056 + 1561199057 04 le 119905 lt 082356911640625 + 4218715625119905 minus 762362501199052 + 2532501199053 minus 431001199054 + 231001199055 minus 71201199056 + 1561199057 08 le 119905 le 1

(94)

Although the above-mentioned problem is a nonlinear delaydifferential equation the method developed in the presentpaper can easily be implemented Because 120574 = 5 and 120582 = 2we take119873 = 3 Furthermore we take119872 = 8 Let

119909 (119905) = 119862119879Φ (119905) (95)

Similarly

119909 (0) = V1198791Φ (119905) 119905 = V1198792Φ (119905) (96)

int1199050119906 (120591) 119889120591 =

121199052 = V1198793Φ (119905) 0 le 119905 lt 04minus 225 + 25119905 = V1198794Φ (119905) 04 le 119905 lt 1 (97)

where V1 V2 V3 and V4 are vectors of order119872119873times1 Also wehave

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (98)1199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) 119863119879119862= 119862119879119863Φ (119905) (99)

1199093 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905)119882= 1198621198791198632Φ (119905) (100)

in which

119882 = 119863119879119862 (101)

Using (99) gives

1199051199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) V2= 119862119879119863V2Φ (119905) (102)

Integrating both sides of (89) with respect to 119905 and using (95)(97) (98) (99) (100) and (102) imply that

119862119879 (119868119872119873 minus 119863V2119875 minus 1198632119875) minus (V1198791 + V1198793 + V1198794 )= 0 (103)

By solving (103) and using (95) the exact value of 119909(119905)will beobtained

10 International Journal of Differential Equations

Table 1 Computational results of maximum error for Example 3Present method CPU time (s) 119864119873 = 20119872 = 1 0112 1165119890 minus 02119873 = 20119872 = 2 0202 3231119890 minus 03119873 = 20119872 = 4 0398 2573119890 minus 06119873 = 20119872 = 6 0769 4836119890 minus 08119873 = 20119872 = 8 0853 3942119890 minus 11119873 = 20119872 = 10 1282 2376119890 minus 13119873 = 20119872 = 12 1841 5875119890 minus 16119873 = 20119872 = 14 2167 3624119890 minus 17119873 = 20119872 = 16 2536 5117119890 minus 2055 Example 5 As the final example we consider the time-varying multidelay system defined by (119905) = 119905119909 (119905 minus 1198861 (119905)) + 11990521199092 (119905 minus 1198862 (119905)) + 119906 (119905) 0 le 119905 le 1

119909 (0) = 1119906 (119905) =

119905 0 le 119905 lt 12 12 12 le 119905 le 1(104)

where

1198861 (119905) = 12 0 le 119905 lt 34 14 34 le 119905 le 1

1198862 (119905) = 14 0 le 119905 lt 12 34 12 le 119905 le 1

(105)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 14 2037320480 + 121199052 + 11321199053 minus 1161199054 + 1101199055 14 le 119905 lt 12 4819161440 + 12119905 + 9161199052 minus 161199053 + 181199054 12 le 119905 lt 34 12897431966080 + 12119905 + 14287409601199052 + 1873841199053 minus 12561199054 + 1241199055 + 1481199056 34 le 119905 le 1(106)

Although the above-mentioned system involves multiplepiecewise constant delays the proposed procedure can beapplied To employ our approach we select 119873 = 4 Bychoosing119872 = 7 the exact solution of 119909(119905) will be obtained6 Conclusion

An efficient and flexible framework has been successfullydeveloped for solving piecewise constant delay systems Thefoundation of the proposed method is based on a hybrid ofblock-pulse functions and Taylorrsquos polynomials The opera-tional matrix of delay has been constructed The operationalmatrices of integration delay and product associated withthe hybrid functions were utilized to transform the problemunder consideration into a system of algebraic equations Itis worth emphasizing that the exact solutions of Examples1 2 4 and 5 cannot be produced solely either by block-pulse functions or by Taylorrsquos polynomials After determiningthe suitable value of 119873 the number of subintervals smallvalues for 119872 the degree of the Taylorrsquos polynomials areneeded to achieve a specific accuracy The simulation resultsdemonstrate the effectiveness of the suggested procedure Inaddition the proposed numerical scheme can be extended

to a class of nonlinear piecewise constant delay systems butsome modifications are required

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this manuscript

References

[1] RDDriverOrdinary and delay differential equations Springer-Verlag New York NY USA 1977

[2] M Malek-Zavarei and M Jamshidi Time-delay systems vol 9of North-Holland Systems and Control Series North-HollandPublishing Co Amsterdam Netherlands 1987

[3] K B Datta and B M Mohan Orthogonal functions in systemsand control vol 9 ofAdvanced Series in Electrical and ComputerEngineering World Scientific Publishing Co 1995

[4] A Bellen ldquoOne-step collocation for delay differential equa-tionsrdquo Journal of Computational and Applied Mathematics vol10 no 3 pp 275ndash283 1984

[5] C Hwang and M Y Chen ldquoAnalysis and parameter identifi-cation of time-delay systems via shifted Legendre polynomialsrdquoInternational Journal of Control vol 41 no 2 pp 403ndash415 1985

International Journal of Differential Equations 11

[6] I R Horng and J H Chou ldquoAnalysis parameter estimation andoptimal control of time-delay systems via Chebyshev seriesrdquoInternational Journal of Control vol 41 no 5 pp 1221ndash12341985

[7] A Bellen and M Zennaro ldquoNumerical solution of delaydifferential equations by uniform corrections to an implicitRunge-Kutta methodrdquo Numerische Mathematik vol 47 no 2pp 301ndash316 1985

[8] I-R Horng and J-H Chou ldquoAnalysis and parameter identi-fication of time-delay systems via shifted Jacobi polynomialsrdquoInternational Journal of Control vol 44 no 4 pp 935ndash942 1986

[9] M Razzaghi and M Razzaghi ldquoTaylor series analysis of time-varying multi-delay systemsrdquo International Journal of Controlvol 50 no 1 pp 183ndash192 1989

[10] B M Mohan and K B Datta ldquoAnalysis of linear time-invarianttime-delay systems via orthogonal functionsrdquo InternationalJournal of Systems Science vol 26 no 1 pp 91ndash111 1995

[11] A Bellen and S Maset ldquoNumerical solution of constantcoefficient linear delay differential equations as abstract Cauchyproblemsrdquo Numerische Mathematik vol 84 no 3 pp 351ndash3742000

[12] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering vol 7 no 4 pp 337ndash353 2001

[13] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004

[14] H R Marzban and M Razzaghi ldquoAnalysis of time-delaysystems via hybrid of block-pulse functions and Taylor seriesrdquoJournal of Vibration and Control vol 11 no 12 pp 1455ndash14682005

[15] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006

[16] H R Marzban ldquoOptimal control of linear multi-delay systemsbased on a multi-interval decomposition schemerdquo OptimalControl Applications and Methods vol 37 no 1 pp 190ndash2112016

[17] W L Chen and B S Jeng ldquoAnalysis of piecewise constantdelay systems via block-pulse functionsrdquo International Journalof Systems Science vol 12 no 5 pp 625ndash633 1981

[18] W-L Chen and C-H Meng ldquoA general procedure of solvingthe linear delay system via block pulse functionsrdquo Computersand Electrical Engineering vol 9 no 3-4 pp 153ndash166 1982

[19] HRMarzban andM Shahsiah ldquoSolution of piecewise constantdelay systems using hybrid of block-pulse and Chebyshevpolynomialsrdquo Optimal Control Applications and Methods vol32 no 6 pp 647ndash659 2011

[20] H R Marzban and S M Hoseini ldquoAnalysis of linear piecewiseconstant delay systems using a hybrid numerical schemerdquoAdvances in Numerical Analysis vol 2016 Article ID 8903184 9pages 2016

[21] H RMarzban and SMHoseini ldquoNumerical treatment of non-linear optimal control problems involving piecewise constantdelayrdquo IMA Journal of Mathematical Control and Information2015

[22] Z H Jiang and W Schaufelberger Block pulse functions andtheir applications in control systems vol 179 of Lecture Notesin Control and Information Sciences Springer Berlin Germany1992

[23] H R Marzban ldquoParameter identification of linear multi-delaysystems via a hybrid of block-pulse functions and Taylorrsquospolynomialsrdquo International Journal of Control 2016

[24] P LancasterTheory of matrices Academic Press New York NYUSA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 9

Similarly

119909 (0) = 119906 (119905) = 119889119879Φ (119905) 119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (86)

By integrating both sides of (79) with respect to 119905 and using(85)-(86) we conclude that119862119879 = (119889119879 + 2119889119879119875) (119868119872119873 + 5119875 + 5119863119875)minus1 (87)

where 119868119872119873 is the119872119873-dimensional identity matrix Define119864 = max 1003816100381610038161003816119909119890 (119905) minus 119909 (119905)1003816100381610038161003816 0 le 119905 le 1 (88)

in which 119909(119905) and 119909119890(119905) denote the approximate solutiondetermined by the present approach and the exact solutionrespectively In Table 1 the numerical results of themaximumerror denoted by 119864 for 119873 = 20 and various values of 119872are reported The simulation results obtained by the currentapproach show a significant improvement in the accuracy ofthe solution compared to the method developed in [19]

54 Example 4 As a more complicated problem considerthe following problem

(119905) = 1199051199092 (119905 minus 119886 (119905)) + 1199093 (119905 minus 119886 (119905)) + 119906 (119905) 0 le 119905 le 1 (89)

119909 (119905) = 0 119905 lt 0 (90)119909 (0) = 1 (91)

119906 (119905) = 119905 0 le 119905 lt 0404 04 le 119905 le 1 (92)

119886 (119905) = 04 0 le 119905 lt 0808 08 le 119905 le 1 (93)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 046632511640625 + 2593315625119905 minus 72962501199052 + 1172501199053 minus 31001199054 + 131001199055 minus 11201199056 + 1561199057 04 le 119905 lt 082356911640625 + 4218715625119905 minus 762362501199052 + 2532501199053 minus 431001199054 + 231001199055 minus 71201199056 + 1561199057 08 le 119905 le 1

(94)

Although the above-mentioned problem is a nonlinear delaydifferential equation the method developed in the presentpaper can easily be implemented Because 120574 = 5 and 120582 = 2we take119873 = 3 Furthermore we take119872 = 8 Let

119909 (119905) = 119862119879Φ (119905) (95)

Similarly

119909 (0) = V1198791Φ (119905) 119905 = V1198792Φ (119905) (96)

int1199050119906 (120591) 119889120591 =

121199052 = V1198793Φ (119905) 0 le 119905 lt 04minus 225 + 25119905 = V1198794Φ (119905) 04 le 119905 lt 1 (97)

where V1 V2 V3 and V4 are vectors of order119872119873times1 Also wehave

119909 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905) (98)1199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) 119863119879119862= 119862119879119863Φ (119905) (99)

1199093 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905)119882= 1198621198791198632Φ (119905) (100)

in which

119882 = 119863119879119862 (101)

Using (99) gives

1199051199092 (119905 minus 119886 (119905)) = 119862119879119863Φ (119905)Φ119879 (119905) V2= 119862119879119863V2Φ (119905) (102)

Integrating both sides of (89) with respect to 119905 and using (95)(97) (98) (99) (100) and (102) imply that

119862119879 (119868119872119873 minus 119863V2119875 minus 1198632119875) minus (V1198791 + V1198793 + V1198794 )= 0 (103)

By solving (103) and using (95) the exact value of 119909(119905)will beobtained

10 International Journal of Differential Equations

Table 1 Computational results of maximum error for Example 3Present method CPU time (s) 119864119873 = 20119872 = 1 0112 1165119890 minus 02119873 = 20119872 = 2 0202 3231119890 minus 03119873 = 20119872 = 4 0398 2573119890 minus 06119873 = 20119872 = 6 0769 4836119890 minus 08119873 = 20119872 = 8 0853 3942119890 minus 11119873 = 20119872 = 10 1282 2376119890 minus 13119873 = 20119872 = 12 1841 5875119890 minus 16119873 = 20119872 = 14 2167 3624119890 minus 17119873 = 20119872 = 16 2536 5117119890 minus 2055 Example 5 As the final example we consider the time-varying multidelay system defined by (119905) = 119905119909 (119905 minus 1198861 (119905)) + 11990521199092 (119905 minus 1198862 (119905)) + 119906 (119905) 0 le 119905 le 1

119909 (0) = 1119906 (119905) =

119905 0 le 119905 lt 12 12 12 le 119905 le 1(104)

where

1198861 (119905) = 12 0 le 119905 lt 34 14 34 le 119905 le 1

1198862 (119905) = 14 0 le 119905 lt 12 34 12 le 119905 le 1

(105)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 14 2037320480 + 121199052 + 11321199053 minus 1161199054 + 1101199055 14 le 119905 lt 12 4819161440 + 12119905 + 9161199052 minus 161199053 + 181199054 12 le 119905 lt 34 12897431966080 + 12119905 + 14287409601199052 + 1873841199053 minus 12561199054 + 1241199055 + 1481199056 34 le 119905 le 1(106)

Although the above-mentioned system involves multiplepiecewise constant delays the proposed procedure can beapplied To employ our approach we select 119873 = 4 Bychoosing119872 = 7 the exact solution of 119909(119905) will be obtained6 Conclusion

An efficient and flexible framework has been successfullydeveloped for solving piecewise constant delay systems Thefoundation of the proposed method is based on a hybrid ofblock-pulse functions and Taylorrsquos polynomials The opera-tional matrix of delay has been constructed The operationalmatrices of integration delay and product associated withthe hybrid functions were utilized to transform the problemunder consideration into a system of algebraic equations Itis worth emphasizing that the exact solutions of Examples1 2 4 and 5 cannot be produced solely either by block-pulse functions or by Taylorrsquos polynomials After determiningthe suitable value of 119873 the number of subintervals smallvalues for 119872 the degree of the Taylorrsquos polynomials areneeded to achieve a specific accuracy The simulation resultsdemonstrate the effectiveness of the suggested procedure Inaddition the proposed numerical scheme can be extended

to a class of nonlinear piecewise constant delay systems butsome modifications are required

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this manuscript

References

[1] RDDriverOrdinary and delay differential equations Springer-Verlag New York NY USA 1977

[2] M Malek-Zavarei and M Jamshidi Time-delay systems vol 9of North-Holland Systems and Control Series North-HollandPublishing Co Amsterdam Netherlands 1987

[3] K B Datta and B M Mohan Orthogonal functions in systemsand control vol 9 ofAdvanced Series in Electrical and ComputerEngineering World Scientific Publishing Co 1995

[4] A Bellen ldquoOne-step collocation for delay differential equa-tionsrdquo Journal of Computational and Applied Mathematics vol10 no 3 pp 275ndash283 1984

[5] C Hwang and M Y Chen ldquoAnalysis and parameter identifi-cation of time-delay systems via shifted Legendre polynomialsrdquoInternational Journal of Control vol 41 no 2 pp 403ndash415 1985

International Journal of Differential Equations 11

[6] I R Horng and J H Chou ldquoAnalysis parameter estimation andoptimal control of time-delay systems via Chebyshev seriesrdquoInternational Journal of Control vol 41 no 5 pp 1221ndash12341985

[7] A Bellen and M Zennaro ldquoNumerical solution of delaydifferential equations by uniform corrections to an implicitRunge-Kutta methodrdquo Numerische Mathematik vol 47 no 2pp 301ndash316 1985

[8] I-R Horng and J-H Chou ldquoAnalysis and parameter identi-fication of time-delay systems via shifted Jacobi polynomialsrdquoInternational Journal of Control vol 44 no 4 pp 935ndash942 1986

[9] M Razzaghi and M Razzaghi ldquoTaylor series analysis of time-varying multi-delay systemsrdquo International Journal of Controlvol 50 no 1 pp 183ndash192 1989

[10] B M Mohan and K B Datta ldquoAnalysis of linear time-invarianttime-delay systems via orthogonal functionsrdquo InternationalJournal of Systems Science vol 26 no 1 pp 91ndash111 1995

[11] A Bellen and S Maset ldquoNumerical solution of constantcoefficient linear delay differential equations as abstract Cauchyproblemsrdquo Numerische Mathematik vol 84 no 3 pp 351ndash3742000

[12] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering vol 7 no 4 pp 337ndash353 2001

[13] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004

[14] H R Marzban and M Razzaghi ldquoAnalysis of time-delaysystems via hybrid of block-pulse functions and Taylor seriesrdquoJournal of Vibration and Control vol 11 no 12 pp 1455ndash14682005

[15] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006

[16] H R Marzban ldquoOptimal control of linear multi-delay systemsbased on a multi-interval decomposition schemerdquo OptimalControl Applications and Methods vol 37 no 1 pp 190ndash2112016

[17] W L Chen and B S Jeng ldquoAnalysis of piecewise constantdelay systems via block-pulse functionsrdquo International Journalof Systems Science vol 12 no 5 pp 625ndash633 1981

[18] W-L Chen and C-H Meng ldquoA general procedure of solvingthe linear delay system via block pulse functionsrdquo Computersand Electrical Engineering vol 9 no 3-4 pp 153ndash166 1982

[19] HRMarzban andM Shahsiah ldquoSolution of piecewise constantdelay systems using hybrid of block-pulse and Chebyshevpolynomialsrdquo Optimal Control Applications and Methods vol32 no 6 pp 647ndash659 2011

[20] H R Marzban and S M Hoseini ldquoAnalysis of linear piecewiseconstant delay systems using a hybrid numerical schemerdquoAdvances in Numerical Analysis vol 2016 Article ID 8903184 9pages 2016

[21] H RMarzban and SMHoseini ldquoNumerical treatment of non-linear optimal control problems involving piecewise constantdelayrdquo IMA Journal of Mathematical Control and Information2015

[22] Z H Jiang and W Schaufelberger Block pulse functions andtheir applications in control systems vol 179 of Lecture Notesin Control and Information Sciences Springer Berlin Germany1992

[23] H R Marzban ldquoParameter identification of linear multi-delaysystems via a hybrid of block-pulse functions and Taylorrsquospolynomialsrdquo International Journal of Control 2016

[24] P LancasterTheory of matrices Academic Press New York NYUSA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Journal of Differential Equations

Table 1 Computational results of maximum error for Example 3Present method CPU time (s) 119864119873 = 20119872 = 1 0112 1165119890 minus 02119873 = 20119872 = 2 0202 3231119890 minus 03119873 = 20119872 = 4 0398 2573119890 minus 06119873 = 20119872 = 6 0769 4836119890 minus 08119873 = 20119872 = 8 0853 3942119890 minus 11119873 = 20119872 = 10 1282 2376119890 minus 13119873 = 20119872 = 12 1841 5875119890 minus 16119873 = 20119872 = 14 2167 3624119890 minus 17119873 = 20119872 = 16 2536 5117119890 minus 2055 Example 5 As the final example we consider the time-varying multidelay system defined by (119905) = 119905119909 (119905 minus 1198861 (119905)) + 11990521199092 (119905 minus 1198862 (119905)) + 119906 (119905) 0 le 119905 le 1

119909 (0) = 1119906 (119905) =

119905 0 le 119905 lt 12 12 12 le 119905 le 1(104)

where

1198861 (119905) = 12 0 le 119905 lt 34 14 34 le 119905 le 1

1198862 (119905) = 14 0 le 119905 lt 12 34 12 le 119905 le 1

(105)

The analytical solution to this problem is described by

119909 (119905) =

1 + 121199052 0 le 119905 lt 14 2037320480 + 121199052 + 11321199053 minus 1161199054 + 1101199055 14 le 119905 lt 12 4819161440 + 12119905 + 9161199052 minus 161199053 + 181199054 12 le 119905 lt 34 12897431966080 + 12119905 + 14287409601199052 + 1873841199053 minus 12561199054 + 1241199055 + 1481199056 34 le 119905 le 1(106)

Although the above-mentioned system involves multiplepiecewise constant delays the proposed procedure can beapplied To employ our approach we select 119873 = 4 Bychoosing119872 = 7 the exact solution of 119909(119905) will be obtained6 Conclusion

An efficient and flexible framework has been successfullydeveloped for solving piecewise constant delay systems Thefoundation of the proposed method is based on a hybrid ofblock-pulse functions and Taylorrsquos polynomials The opera-tional matrix of delay has been constructed The operationalmatrices of integration delay and product associated withthe hybrid functions were utilized to transform the problemunder consideration into a system of algebraic equations Itis worth emphasizing that the exact solutions of Examples1 2 4 and 5 cannot be produced solely either by block-pulse functions or by Taylorrsquos polynomials After determiningthe suitable value of 119873 the number of subintervals smallvalues for 119872 the degree of the Taylorrsquos polynomials areneeded to achieve a specific accuracy The simulation resultsdemonstrate the effectiveness of the suggested procedure Inaddition the proposed numerical scheme can be extended

to a class of nonlinear piecewise constant delay systems butsome modifications are required

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this manuscript

References

[1] RDDriverOrdinary and delay differential equations Springer-Verlag New York NY USA 1977

[2] M Malek-Zavarei and M Jamshidi Time-delay systems vol 9of North-Holland Systems and Control Series North-HollandPublishing Co Amsterdam Netherlands 1987

[3] K B Datta and B M Mohan Orthogonal functions in systemsand control vol 9 ofAdvanced Series in Electrical and ComputerEngineering World Scientific Publishing Co 1995

[4] A Bellen ldquoOne-step collocation for delay differential equa-tionsrdquo Journal of Computational and Applied Mathematics vol10 no 3 pp 275ndash283 1984

[5] C Hwang and M Y Chen ldquoAnalysis and parameter identifi-cation of time-delay systems via shifted Legendre polynomialsrdquoInternational Journal of Control vol 41 no 2 pp 403ndash415 1985

International Journal of Differential Equations 11

[6] I R Horng and J H Chou ldquoAnalysis parameter estimation andoptimal control of time-delay systems via Chebyshev seriesrdquoInternational Journal of Control vol 41 no 5 pp 1221ndash12341985

[7] A Bellen and M Zennaro ldquoNumerical solution of delaydifferential equations by uniform corrections to an implicitRunge-Kutta methodrdquo Numerische Mathematik vol 47 no 2pp 301ndash316 1985

[8] I-R Horng and J-H Chou ldquoAnalysis and parameter identi-fication of time-delay systems via shifted Jacobi polynomialsrdquoInternational Journal of Control vol 44 no 4 pp 935ndash942 1986

[9] M Razzaghi and M Razzaghi ldquoTaylor series analysis of time-varying multi-delay systemsrdquo International Journal of Controlvol 50 no 1 pp 183ndash192 1989

[10] B M Mohan and K B Datta ldquoAnalysis of linear time-invarianttime-delay systems via orthogonal functionsrdquo InternationalJournal of Systems Science vol 26 no 1 pp 91ndash111 1995

[11] A Bellen and S Maset ldquoNumerical solution of constantcoefficient linear delay differential equations as abstract Cauchyproblemsrdquo Numerische Mathematik vol 84 no 3 pp 351ndash3742000

[12] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering vol 7 no 4 pp 337ndash353 2001

[13] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004

[14] H R Marzban and M Razzaghi ldquoAnalysis of time-delaysystems via hybrid of block-pulse functions and Taylor seriesrdquoJournal of Vibration and Control vol 11 no 12 pp 1455ndash14682005

[15] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006

[16] H R Marzban ldquoOptimal control of linear multi-delay systemsbased on a multi-interval decomposition schemerdquo OptimalControl Applications and Methods vol 37 no 1 pp 190ndash2112016

[17] W L Chen and B S Jeng ldquoAnalysis of piecewise constantdelay systems via block-pulse functionsrdquo International Journalof Systems Science vol 12 no 5 pp 625ndash633 1981

[18] W-L Chen and C-H Meng ldquoA general procedure of solvingthe linear delay system via block pulse functionsrdquo Computersand Electrical Engineering vol 9 no 3-4 pp 153ndash166 1982

[19] HRMarzban andM Shahsiah ldquoSolution of piecewise constantdelay systems using hybrid of block-pulse and Chebyshevpolynomialsrdquo Optimal Control Applications and Methods vol32 no 6 pp 647ndash659 2011

[20] H R Marzban and S M Hoseini ldquoAnalysis of linear piecewiseconstant delay systems using a hybrid numerical schemerdquoAdvances in Numerical Analysis vol 2016 Article ID 8903184 9pages 2016

[21] H RMarzban and SMHoseini ldquoNumerical treatment of non-linear optimal control problems involving piecewise constantdelayrdquo IMA Journal of Mathematical Control and Information2015

[22] Z H Jiang and W Schaufelberger Block pulse functions andtheir applications in control systems vol 179 of Lecture Notesin Control and Information Sciences Springer Berlin Germany1992

[23] H R Marzban ldquoParameter identification of linear multi-delaysystems via a hybrid of block-pulse functions and Taylorrsquospolynomialsrdquo International Journal of Control 2016

[24] P LancasterTheory of matrices Academic Press New York NYUSA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 11

[6] I R Horng and J H Chou ldquoAnalysis parameter estimation andoptimal control of time-delay systems via Chebyshev seriesrdquoInternational Journal of Control vol 41 no 5 pp 1221ndash12341985

[7] A Bellen and M Zennaro ldquoNumerical solution of delaydifferential equations by uniform corrections to an implicitRunge-Kutta methodrdquo Numerische Mathematik vol 47 no 2pp 301ndash316 1985

[8] I-R Horng and J-H Chou ldquoAnalysis and parameter identi-fication of time-delay systems via shifted Jacobi polynomialsrdquoInternational Journal of Control vol 44 no 4 pp 935ndash942 1986

[9] M Razzaghi and M Razzaghi ldquoTaylor series analysis of time-varying multi-delay systemsrdquo International Journal of Controlvol 50 no 1 pp 183ndash192 1989

[10] B M Mohan and K B Datta ldquoAnalysis of linear time-invarianttime-delay systems via orthogonal functionsrdquo InternationalJournal of Systems Science vol 26 no 1 pp 91ndash111 1995

[11] A Bellen and S Maset ldquoNumerical solution of constantcoefficient linear delay differential equations as abstract Cauchyproblemsrdquo Numerische Mathematik vol 84 no 3 pp 351ndash3742000

[12] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering vol 7 no 4 pp 337ndash353 2001

[13] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004

[14] H R Marzban and M Razzaghi ldquoAnalysis of time-delaysystems via hybrid of block-pulse functions and Taylor seriesrdquoJournal of Vibration and Control vol 11 no 12 pp 1455ndash14682005

[15] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006

[16] H R Marzban ldquoOptimal control of linear multi-delay systemsbased on a multi-interval decomposition schemerdquo OptimalControl Applications and Methods vol 37 no 1 pp 190ndash2112016

[17] W L Chen and B S Jeng ldquoAnalysis of piecewise constantdelay systems via block-pulse functionsrdquo International Journalof Systems Science vol 12 no 5 pp 625ndash633 1981

[18] W-L Chen and C-H Meng ldquoA general procedure of solvingthe linear delay system via block pulse functionsrdquo Computersand Electrical Engineering vol 9 no 3-4 pp 153ndash166 1982

[19] HRMarzban andM Shahsiah ldquoSolution of piecewise constantdelay systems using hybrid of block-pulse and Chebyshevpolynomialsrdquo Optimal Control Applications and Methods vol32 no 6 pp 647ndash659 2011

[20] H R Marzban and S M Hoseini ldquoAnalysis of linear piecewiseconstant delay systems using a hybrid numerical schemerdquoAdvances in Numerical Analysis vol 2016 Article ID 8903184 9pages 2016

[21] H RMarzban and SMHoseini ldquoNumerical treatment of non-linear optimal control problems involving piecewise constantdelayrdquo IMA Journal of Mathematical Control and Information2015

[22] Z H Jiang and W Schaufelberger Block pulse functions andtheir applications in control systems vol 179 of Lecture Notesin Control and Information Sciences Springer Berlin Germany1992

[23] H R Marzban ldquoParameter identification of linear multi-delaysystems via a hybrid of block-pulse functions and Taylorrsquospolynomialsrdquo International Journal of Control 2016

[24] P LancasterTheory of matrices Academic Press New York NYUSA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

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Discrete MathematicsJournal of

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Stochastic AnalysisInternational Journal of