Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 542839 9 pageshttpdxdoiorg1011552013542839

Research ArticleNew Wavelets Collocation Method forSolving Second-Order Multipoint Boundary Value ProblemsUsing Chebyshev Polynomials of Third and Fourth Kinds

W M Abd-Elhameed12 E H Doha2 and Y H Youssri2

1 Department of Mathematics Faculty of Science King Abdulaziz University Jeddah Saudi Arabia2Department of Mathematics Faculty of Science Cairo University Giza 12613 Egypt

Correspondence should be addressed to W M Abd-Elhameed walee 9yahoocom

Received 7 August 2013 Revised 13 September 2013 Accepted 13 September 2013

Academic Editor Soheil Salahshour

Copyright copy 2013 W M Abd-Elhameed et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper is concernedwith introducing twowavelets collocation algorithms for solving linear and nonlinearmultipoint boundaryvalue problems The principal idea for obtaining spectral numerical solutions for such equations is employing third- and fourth-kind Chebyshev wavelets along with the spectral collocation method to transform the differential equation with its boundaryconditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients which can be efficientlysolved Convergence analysis and some specific numerical examples are discussed to demonstrate the validity and applicability ofthe proposed algorithms The obtained numerical results are comparing favorably with the analytical known solutions

1 Introduction

Spectral methods are one of the principal methods of dis-cretization for the numerical solution of differential equa-tions The main advantage of these methods lies in theiraccuracy for a given number of unknowns (see eg [1ndash4]) For smooth problems in simple geometries they offerexponential rates of convergencespectral accuracy In con-trast finite difference and finite-element methods yield onlyalgebraic convergence ratesThe threemost widely used spec-tral versions are the Galerkin collocation and tau methodsCollocationmethods [5 6] have become increasingly popularfor solving differential equations also they are very useful inproviding highly accurate solutions to nonlinear differentialequations

Many practical problems arising in numerous branches ofscience and engineering require solving high even-order andhigh odd-order boundary value problems Legendre poly-nomials have been previously used for obtaining numericalspectral solutions for handling some of these kinds of prob-lems (see eg [7 8]) In [9] the author has constructed somealgorithms by selecting suitable combinations of Legendre

polynomials for solving the differentiated forms of high-odd-order boundary value problems with the aid of Petrov-Galerkin method while in the two papers [10 11] the authorshandled third- and fifth-order differential equations usingJacobi tau and Jacobi collocation methods

Multipoint boundary value problems (BVPs) arise in avariety of applied mathematics and physics For instance thevibrations of a guy wire of uniform cross-section composedof119873 parts of different densities can be set up as a multipointBVP as in [12] also many problems in the theory ofelastic stability can be handled by the method of multipointproblems [13] The existence and multiplicity of solutions ofmultipoint boundary value problems have been studied bymany authors see [14ndash17] and the references therein Fortwo-point BVPs there are many solution methods such asorthonormalization invariant imbedding algorithms finitedifference and collocation methods (see [18ndash20]) Howeverthere seems to be little discussion about numerical solutionsof multipoint boundary value problems

Second-ordermultipoint boundary value problems (BVP)arise in the mathematical modeling of deflection of can-tilever beams under concentrated load [21 22] deformation

2 Abstract and Applied Analysis

of beams and plate deflection theory [23] obstacle problems[24] Troeschrsquos problem relating to the confinement of aplasma column by radiation pressure [25 26] temperaturedistribution of the radiation fin of trapezoidal profile [2127] and a number of other engineering applications Manyauthors have used numerical and approximate methodsto solve second-order BVPs The details about the relatednumerical methods can be found in a large number of papers(see for instance [21 23 24 28]) The Walsh wavelets andthe semiorthogonal B-spline wavelets are used in [23 29]to construct some numerical algorithms for the solution ofsecond-order BVPs with Dirichlet and Neumann boundaryconditions Na [21] has found the numerical solution ofsecond- third- and fourth-order BVPs by converting theminto initial value problems and then applying a class ofmethods like nonlinear shootingmethod of reduced physicalparameters method of invariant imbedding and so forthThe presented approach in this paper can be applied to bothBVPs and IVPs with a slight modification but without thetransformation of BVPs into IVPs or vice versa

Wavelets theory is a relatively new and an emerging areain mathematical research It has been applied to a wide rangeof engineering disciplines particularly wavelets are verysuccessfully used in signal analysis for wave form repre-sentation and segmentations time frequency analysis andfast algorithms for easy implementation Wavelets permitthe accurate representation of a variety of functions andoperatorsMoreover wavelets establish a connectionwith fastnumerical algorithms (see [30 31])

The application of Legendre wavelets for solving differen-tial and integral equations is thoroughly considered by manyauthors (see for instance [32 33]) Also Chebyshev waveletsare used for solving some fractional and integral equations(see [34 35])

Chebyshev polynomials have become increasingly crucialin numerical analysis from both theoretical and practicalpoints of view It is well known that there are four kindsof Chebyshev polynomials and all of them are special casesof the more widest class of Jacobi polynomials The firstand second kinds are special cases of the symmetric Jacobipolynomials (ie ultraspherical polynomials) while the thirdand fourth kinds are special cases of the nonsymmetric Jacobipolynomials In the literature there is a great concentrationon the first and second kinds of Chebyshev polynomials119879

119899(119909)

and 119880119899(119909) and their various uses in numerous applications

(see for instance [36]) However there are few articles thatconcentrate on the other two types of Chebyshev polynomi-als namely third and fourth kinds 119881

119899(119909) and 119882

119899(119909) either

from theoretical or practical point of view and their usesin various applications (see eg [37]) This motivates ourinterest in such polynomialsWe therefore intend in this workto use them in a marvelous application of multipoint BVPsarising in physics

There are several advantages of using Chebyshev waveletsapproximations based on collocation spectral method Firstunlike most numerical techniques it is now well establishedthat they are characterized by exponentially decaying errorsSecond approximation by wavelets handles singularities inthe problem The effect of any such singularities will appear

in some form in any scheme of the numerical solutionand it is well known that other numerical methods do notperform well near singularities Finally due to their rapidconvergence Chebyshev wavelets collocation method doesnot suffer from the common instability problems associatedwith other numerical methods

Themain aim of this paper is to develop two new spectralalgorithms for solving second-order multipoint BVPs basedon shifted third- and fourth-kind Chebyshev wavelets Themethod reduces the differential equation with its boundaryconditions to a system of algebraic equations in the unknownexpansion coefficients Large systems of algebraic equationsmay lead to greater computational complexity and largestorage requirements However the third- and fourth-kindChebyshev wavelets collocation method reduces drasticallythe computational complexity of solving the resulting alge-braic system

The structure of the paper is as follows In Section 2we give some relevant properties of Chebyshev polynomi-als of third and fourth kinds and their shifted ones InSection 3 the third- and fourth-kind Chebyshev wavelets areconstructed Also in this section we ascertain the conver-gence of the Chebyshev wavelets series expansion Two newshifted Chebyshev wavelets collocation methods for solvingsecond-order linear and nonlinear multipoint boundaryvalue problems are implemented and presented in Section 4In Section 5 some numerical examples are presented toshow the efficiency and the applicability of the presentedalgorithms Some concluding remarks are given in Section 6

2 Some Properties of 119881119896(119909) and 119882

119896(119909)

The Chebyshev polynomials 119881119896(119909) and 119882

119896(119909) of third and

fourth kinds are polynomials of degree 119896 in 119909 definedrespectively by (see [38])

119881119896(119909) =

cos (119896 + (12)) 120579cos (1205792)

119882119896(119909) =

sin (119896 + (12)) 120579sin (1205792)

(1)

where 119909 = cos 120579 also they can be obtained explicitly astwo particular cases of Jacobi polynomials 119875(120572120573)

119896(119909) for the

two nonsymmetric cases correspond to 120573 = minus120572 = plusmn12Explicitly we have

119881119896(119909) =

(2119896

119896)2

(2119896)119875(minus1212)

119896(119909)

119882119896(119909) =

(2119896

119896)2

(2119896)119875(12minus12)

119896(119909)

(2)

It is readily seen that

119882119896(119909) = (minus1)

119896

119881119896(minus119909) (3)

Hence it is sufficient to establish properties and relations for119881119899(119909) and then deduce their corresponding properties and

relations for119882119899(119909) (by replacing 119909 by minus119909)

Abstract and Applied Analysis 3

The polynomials 119881119899(119909) and 119882

119899(119909) are orthogonal on

(minus1 1) that is

int

1

minus1

1205961(119909)119881119896(119909) 119881119895(119909) 119889119909

= int

1

minus1

1205962(119909)119882

119896(119909)119882

119895(119909) 119889119909

= 120587 119896 = 119895

0 119896 = 119895

(4)

where

1205961(119909) = radic

1 + 119909

1 minus 119909 120596

2(119909) = radic

1 minus 119909

1 + 119909 (5)

and they may be generated by using the two recurrencerelations

119881119896(119909) = 2119909119881

119896minus1(119909) minus 119881

119896minus2(119909) 119896 = 2 3 (6)

with the initial values

1198810(119909) = 1 119881

1(119909) = 2119909 minus 1 (7)

119882119896(119909) = 2119909 119882

119896minus1(119909) minus 119882

119896minus2(119909) 119896 = 2 3 (8)

with the initial values

1198820(119909) = 1 119882

1(119909) = 2119909 + 1 (9)

The shifted Chebyshev polynomials of third and fourthkinds are defined on [0 1] respectively as

119881lowast

119899(119909) = 119881

119899(2119909 minus 1) 119882

lowast

119899(119909) = 119882

119899(2119909 minus 1) (10)

All results of Chebyshev polynomials of third and fourthkinds can be easily transformed to give the correspondingresults for their shifted ones

The orthogonality relations of 119881lowast119899(119905) and119882lowast

119899(119905) on [0 1]

are given by

int

1

0

119908lowast

1119881lowast

119898(119905) 119881lowast

119899(119905) 119889119905

= int

1

0

119908lowast

2119882lowast

119898(119905)119882lowast

119899(119905) 119889119905

=

120587

2 119898 = 119899

0 119898 = 119899

(11)

where

120596lowast

1= radic

119905

1 minus 119905 120596

lowast

2= radic

1 minus 119905

119905 (12)

3 Shifted Third- and Fourth-KindChebyshev Wavelets

Wavelets constitute of a family of functions constructed fromdilation and translation of single function called the mother

wavelet When the dilation parameter 119886 and the translationparameter 119887 vary continuously then we have the followingfamily of continuous wavelets

120595119886119887(119905) = |119886|

minus12

120595(119905 minus 119887

119886) 119886 119887 isin R 119886 = 0 (13)

Each of the third- and fourth-kind Chebyshev wavelets120595119899119898(119905) = 120595(119896 119899119898 119905) has four arguments 119896 119899 isin N 119898 is

the order of the polynomial 119881lowast119898(119905) or 119882lowast

119898(119905) and 119905 is the

normalized time They are defined explicitly on the interval[0 1] as

120595119899119898(119905)

=

2(119896+1)2

radic120587119881lowast

119898(2119896

119905 minus 119899)

resp

2(119896+1)2

radic120587119882lowast

119898(2119896

119905 minus 119899) 119905 isin [119899

2119896119899 + 1

2119896] 0 ⩽ 119898 ⩽ 119872

0 ⩽ 119899 ⩽ 2119896

minus 1

0 otherwise

(14)

31 Function Approximation A function 119891(119905) defined over[0 1]may be expanded in terms of Chebyshev wavelets as

119891 (119905) =

infin

sum

119899=0

infin

sum

119898=0

119888119899119898120595119899119898(119905) (15)

where

119888119899119898= (119891 (119905) 120595

119899119898(119905))120596lowast

119894

= int

1

0

120596lowast

119894119891 (119905) 120595

119899119898(119905) 119889119905 (16)

and the weights 119908lowast119894 119894 = 1 2 are given in (12)

Assume that 119891(119905) can be approximated in terms ofChebyshev wavelets as

119891 (119905) ≃

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(119905) (17)

32 Convergence Analysis In this section we state and provea theorem to ascertain that the third- and fourth-kindCheby-shev wavelets expansion of a function 119891(119905) with boundedsecond derivative converges uniformly to 119891(119905)

Theorem 1 Assume that a function 119891(119905) isin 1198712

120596lowast

1

[0 1] 120596lowast1=

radic119905(1 minus 119905) with |11989110158401015840(119905)| ⩽ 119871 can be expanded as an infiniteseries of third-kind Chebyshev wavelets then this series con-verges uniformly to 119891(119905) Explicitly the expansion coefficientsin (16) satisfy the following inequality

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

2radic21205871198711198982

(119899 + 1)52

(1198984 minus 1)

forall119899 ⩾ 0 119898 gt 1 (18)

4 Abstract and Applied Analysis

Proof From (16) it follows that

119888119899119898=2(119896+1)2

radic120587int

(119899+1)2119896

1198992119896

119891 (119905) 119881lowast

119898(2119896

119905 minus 119899) 120596lowast

1(2119896

119905 minus 119899) 119889119905

(19)

If we make use of the substitution 2119896119905minus119899 = cos 120579 in (19) thenwe get

119888119899119898=2(minus119896+1)2

radic120587int

120587

0

119891(cos 120579 + 1198992119896

)

timescos (119898 + (12)) 120579

cos (1205792)radic1 + cos 1205791 minus cos 120579

sin 120579119889120579

=2(minus119896+3)2

radic120587int

120587

0

119891(cos 120579 + 1198992119896

) cos(119898 +1

2) 120579 cos(120579

2)119889120579

=2(minus119896+1)2

radic120587int

120587

0

119891(cos 120579 + 1198992119896

)

times [cos (119898 + 1) 120579 + cos119898120579] 119889120579(20)

which in turn and after performing integration by parts twotimes yields

119888119899119898=

1

251198962radic2120587int

120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579 (21)

where

120574119898(120579) =

1

119898 + 1(sin119898120579119898

minussin (119898 + 2) 120579

119898 + 2)

+1

119898(sin (119898 minus 1) 120579

119898 minus 1minussin (119898 + 1) 120579

119898 + 1)

(22)

Now we have

10038161003816100381610038161198881198991198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1

251198962radic2120587int

120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

=1

251198962radic2120587

10038161003816100381610038161003816100381610038161003816

int

120587

0

11989110158401015840

(cos 120579 + 1198992119896

) 120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

⩽119871

251198962radic2120587int

120587

0

1003816100381610038161003816120574119898 (120579)1003816100381610038161003816 119889120579

⩽119871radic120587

2(5119896+1)2[

1

119898 + 1(1

119898+

1

119898 + 2)

+1

119898(

1

119898 minus 1+

1

119898 + 1)]

=119871radic2120587

251198962[

1

1198982 + 2119898+

1

1198982 minus 1]

lt2119871radic2120587

251198962(

1198982

1198984 minus 1)

(23)

Finally since 119899 ⩽ 2119896 minus 1 we have

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

2radic21205871198711198982

(119899 + 1)52

(1198984 minus 1)

(24)

Remark 2 The estimation in (18) is also valid for the coef-ficients of fourth-kind Chebyshev wavelets expansion Theproof is similar to the proof of Theorem 1

4 Solution of Multipoint BVPs

In this section we present two Chebyshev wavelets col-location methods namely third-kind Chebyshev waveletscollocation method (3CWCM) and fourth-kind Chebyshevwavelets collocationmethod (4CWCM) to numerically solvethe following multipoint boundary value problem (BVP)

119886 (119909) 11991010158401015840

(119909) + 119887 (119909) 1199101015840

(119909) + 119888 (119909) 119910 (119909)

+ 119891 (119909 1199101015840

) + 119892 (119909 119910) = 0 0 ⩽ 119909 ⩽ 1

(25)

1205720119910 (0) + 120572

11199101015840

(0) =

1198980

sum

119894=1

120582119894119910 (120585119894) + 1205750

1205730119910 (1) + 120573

11199101015840

(1) =

1198981

sum

119894=1

120583119894119910 (120578119894) + 1205751

(26)

where 119886(119909) 119887(119909) and 119888(119909) are piecewise continuous on [0 1]119886(0)119886(1) may equal zero 0 lt 120585

119894 120578119894lt 1 120572

119894 120573119894 120582119894 120583119894 and 120575

119894

are constants such that (12057220+1205722

1)(1205732

0+1205732

1) = 0 119891 is a nonlinear

function of 1199101015840 and 119892 is a nonlinear function in 119910Consider an approximate solution to (25) and (26) which

is given in terms of Chebyshev wavelets as

119910119896119872

(119909) =

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(119909) (27)

then the substitution of (27) into (25) enables one to write theresidual of (25) in the form

119877 (119909) =

2119896

minus1

sum

119899=0

119872

sum

119898=2

119888119899119898119886 (119909) 120595

10158401015840

119899119898(119909)

+

2119896

minus1

sum

119899=0

119872

sum

119898=1

119888119899119898119887 (119909) 120595

1015840

119899119898(119909)

+

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898119888 (119909) 120595

119899119898(119909)

+ 119891(119909

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(119909))

+ 119892(119909

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(119909))

(28)

Abstract and Applied Analysis 5

Now the application of the typical collocation method (seeeg [5]) gives

119877 (119909119894) = 0 119894 = 1 2 2

119896

(119872 + 1) minus 2 (29)

where 119909119894are the first (2119896(119872 + 1) minus 2) roots of 119881lowast

2119896(119872+1)

(119909) or119882lowast

2119896(119872+1)

(119909) Moreover the use of the boundary conditions(26) gives

1205720

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(0)

+ 1205721

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(0)

=

1198980

sum

119894=1

2119896

minus1

sum

119899=0

119872

sum

119898=0

120582119894119888119899119898120595119899119898(120585119894) + 1205750

1205730

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(1) + 120573

1

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(1)

=

1198981

sum

119894=1

2119896

minus1

sum

119899=0

119872

sum

119898=0

120583119894119888119899119898120595119899119898(120578119894) + 1205751

(30)

Equations (29) and (30) generate 2119896(119872 + 1) equations inthe unknown expansion coefficients 119888

119899119898 which can be

solved with the aid of the well-known Newtonrsquos iterativemethod Consequently we get the desired approximate solu-tion 119910

119896119872(119909) given by (27)

5 Numerical Examples

In this section the presented algorithms in Section 4 areapplied to solve both of linear and nonlinear multipointBVPs Some examples are considered to illustrate the effi-ciency and applicability of the two proposed algorithms

Example 1 Consider the second-order nonlinear BVP (see[6 28])

11991010158401015840

+3

8119910 +

2

1089(1199101015840

)2

+ 1 = 0 0 lt 119909 lt 1

119910 (0) = 0 119910 (1

3) = 119910 (1)

(31)

The two proposed methods are applied to the problem forthe case corresponding to 119896 = 0 and119872 = 8 The numericalsolutions are shown in Table 1 Due to nonavailability of theexact solution we compare our results with Haar waveletsmethod [6] ADM solution [28] and ODEs Solver fromMathematica which is carried out by using Runge-Kuttamethod This comparison is also shown in Table 1

Example 2 Consider the second-order linear BVP (see [3940])

11991010158401015840

= sinh119909 minus 2 0 lt 119909 lt 1

1199101015840

(0) = 0 119910 (1) = 3119910 (3

5)

(32)

The exact solution of problem (32) is given by

119910 (119909) =1

2(sinh 1 minus 3 sinh 3

5) + sinh119909 minus 1199092 minus 119909 + 11

25

(33)

In Table 2 the maximum absolute error 119864 is listed for119896 = 1 and various values of 119872 while in Table 3 wegive a comparison between the best errors resulted fromthe application of various methods for Example 2 while inFigure 1 we give a comparison between the exact solution of(32) with three approximate solutions

Example 3 Consider the second-order singular nonlinearBVP (see [40 41])

119909 (1 minus 119909) 11991010158401015840

+ 61199101015840

+ 2119910 + 1199102

= 6 cosh 119909 + (2 + 119909 minus 1199092 + sinh119909) sinh119909

0 lt 119909 lt 1

119910 (0) + 119910 (2

3) = sinh(2

3)

119910 (1) +1

2119910 (

4

5) =

1

2sinh(4

5) + sinh 1

(34)

with the exact solution 119910(119909) = sinh119909 In Table 4 the maxi-mum absolute error 119864 is listed for 119896 = 0 and various valuesof 119872 while in Table 5 we give a comparison between thebest errors resulted from the application of various methodsfor Example 3 This table shows that our two algorithms aremore accurate if compared with the two methods developedin [40 41]

Example 4 Consider the second-order nonlinear BVP (see[42])

11991010158401015840

+ (1 + 119909 + 1199093

) 1199102

= 119891 (119909) 0 lt 119909 lt 1

119910 (0) =1

6119910 (

2

9) +

1

3119910 (

7

9) minus 00286634

119910 (1) =1

5119910 (

2

9) +

1

2119910 (

7

9) minus 00401287

(35)

where

119891 (119909) =1

9[ minus 6 cos (119909 minus 1199092) + sin (119909 minus 1199092)

times (minus3(1 minus 2119909)2

+ (1 + 119909 + 1199093

) sin (119909 minus 1199092))]

(36)

6 Abstract and Applied Analysis

00 02 04 06 08 10

0010

0050

0020

0030

0015

0070

x

ln|y(x)|

Exactk = 0M = 4

k = 0M = 3

k = 0M = 2

Figure 1 Different solutions of Example 2

The exact solution of (35) is given by 119910(119909) = (13) sin(119909 minus1199092

) In Table 6 the maximum absolute error 119864 is listed for119896 = 2 and various values of 119872 and in Table 7 we give acomparison between best errors resulted from the applicationof various methods for Example 4 This table shows thatour two algorithms are more accurate if compared with themethod developed in [42]

Example 5 Consider the second-order singular linear BVP

11991010158401015840

+ 119891 (119909) 119910 = 119892 (119909) 0 lt 119909 lt 1

119910 (0) + 16119910 (1

4) = 3

4

radic119890

119910 (1) + 16119910 (3

4) = 3

4

radic1198903

(37)

where

119891 (119909) =

3119909 0 ⩽ 119909 ⩽1

2

21199091

2lt 119909 ⩽ 1

(38)

and 119892(119909) is chosen such that the exact solution of (37) is119910(119909) = 119909(1 minus 119909)119890

119909 In Table 8 the maximum absolute error 119864is listed for 119896 = 1 and various values of119872 while in Figure 2we give a comparison between the exact solution of (37) withthree approximate solutions

Example 6 Consider the following nonlinear second-orderBVP

11991010158401015840

+ (1199101015840

)2

minus 64119910 = 32 0 lt 119909 lt 1 (39)

119910 (0) + 119910 (1

4) = 1 (40)

4119910 (1

2) minus 119910 (1) = 0 (41)

with the exact solution 119910(119909) = 161199092 We solve (39) using

3CWCM for the case corresponding to 119896 = 0 and 119872 = 3

00 02 04 06 08 1000

01

02

03

04

05

x

Exactk = 1M = 3

k = 1M = 2

k = 1M = 1

y(x)

Figure 2 Different solutions of Example 5

to obtain an approximate solution of 119910(119909) If we make use of(27) then the approximate solution 119910

03(119909) can be expanded

in terms of third-kind Chebyshev wavelets as

11991003(119909) = 119888

00radic2

120587+ 11988801radic2

120587(4119909 minus 3)

+ 11988802radic2

120587(161199092

minus 20119909 + 5)

(42)

If we set

V119894= radic

2

1205871198880119894 119894 = 0 1 2 (43)

then (42) reduces to the form

11991003(119909) = V

0+ V1(4119909 minus 3) + V

2(161199092

minus 20119909 + 5) (44)

If we substitute (44) into (39) then the residual of (39) isgiven by

119877 (119909) = 2V2+ [V1+ V2(8119909 minus 5)]

2

minus 4 [V0+ V1(4119909 minus 3)

+ V2(161199092

minus 20119909 + 5)] minus 2

(45)

We enforce the residual to vanish at the first root of 119881lowast3(119909) =

641199093

minus1121199092

+56119909minus7 namely at 1199091= 018825509907063323

to get

610388V22+ 05V2

1minus 349396V

2V1minus 260388V

2

+ 449396V1minus 2V0= 1

(46)

Furthermore the use of the boundary conditions (40) and(41) yields

2V0minus 5V1+ 6V2= 1

3V0minus 5V1minus 5V2= 0

(47)

Abstract and Applied Analysis 7

Table 1 Comparison between different solutions for Example 1

119909 3CWCM 4CWCM Haar method [6] ADMmethod [28] ODEs solver fromMathematica01 006560 006560 006561 00656 00656003 016587 016587 016588 01658 01658705 022369 022369 022369 02236 02236907 023820 023820 023821 02382 02382009 020920 020920 020910 02092 020920

Table 2 The maximum absolute error 119864 for Example 2

119896 119872 10 11 12 13 14 15

0 3CWCM 5173 sdot 10minus9

7184 sdot 10minus11

4418 sdot 10minus12

5018 sdot 10minus14

2442 sdot 10minus15

2220 sdot 10minus16

4CWCM 3324 sdot 10minus10

2533 sdot 10minus11

2811 sdot 10minus12

1665 sdot 10minus14

1110 sdot 10minus15

2220 sdot 10minus16

119872 4 5 6 7 8 9

1 3CWCM 4644 sdot 10minus6

1001 sdot 10minus8

1840 sdot 10minus9

2547 sdot 10minus10

6247 sdot 10minus11

5681 sdot 10minus12

4CWCM 215 sdot 10minus6

5247 sdot 10minus9

7548 sdot 10minus10

1004 sdot 10minus10

2154 sdot 10minus11

4257 sdot 10minus12

Table 3 The best errors for Example 2

Method in [39] Method in [40] 3CWCM 4CWCM900 sdot 10

minus6

400 sdot 10minus5

222 sdot 10minus16

222 sdot 10minus16

Table 4 The maximum absolute error 119864 for Example 3

119872 8 9 10 11 12 133CWCM 4444 sdot 10

minus10

2759 sdot 10minus11

4389 sdot 10minus13

1643 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

4CWCM 4478 sdot 10minus9

2179 sdot 10minus10

1527 sdot 10minus12

2975 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

Table 5 The best errors for Example 3

Method in [40] Method in [41] 3CWCM 4CWCM300 sdot 10

minus8

660 sdot 10minus7

222 sdot 10minus16

222 sdot 10minus16

Table 6 The maximum absolute error 119864 for Example 4

119872 4 5 6 7 8 93CWCM 2881 sdot 10

minus3

3441 sdot 10minus3

1212 sdot 10minus5

1933 sdot 10minus5

1990 sdot 10minus6

3010 sdot 10minus8

4CWCM 1241 sdot 10minus3

8542 sdot 10minus4

7526 sdot 10minus6

1002 sdot 10minus6

6321 sdot 10minus7

2354 sdot 10minus9

Table 7 The best errors for Example 4

Method in [42] 3CWCM 4CWCM800 sdot 10

minus6

3010 sdot 10minus8

235 sdot 10minus9

Table 8 The maximum absolute error 119864 for Example 5

119872 7 8 9 10 11 12 13 143CWCM 71 sdot 10

minus6

32 sdot 10minus7

13 sdot 10minus8

74 sdot 10minus10

13 sdot 10minus11

35 sdot 10minus13

93 sdot 10minus15

81 sdot 10minus16

4CWCM 28 sdot 10minus5

15 sdot 10minus6

63 sdot 10minus8

24 sdot 10minus9

78 sdot 10minus11

23 sdot 10minus12

63 sdot 10minus14

17 sdot 10minus15

8 Abstract and Applied Analysis

The solution of the nonlinear system of (46) and (47) gives

V0= 10 V

1= 5 V

2= 1 (48)

and consequently

11991003(119909) = 10 + 5 (4119909 minus 3) + (16119909

2

minus 20119909 + 5) = 161199092

(49)

which is the exact solution

Remark 3 It is worth noting here that the obtained numericalresults in the previous solved six examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecomparing favorably with the known analytical solutions

6 Concluding Remarks

In this paper two algorithms for obtaining numerical spectralwavelets solutions for second-order multipoint linear andnonlinear boundary value problems are analyzed and dis-cussed Chebyshev polynomials of third and fourth kindsare used One of the advantages of the developed algorithmsis their availability for application on singular boundaryvalue problems Another advantage is that high accurateapproximate solutions are achieved using a few number ofterms of the approximate expansionThe obtained numericalresults are comparing favorably with the analytical ones

Acknowledgment

The authors would like to thank the referee for his valuablecomments and suggestions which improved the paper in itspresent form

References

[1] E H Doha W M Abd-Elhameed and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for the integrated formsof third- and fifth-order elliptic differential equations usinggeneral parameters generalized Jacobi polynomialsrdquo AppliedMathematics and Computation vol 218 no 15 pp 7727ndash77402012

[2] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[3] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013

[4] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[5] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Springer Berlin Germany 2006

[6] Siraj-ul-Islam I Aziz and B Sarler ldquoThe numerical solution ofsecond-order boundary-value problems by collocation methodwith theHaarwaveletsrdquoMathematical andComputerModellingvol 52 no 9-10 pp 1577ndash1590 2010

[7] A H Bhrawy A S Alofi and S I El-Soubhy ldquoAn exten-sion of the Legendre-Galerkin method for solving sixth-orderdifferential equations with variable polynomial coefficientsrdquoMathematical Problems in Engineering vol 2012 Article ID896575 13 pages 2012

[8] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[9] WMAbd-Elhameed ldquoEfficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of (2119899 + 1)th-order linear differential equationsrdquo Journal of the EgyptianMathematical Society vol 17 no 2 pp 189ndash211 2009

[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011

[11] A H Bhrawy A S Alofi and S I El-Soubhy ldquoSpectral shiftedJacobi tau and collocation methods for solving fifth-orderboundary value problemsrdquo Abstract and Applied Analysis vol2011 Article ID 823273 14 pages 2011

[12] M Moshiinsky ldquoSobre los problemas de condiciones a lafrontiera en una dimension de caracteristicas discontinuasrdquoBoletin de La Sociedad Matematica Mexicana vol 7 article 1251950

[13] S P TimoshenkoTheory of Elastic StabilityMcGraw-Hill BookNew York NY USA 2nd edition 1961

[14] R P Agarwal and I Kiguradze ldquoOnmulti-point boundary valueproblems for linear ordinary differential equations with singu-laritiesrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 131ndash151 2004

[15] Z Du ldquoSolvability of functional differential equations withmulti-point boundary value problems at resonancerdquoComputersamp Mathematics with Applications vol 55 no 11 pp 2653ndash26612008

[16] W Feng and J R L Webb ldquoSolvability of 119898-point boundaryvalue problemswith nonlinear growthrdquo Journal ofMathematicalAnalysis and Applications vol 212 no 2 pp 467ndash480 1997

[17] H B Thompson and C Tisdell ldquoThree-point boundary valueproblems for second-order ordinary differential equationsrdquoMathematical and Computer Modelling vol 34 no 3-4 pp 311ndash318 2001

[18] M R Scott and H A Watts ldquoSUPORTmdasha computer code fortwo-point boundary-value problems via orthonormalizationrdquoSandia Labs Report 75-0198 Sandia Laboratories AlbuquerqueNM USA 1975

[19] M R Scott and H A Watts ldquoComputational solution of lineartwo-point boundary value problems via orthonormalizationrdquoSIAM Journal on Numerical Analysis vol 14 no 1 pp 40ndash701977

[20] M R Scott and W H Vandevender ldquoA comparison of severalinvariant imbedding algorithms for the solution of two-pointboundary-value problemsrdquo Applied Mathematics and Compu-tation vol 1 no 3 pp 187ndash218 1975

[21] T YNaComputationalMethods in Engineering BoundaryValueProblems vol 145 Academic Press New York NY USA 1979

Abstract and Applied Analysis 9

[22] K E Bisshopp andDCDrucker ldquoLarge deflection of cantileverbeamsrdquo Quarterly of Applied Mathematics vol 3 pp 272ndash2751945

[23] W Glabisz ldquoThe use of Walsh-wavelet packets in linear bound-ary value problemsrdquo Computers amp Structures vol 82 no 2-3pp 131ndash141 2004

[24] Siraj-ul-Islam M A Noor I A Tirmizi and M A KhanldquoQuadratic non-polynomial spline approach to the solution ofa system of second-order boundary-value problemsrdquo AppliedMathematics and Computation vol 179 no 1 pp 153ndash160 2006

[25] S Robert and J Shipman ldquoSolution of Troeschrsquos two-pointboundary value problems by shooting techniquesrdquo Journal ofComputational Physics vol 10 pp 232ndash241 1972

[26] E Wiebel Confinement of a Plasma Column by RadiationPressure in the Plasma in a Magnetic Field Stanford UniversityPress Stanfor Calif USA 1958

[27] H H Keller and E S Holdrege ldquoRadiation heat transfer forannular fins of trapezoidal profilerdquo International Journal of HighPerformance Computing Applications vol 92 pp 113ndash116 1970

[28] M Tatari andM Dehgan ldquoThe use of the Adomian decomposi-tion method for solving multipoint boundary value problemsrdquoPhysica Scripta vol 73 pp 672ndash676 2006

[29] M Lakestani andM Dehghan ldquoThe solution of a second-ordernonlinear differential equation with Neumann boundary con-ditions using semi-orthogonal B-spline waveletsrdquo InternationalJournal of Computer Mathematics vol 83 no 8-9 pp 685ndash6942006

[30] A Constantmldes Applied Numerical Methods with PersonalComputers McGraw-Hill New York NY USA 1987

[31] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[32] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[33] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[34] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[35] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[36] E H Doha W M Abd- Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[37] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[38] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp Hall New York NY USA 2003

[39] Y Lin J Niu and M Cui ldquoA numerical solution to nonlinearsecond order three-point boundary value problems in thereproducing kernel spacerdquo Applied Mathematics and Computa-tion vol 218 no 14 pp 7362ndash7368 2012

[40] F Geng ldquoSolving singular second order three-point bound-ary value problems using reproducing kernel Hilbert spacemethodrdquo Applied Mathematics and Computation vol 215 no6 pp 2095ndash2102 2009

[41] F Z Geng ldquoA numerical algorithm for nonlinear multi-point boundary value problemsrdquo Journal of Computational andApplied Mathematics vol 236 no 7 pp 1789ndash1794 2012

[42] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

of beams and plate deflection theory [23] obstacle problems[24] Troeschrsquos problem relating to the confinement of aplasma column by radiation pressure [25 26] temperaturedistribution of the radiation fin of trapezoidal profile [2127] and a number of other engineering applications Manyauthors have used numerical and approximate methodsto solve second-order BVPs The details about the relatednumerical methods can be found in a large number of papers(see for instance [21 23 24 28]) The Walsh wavelets andthe semiorthogonal B-spline wavelets are used in [23 29]to construct some numerical algorithms for the solution ofsecond-order BVPs with Dirichlet and Neumann boundaryconditions Na [21] has found the numerical solution ofsecond- third- and fourth-order BVPs by converting theminto initial value problems and then applying a class ofmethods like nonlinear shootingmethod of reduced physicalparameters method of invariant imbedding and so forthThe presented approach in this paper can be applied to bothBVPs and IVPs with a slight modification but without thetransformation of BVPs into IVPs or vice versa

Wavelets theory is a relatively new and an emerging areain mathematical research It has been applied to a wide rangeof engineering disciplines particularly wavelets are verysuccessfully used in signal analysis for wave form repre-sentation and segmentations time frequency analysis andfast algorithms for easy implementation Wavelets permitthe accurate representation of a variety of functions andoperatorsMoreover wavelets establish a connectionwith fastnumerical algorithms (see [30 31])

The application of Legendre wavelets for solving differen-tial and integral equations is thoroughly considered by manyauthors (see for instance [32 33]) Also Chebyshev waveletsare used for solving some fractional and integral equations(see [34 35])

Chebyshev polynomials have become increasingly crucialin numerical analysis from both theoretical and practicalpoints of view It is well known that there are four kindsof Chebyshev polynomials and all of them are special casesof the more widest class of Jacobi polynomials The firstand second kinds are special cases of the symmetric Jacobipolynomials (ie ultraspherical polynomials) while the thirdand fourth kinds are special cases of the nonsymmetric Jacobipolynomials In the literature there is a great concentrationon the first and second kinds of Chebyshev polynomials119879

119899(119909)

and 119880119899(119909) and their various uses in numerous applications

(see for instance [36]) However there are few articles thatconcentrate on the other two types of Chebyshev polynomi-als namely third and fourth kinds 119881

119899(119909) and 119882

119899(119909) either

from theoretical or practical point of view and their usesin various applications (see eg [37]) This motivates ourinterest in such polynomialsWe therefore intend in this workto use them in a marvelous application of multipoint BVPsarising in physics

There are several advantages of using Chebyshev waveletsapproximations based on collocation spectral method Firstunlike most numerical techniques it is now well establishedthat they are characterized by exponentially decaying errorsSecond approximation by wavelets handles singularities inthe problem The effect of any such singularities will appear

in some form in any scheme of the numerical solutionand it is well known that other numerical methods do notperform well near singularities Finally due to their rapidconvergence Chebyshev wavelets collocation method doesnot suffer from the common instability problems associatedwith other numerical methods

Themain aim of this paper is to develop two new spectralalgorithms for solving second-order multipoint BVPs basedon shifted third- and fourth-kind Chebyshev wavelets Themethod reduces the differential equation with its boundaryconditions to a system of algebraic equations in the unknownexpansion coefficients Large systems of algebraic equationsmay lead to greater computational complexity and largestorage requirements However the third- and fourth-kindChebyshev wavelets collocation method reduces drasticallythe computational complexity of solving the resulting alge-braic system

The structure of the paper is as follows In Section 2we give some relevant properties of Chebyshev polynomi-als of third and fourth kinds and their shifted ones InSection 3 the third- and fourth-kind Chebyshev wavelets areconstructed Also in this section we ascertain the conver-gence of the Chebyshev wavelets series expansion Two newshifted Chebyshev wavelets collocation methods for solvingsecond-order linear and nonlinear multipoint boundaryvalue problems are implemented and presented in Section 4In Section 5 some numerical examples are presented toshow the efficiency and the applicability of the presentedalgorithms Some concluding remarks are given in Section 6

2 Some Properties of 119881119896(119909) and 119882

119896(119909)

The Chebyshev polynomials 119881119896(119909) and 119882

119896(119909) of third and

fourth kinds are polynomials of degree 119896 in 119909 definedrespectively by (see [38])

119881119896(119909) =

cos (119896 + (12)) 120579cos (1205792)

119882119896(119909) =

sin (119896 + (12)) 120579sin (1205792)

(1)

where 119909 = cos 120579 also they can be obtained explicitly astwo particular cases of Jacobi polynomials 119875(120572120573)

119896(119909) for the

two nonsymmetric cases correspond to 120573 = minus120572 = plusmn12Explicitly we have

119881119896(119909) =

(2119896

119896)2

(2119896)119875(minus1212)

119896(119909)

119882119896(119909) =

(2119896

119896)2

(2119896)119875(12minus12)

119896(119909)

(2)

It is readily seen that

119882119896(119909) = (minus1)

119896

119881119896(minus119909) (3)

Hence it is sufficient to establish properties and relations for119881119899(119909) and then deduce their corresponding properties and

relations for119882119899(119909) (by replacing 119909 by minus119909)

Abstract and Applied Analysis 3

The polynomials 119881119899(119909) and 119882

119899(119909) are orthogonal on

(minus1 1) that is

int

1

minus1

1205961(119909)119881119896(119909) 119881119895(119909) 119889119909

= int

1

minus1

1205962(119909)119882

119896(119909)119882

119895(119909) 119889119909

= 120587 119896 = 119895

0 119896 = 119895

(4)

where

1205961(119909) = radic

1 + 119909

1 minus 119909 120596

2(119909) = radic

1 minus 119909

1 + 119909 (5)

and they may be generated by using the two recurrencerelations

119881119896(119909) = 2119909119881

119896minus1(119909) minus 119881

119896minus2(119909) 119896 = 2 3 (6)

with the initial values

1198810(119909) = 1 119881

1(119909) = 2119909 minus 1 (7)

119882119896(119909) = 2119909 119882

119896minus1(119909) minus 119882

119896minus2(119909) 119896 = 2 3 (8)

with the initial values

1198820(119909) = 1 119882

1(119909) = 2119909 + 1 (9)

The shifted Chebyshev polynomials of third and fourthkinds are defined on [0 1] respectively as

119881lowast

119899(119909) = 119881

119899(2119909 minus 1) 119882

lowast

119899(119909) = 119882

119899(2119909 minus 1) (10)

All results of Chebyshev polynomials of third and fourthkinds can be easily transformed to give the correspondingresults for their shifted ones

The orthogonality relations of 119881lowast119899(119905) and119882lowast

119899(119905) on [0 1]

are given by

int

1

0

119908lowast

1119881lowast

119898(119905) 119881lowast

119899(119905) 119889119905

= int

1

0

119908lowast

2119882lowast

119898(119905)119882lowast

119899(119905) 119889119905

=

120587

2 119898 = 119899

0 119898 = 119899

(11)

where

120596lowast

1= radic

119905

1 minus 119905 120596

lowast

2= radic

1 minus 119905

119905 (12)

3 Shifted Third- and Fourth-KindChebyshev Wavelets

Wavelets constitute of a family of functions constructed fromdilation and translation of single function called the mother

wavelet When the dilation parameter 119886 and the translationparameter 119887 vary continuously then we have the followingfamily of continuous wavelets

120595119886119887(119905) = |119886|

minus12

120595(119905 minus 119887

119886) 119886 119887 isin R 119886 = 0 (13)

Each of the third- and fourth-kind Chebyshev wavelets120595119899119898(119905) = 120595(119896 119899119898 119905) has four arguments 119896 119899 isin N 119898 is

the order of the polynomial 119881lowast119898(119905) or 119882lowast

119898(119905) and 119905 is the

normalized time They are defined explicitly on the interval[0 1] as

120595119899119898(119905)

=

2(119896+1)2

radic120587119881lowast

119898(2119896

119905 minus 119899)

resp

2(119896+1)2

radic120587119882lowast

119898(2119896

119905 minus 119899) 119905 isin [119899

2119896119899 + 1

2119896] 0 ⩽ 119898 ⩽ 119872

0 ⩽ 119899 ⩽ 2119896

minus 1

0 otherwise

(14)

31 Function Approximation A function 119891(119905) defined over[0 1]may be expanded in terms of Chebyshev wavelets as

119891 (119905) =

infin

sum

119899=0

infin

sum

119898=0

119888119899119898120595119899119898(119905) (15)

where

119888119899119898= (119891 (119905) 120595

119899119898(119905))120596lowast

119894

= int

1

0

120596lowast

119894119891 (119905) 120595

119899119898(119905) 119889119905 (16)

and the weights 119908lowast119894 119894 = 1 2 are given in (12)

Assume that 119891(119905) can be approximated in terms ofChebyshev wavelets as

119891 (119905) ≃

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(119905) (17)

32 Convergence Analysis In this section we state and provea theorem to ascertain that the third- and fourth-kindCheby-shev wavelets expansion of a function 119891(119905) with boundedsecond derivative converges uniformly to 119891(119905)

Theorem 1 Assume that a function 119891(119905) isin 1198712

120596lowast

1

[0 1] 120596lowast1=

radic119905(1 minus 119905) with |11989110158401015840(119905)| ⩽ 119871 can be expanded as an infiniteseries of third-kind Chebyshev wavelets then this series con-verges uniformly to 119891(119905) Explicitly the expansion coefficientsin (16) satisfy the following inequality

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

2radic21205871198711198982

(119899 + 1)52

(1198984 minus 1)

forall119899 ⩾ 0 119898 gt 1 (18)

4 Abstract and Applied Analysis

Proof From (16) it follows that

119888119899119898=2(119896+1)2

radic120587int

(119899+1)2119896

1198992119896

119891 (119905) 119881lowast

119898(2119896

119905 minus 119899) 120596lowast

1(2119896

119905 minus 119899) 119889119905

(19)

If we make use of the substitution 2119896119905minus119899 = cos 120579 in (19) thenwe get

119888119899119898=2(minus119896+1)2

radic120587int

120587

0

119891(cos 120579 + 1198992119896

)

timescos (119898 + (12)) 120579

cos (1205792)radic1 + cos 1205791 minus cos 120579

sin 120579119889120579

=2(minus119896+3)2

radic120587int

120587

0

119891(cos 120579 + 1198992119896

) cos(119898 +1

2) 120579 cos(120579

2)119889120579

=2(minus119896+1)2

radic120587int

120587

0

119891(cos 120579 + 1198992119896

)

times [cos (119898 + 1) 120579 + cos119898120579] 119889120579(20)

which in turn and after performing integration by parts twotimes yields

119888119899119898=

1

251198962radic2120587int

120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579 (21)

where

120574119898(120579) =

1

119898 + 1(sin119898120579119898

minussin (119898 + 2) 120579

119898 + 2)

+1

119898(sin (119898 minus 1) 120579

119898 minus 1minussin (119898 + 1) 120579

119898 + 1)

(22)

Now we have

10038161003816100381610038161198881198991198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1

251198962radic2120587int

120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

=1

251198962radic2120587

10038161003816100381610038161003816100381610038161003816

int

120587

0

11989110158401015840

(cos 120579 + 1198992119896

) 120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

⩽119871

251198962radic2120587int

120587

0

1003816100381610038161003816120574119898 (120579)1003816100381610038161003816 119889120579

⩽119871radic120587

2(5119896+1)2[

1

119898 + 1(1

119898+

1

119898 + 2)

+1

119898(

1

119898 minus 1+

1

119898 + 1)]

=119871radic2120587

251198962[

1

1198982 + 2119898+

1

1198982 minus 1]

lt2119871radic2120587

251198962(

1198982

1198984 minus 1)

(23)

Finally since 119899 ⩽ 2119896 minus 1 we have

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

2radic21205871198711198982

(119899 + 1)52

(1198984 minus 1)

(24)

Remark 2 The estimation in (18) is also valid for the coef-ficients of fourth-kind Chebyshev wavelets expansion Theproof is similar to the proof of Theorem 1

4 Solution of Multipoint BVPs

In this section we present two Chebyshev wavelets col-location methods namely third-kind Chebyshev waveletscollocation method (3CWCM) and fourth-kind Chebyshevwavelets collocationmethod (4CWCM) to numerically solvethe following multipoint boundary value problem (BVP)

119886 (119909) 11991010158401015840

(119909) + 119887 (119909) 1199101015840

(119909) + 119888 (119909) 119910 (119909)

+ 119891 (119909 1199101015840

) + 119892 (119909 119910) = 0 0 ⩽ 119909 ⩽ 1

(25)

1205720119910 (0) + 120572

11199101015840

(0) =

1198980

sum

119894=1

120582119894119910 (120585119894) + 1205750

1205730119910 (1) + 120573

11199101015840

(1) =

1198981

sum

119894=1

120583119894119910 (120578119894) + 1205751

(26)

where 119886(119909) 119887(119909) and 119888(119909) are piecewise continuous on [0 1]119886(0)119886(1) may equal zero 0 lt 120585

119894 120578119894lt 1 120572

119894 120573119894 120582119894 120583119894 and 120575

119894

are constants such that (12057220+1205722

1)(1205732

0+1205732

1) = 0 119891 is a nonlinear

function of 1199101015840 and 119892 is a nonlinear function in 119910Consider an approximate solution to (25) and (26) which

is given in terms of Chebyshev wavelets as

119910119896119872

(119909) =

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(119909) (27)

then the substitution of (27) into (25) enables one to write theresidual of (25) in the form

119877 (119909) =

2119896

minus1

sum

119899=0

119872

sum

119898=2

119888119899119898119886 (119909) 120595

10158401015840

119899119898(119909)

+

2119896

minus1

sum

119899=0

119872

sum

119898=1

119888119899119898119887 (119909) 120595

1015840

119899119898(119909)

+

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898119888 (119909) 120595

119899119898(119909)

+ 119891(119909

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(119909))

+ 119892(119909

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(119909))

(28)

Abstract and Applied Analysis 5

Now the application of the typical collocation method (seeeg [5]) gives

119877 (119909119894) = 0 119894 = 1 2 2

119896

(119872 + 1) minus 2 (29)

where 119909119894are the first (2119896(119872 + 1) minus 2) roots of 119881lowast

2119896(119872+1)

(119909) or119882lowast

2119896(119872+1)

(119909) Moreover the use of the boundary conditions(26) gives

1205720

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(0)

+ 1205721

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(0)

=

1198980

sum

119894=1

2119896

minus1

sum

119899=0

119872

sum

119898=0

120582119894119888119899119898120595119899119898(120585119894) + 1205750

1205730

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(1) + 120573

1

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(1)

=

1198981

sum

119894=1

2119896

minus1

sum

119899=0

119872

sum

119898=0

120583119894119888119899119898120595119899119898(120578119894) + 1205751

(30)

Equations (29) and (30) generate 2119896(119872 + 1) equations inthe unknown expansion coefficients 119888

119899119898 which can be

solved with the aid of the well-known Newtonrsquos iterativemethod Consequently we get the desired approximate solu-tion 119910

119896119872(119909) given by (27)

5 Numerical Examples

In this section the presented algorithms in Section 4 areapplied to solve both of linear and nonlinear multipointBVPs Some examples are considered to illustrate the effi-ciency and applicability of the two proposed algorithms

Example 1 Consider the second-order nonlinear BVP (see[6 28])

11991010158401015840

+3

8119910 +

2

1089(1199101015840

)2

+ 1 = 0 0 lt 119909 lt 1

119910 (0) = 0 119910 (1

3) = 119910 (1)

(31)

The two proposed methods are applied to the problem forthe case corresponding to 119896 = 0 and119872 = 8 The numericalsolutions are shown in Table 1 Due to nonavailability of theexact solution we compare our results with Haar waveletsmethod [6] ADM solution [28] and ODEs Solver fromMathematica which is carried out by using Runge-Kuttamethod This comparison is also shown in Table 1

Example 2 Consider the second-order linear BVP (see [3940])

11991010158401015840

= sinh119909 minus 2 0 lt 119909 lt 1

1199101015840

(0) = 0 119910 (1) = 3119910 (3

5)

(32)

The exact solution of problem (32) is given by

119910 (119909) =1

2(sinh 1 minus 3 sinh 3

5) + sinh119909 minus 1199092 minus 119909 + 11

25

(33)

In Table 2 the maximum absolute error 119864 is listed for119896 = 1 and various values of 119872 while in Table 3 wegive a comparison between the best errors resulted fromthe application of various methods for Example 2 while inFigure 1 we give a comparison between the exact solution of(32) with three approximate solutions

Example 3 Consider the second-order singular nonlinearBVP (see [40 41])

119909 (1 minus 119909) 11991010158401015840

+ 61199101015840

+ 2119910 + 1199102

= 6 cosh 119909 + (2 + 119909 minus 1199092 + sinh119909) sinh119909

0 lt 119909 lt 1

119910 (0) + 119910 (2

3) = sinh(2

3)

119910 (1) +1

2119910 (

4

5) =

1

2sinh(4

5) + sinh 1

(34)

with the exact solution 119910(119909) = sinh119909 In Table 4 the maxi-mum absolute error 119864 is listed for 119896 = 0 and various valuesof 119872 while in Table 5 we give a comparison between thebest errors resulted from the application of various methodsfor Example 3 This table shows that our two algorithms aremore accurate if compared with the two methods developedin [40 41]

Example 4 Consider the second-order nonlinear BVP (see[42])

11991010158401015840

+ (1 + 119909 + 1199093

) 1199102

= 119891 (119909) 0 lt 119909 lt 1

119910 (0) =1

6119910 (

2

9) +

1

3119910 (

7

9) minus 00286634

119910 (1) =1

5119910 (

2

9) +

1

2119910 (

7

9) minus 00401287

(35)

where

119891 (119909) =1

9[ minus 6 cos (119909 minus 1199092) + sin (119909 minus 1199092)

times (minus3(1 minus 2119909)2

+ (1 + 119909 + 1199093

) sin (119909 minus 1199092))]

(36)

6 Abstract and Applied Analysis

00 02 04 06 08 10

0010

0050

0020

0030

0015

0070

x

ln|y(x)|

Exactk = 0M = 4

k = 0M = 3

k = 0M = 2

Figure 1 Different solutions of Example 2

The exact solution of (35) is given by 119910(119909) = (13) sin(119909 minus1199092

) In Table 6 the maximum absolute error 119864 is listed for119896 = 2 and various values of 119872 and in Table 7 we give acomparison between best errors resulted from the applicationof various methods for Example 4 This table shows thatour two algorithms are more accurate if compared with themethod developed in [42]

Example 5 Consider the second-order singular linear BVP

11991010158401015840

+ 119891 (119909) 119910 = 119892 (119909) 0 lt 119909 lt 1

119910 (0) + 16119910 (1

4) = 3

4

radic119890

119910 (1) + 16119910 (3

4) = 3

4

radic1198903

(37)

where

119891 (119909) =

3119909 0 ⩽ 119909 ⩽1

2

21199091

2lt 119909 ⩽ 1

(38)

and 119892(119909) is chosen such that the exact solution of (37) is119910(119909) = 119909(1 minus 119909)119890

119909 In Table 8 the maximum absolute error 119864is listed for 119896 = 1 and various values of119872 while in Figure 2we give a comparison between the exact solution of (37) withthree approximate solutions

Example 6 Consider the following nonlinear second-orderBVP

11991010158401015840

+ (1199101015840

)2

minus 64119910 = 32 0 lt 119909 lt 1 (39)

119910 (0) + 119910 (1

4) = 1 (40)

4119910 (1

2) minus 119910 (1) = 0 (41)

with the exact solution 119910(119909) = 161199092 We solve (39) using

3CWCM for the case corresponding to 119896 = 0 and 119872 = 3

00 02 04 06 08 1000

01

02

03

04

05

x

Exactk = 1M = 3

k = 1M = 2

k = 1M = 1

y(x)

Figure 2 Different solutions of Example 5

to obtain an approximate solution of 119910(119909) If we make use of(27) then the approximate solution 119910

03(119909) can be expanded

in terms of third-kind Chebyshev wavelets as

11991003(119909) = 119888

00radic2

120587+ 11988801radic2

120587(4119909 minus 3)

+ 11988802radic2

120587(161199092

minus 20119909 + 5)

(42)

If we set

V119894= radic

2

1205871198880119894 119894 = 0 1 2 (43)

then (42) reduces to the form

11991003(119909) = V

0+ V1(4119909 minus 3) + V

2(161199092

minus 20119909 + 5) (44)

If we substitute (44) into (39) then the residual of (39) isgiven by

119877 (119909) = 2V2+ [V1+ V2(8119909 minus 5)]

2

minus 4 [V0+ V1(4119909 minus 3)

+ V2(161199092

minus 20119909 + 5)] minus 2

(45)

We enforce the residual to vanish at the first root of 119881lowast3(119909) =

641199093

minus1121199092

+56119909minus7 namely at 1199091= 018825509907063323

to get

610388V22+ 05V2

1minus 349396V

2V1minus 260388V

2

+ 449396V1minus 2V0= 1

(46)

Furthermore the use of the boundary conditions (40) and(41) yields

2V0minus 5V1+ 6V2= 1

3V0minus 5V1minus 5V2= 0

(47)

Abstract and Applied Analysis 7

Table 1 Comparison between different solutions for Example 1

119909 3CWCM 4CWCM Haar method [6] ADMmethod [28] ODEs solver fromMathematica01 006560 006560 006561 00656 00656003 016587 016587 016588 01658 01658705 022369 022369 022369 02236 02236907 023820 023820 023821 02382 02382009 020920 020920 020910 02092 020920

Table 2 The maximum absolute error 119864 for Example 2

119896 119872 10 11 12 13 14 15

0 3CWCM 5173 sdot 10minus9

7184 sdot 10minus11

4418 sdot 10minus12

5018 sdot 10minus14

2442 sdot 10minus15

2220 sdot 10minus16

4CWCM 3324 sdot 10minus10

2533 sdot 10minus11

2811 sdot 10minus12

1665 sdot 10minus14

1110 sdot 10minus15

2220 sdot 10minus16

119872 4 5 6 7 8 9

1 3CWCM 4644 sdot 10minus6

1001 sdot 10minus8

1840 sdot 10minus9

2547 sdot 10minus10

6247 sdot 10minus11

5681 sdot 10minus12

4CWCM 215 sdot 10minus6

5247 sdot 10minus9

7548 sdot 10minus10

1004 sdot 10minus10

2154 sdot 10minus11

4257 sdot 10minus12

Table 3 The best errors for Example 2

Method in [39] Method in [40] 3CWCM 4CWCM900 sdot 10

minus6

400 sdot 10minus5

222 sdot 10minus16

222 sdot 10minus16

Table 4 The maximum absolute error 119864 for Example 3

119872 8 9 10 11 12 133CWCM 4444 sdot 10

minus10

2759 sdot 10minus11

4389 sdot 10minus13

1643 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

4CWCM 4478 sdot 10minus9

2179 sdot 10minus10

1527 sdot 10minus12

2975 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

Table 5 The best errors for Example 3

Method in [40] Method in [41] 3CWCM 4CWCM300 sdot 10

minus8

660 sdot 10minus7

222 sdot 10minus16

222 sdot 10minus16

Table 6 The maximum absolute error 119864 for Example 4

119872 4 5 6 7 8 93CWCM 2881 sdot 10

minus3

3441 sdot 10minus3

1212 sdot 10minus5

1933 sdot 10minus5

1990 sdot 10minus6

3010 sdot 10minus8

4CWCM 1241 sdot 10minus3

8542 sdot 10minus4

7526 sdot 10minus6

1002 sdot 10minus6

6321 sdot 10minus7

2354 sdot 10minus9

Table 7 The best errors for Example 4

Method in [42] 3CWCM 4CWCM800 sdot 10

minus6

3010 sdot 10minus8

235 sdot 10minus9

Table 8 The maximum absolute error 119864 for Example 5

119872 7 8 9 10 11 12 13 143CWCM 71 sdot 10

minus6

32 sdot 10minus7

13 sdot 10minus8

74 sdot 10minus10

13 sdot 10minus11

35 sdot 10minus13

93 sdot 10minus15

81 sdot 10minus16

4CWCM 28 sdot 10minus5

15 sdot 10minus6

63 sdot 10minus8

24 sdot 10minus9

78 sdot 10minus11

23 sdot 10minus12

63 sdot 10minus14

17 sdot 10minus15

8 Abstract and Applied Analysis

The solution of the nonlinear system of (46) and (47) gives

V0= 10 V

1= 5 V

2= 1 (48)

and consequently

11991003(119909) = 10 + 5 (4119909 minus 3) + (16119909

2

minus 20119909 + 5) = 161199092

(49)

which is the exact solution

Remark 3 It is worth noting here that the obtained numericalresults in the previous solved six examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecomparing favorably with the known analytical solutions

6 Concluding Remarks

In this paper two algorithms for obtaining numerical spectralwavelets solutions for second-order multipoint linear andnonlinear boundary value problems are analyzed and dis-cussed Chebyshev polynomials of third and fourth kindsare used One of the advantages of the developed algorithmsis their availability for application on singular boundaryvalue problems Another advantage is that high accurateapproximate solutions are achieved using a few number ofterms of the approximate expansionThe obtained numericalresults are comparing favorably with the analytical ones

Acknowledgment

The authors would like to thank the referee for his valuablecomments and suggestions which improved the paper in itspresent form

References

[1] E H Doha W M Abd-Elhameed and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for the integrated formsof third- and fifth-order elliptic differential equations usinggeneral parameters generalized Jacobi polynomialsrdquo AppliedMathematics and Computation vol 218 no 15 pp 7727ndash77402012

[2] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[3] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013

[4] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[5] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Springer Berlin Germany 2006

[6] Siraj-ul-Islam I Aziz and B Sarler ldquoThe numerical solution ofsecond-order boundary-value problems by collocation methodwith theHaarwaveletsrdquoMathematical andComputerModellingvol 52 no 9-10 pp 1577ndash1590 2010

[7] A H Bhrawy A S Alofi and S I El-Soubhy ldquoAn exten-sion of the Legendre-Galerkin method for solving sixth-orderdifferential equations with variable polynomial coefficientsrdquoMathematical Problems in Engineering vol 2012 Article ID896575 13 pages 2012

[8] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[9] WMAbd-Elhameed ldquoEfficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of (2119899 + 1)th-order linear differential equationsrdquo Journal of the EgyptianMathematical Society vol 17 no 2 pp 189ndash211 2009

[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011

[11] A H Bhrawy A S Alofi and S I El-Soubhy ldquoSpectral shiftedJacobi tau and collocation methods for solving fifth-orderboundary value problemsrdquo Abstract and Applied Analysis vol2011 Article ID 823273 14 pages 2011

[12] M Moshiinsky ldquoSobre los problemas de condiciones a lafrontiera en una dimension de caracteristicas discontinuasrdquoBoletin de La Sociedad Matematica Mexicana vol 7 article 1251950

[13] S P TimoshenkoTheory of Elastic StabilityMcGraw-Hill BookNew York NY USA 2nd edition 1961

[14] R P Agarwal and I Kiguradze ldquoOnmulti-point boundary valueproblems for linear ordinary differential equations with singu-laritiesrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 131ndash151 2004

[15] Z Du ldquoSolvability of functional differential equations withmulti-point boundary value problems at resonancerdquoComputersamp Mathematics with Applications vol 55 no 11 pp 2653ndash26612008

[16] W Feng and J R L Webb ldquoSolvability of 119898-point boundaryvalue problemswith nonlinear growthrdquo Journal ofMathematicalAnalysis and Applications vol 212 no 2 pp 467ndash480 1997

[17] H B Thompson and C Tisdell ldquoThree-point boundary valueproblems for second-order ordinary differential equationsrdquoMathematical and Computer Modelling vol 34 no 3-4 pp 311ndash318 2001

[18] M R Scott and H A Watts ldquoSUPORTmdasha computer code fortwo-point boundary-value problems via orthonormalizationrdquoSandia Labs Report 75-0198 Sandia Laboratories AlbuquerqueNM USA 1975

[19] M R Scott and H A Watts ldquoComputational solution of lineartwo-point boundary value problems via orthonormalizationrdquoSIAM Journal on Numerical Analysis vol 14 no 1 pp 40ndash701977

[20] M R Scott and W H Vandevender ldquoA comparison of severalinvariant imbedding algorithms for the solution of two-pointboundary-value problemsrdquo Applied Mathematics and Compu-tation vol 1 no 3 pp 187ndash218 1975

[21] T YNaComputationalMethods in Engineering BoundaryValueProblems vol 145 Academic Press New York NY USA 1979

Abstract and Applied Analysis 9

[22] K E Bisshopp andDCDrucker ldquoLarge deflection of cantileverbeamsrdquo Quarterly of Applied Mathematics vol 3 pp 272ndash2751945

[23] W Glabisz ldquoThe use of Walsh-wavelet packets in linear bound-ary value problemsrdquo Computers amp Structures vol 82 no 2-3pp 131ndash141 2004

[24] Siraj-ul-Islam M A Noor I A Tirmizi and M A KhanldquoQuadratic non-polynomial spline approach to the solution ofa system of second-order boundary-value problemsrdquo AppliedMathematics and Computation vol 179 no 1 pp 153ndash160 2006

[25] S Robert and J Shipman ldquoSolution of Troeschrsquos two-pointboundary value problems by shooting techniquesrdquo Journal ofComputational Physics vol 10 pp 232ndash241 1972

[26] E Wiebel Confinement of a Plasma Column by RadiationPressure in the Plasma in a Magnetic Field Stanford UniversityPress Stanfor Calif USA 1958

[27] H H Keller and E S Holdrege ldquoRadiation heat transfer forannular fins of trapezoidal profilerdquo International Journal of HighPerformance Computing Applications vol 92 pp 113ndash116 1970

[28] M Tatari andM Dehgan ldquoThe use of the Adomian decomposi-tion method for solving multipoint boundary value problemsrdquoPhysica Scripta vol 73 pp 672ndash676 2006

[29] M Lakestani andM Dehghan ldquoThe solution of a second-ordernonlinear differential equation with Neumann boundary con-ditions using semi-orthogonal B-spline waveletsrdquo InternationalJournal of Computer Mathematics vol 83 no 8-9 pp 685ndash6942006

[30] A Constantmldes Applied Numerical Methods with PersonalComputers McGraw-Hill New York NY USA 1987

[31] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[32] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[33] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[34] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[35] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[36] E H Doha W M Abd- Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[37] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[38] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp Hall New York NY USA 2003

[39] Y Lin J Niu and M Cui ldquoA numerical solution to nonlinearsecond order three-point boundary value problems in thereproducing kernel spacerdquo Applied Mathematics and Computa-tion vol 218 no 14 pp 7362ndash7368 2012

[40] F Geng ldquoSolving singular second order three-point bound-ary value problems using reproducing kernel Hilbert spacemethodrdquo Applied Mathematics and Computation vol 215 no6 pp 2095ndash2102 2009

[41] F Z Geng ldquoA numerical algorithm for nonlinear multi-point boundary value problemsrdquo Journal of Computational andApplied Mathematics vol 236 no 7 pp 1789ndash1794 2012

[42] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

Abstract and Applied Analysis 3

The polynomials 119881119899(119909) and 119882

119899(119909) are orthogonal on

(minus1 1) that is

int

1

minus1

1205961(119909)119881119896(119909) 119881119895(119909) 119889119909

= int

1

minus1

1205962(119909)119882

119896(119909)119882

119895(119909) 119889119909

= 120587 119896 = 119895

0 119896 = 119895

(4)

where

1205961(119909) = radic

1 + 119909

1 minus 119909 120596

2(119909) = radic

1 minus 119909

1 + 119909 (5)

and they may be generated by using the two recurrencerelations

119881119896(119909) = 2119909119881

119896minus1(119909) minus 119881

119896minus2(119909) 119896 = 2 3 (6)

with the initial values

1198810(119909) = 1 119881

1(119909) = 2119909 minus 1 (7)

119882119896(119909) = 2119909 119882

119896minus1(119909) minus 119882

119896minus2(119909) 119896 = 2 3 (8)

with the initial values

1198820(119909) = 1 119882

1(119909) = 2119909 + 1 (9)

The shifted Chebyshev polynomials of third and fourthkinds are defined on [0 1] respectively as

119881lowast

119899(119909) = 119881

119899(2119909 minus 1) 119882

lowast

119899(119909) = 119882

119899(2119909 minus 1) (10)

All results of Chebyshev polynomials of third and fourthkinds can be easily transformed to give the correspondingresults for their shifted ones

The orthogonality relations of 119881lowast119899(119905) and119882lowast

119899(119905) on [0 1]

are given by

int

1

0

119908lowast

1119881lowast

119898(119905) 119881lowast

119899(119905) 119889119905

= int

1

0

119908lowast

2119882lowast

119898(119905)119882lowast

119899(119905) 119889119905

=

120587

2 119898 = 119899

0 119898 = 119899

(11)

where

120596lowast

1= radic

119905

1 minus 119905 120596

lowast

2= radic

1 minus 119905

119905 (12)

3 Shifted Third- and Fourth-KindChebyshev Wavelets

Wavelets constitute of a family of functions constructed fromdilation and translation of single function called the mother

wavelet When the dilation parameter 119886 and the translationparameter 119887 vary continuously then we have the followingfamily of continuous wavelets

120595119886119887(119905) = |119886|

minus12

120595(119905 minus 119887

119886) 119886 119887 isin R 119886 = 0 (13)

Each of the third- and fourth-kind Chebyshev wavelets120595119899119898(119905) = 120595(119896 119899119898 119905) has four arguments 119896 119899 isin N 119898 is

the order of the polynomial 119881lowast119898(119905) or 119882lowast

119898(119905) and 119905 is the

normalized time They are defined explicitly on the interval[0 1] as

120595119899119898(119905)

=

2(119896+1)2

radic120587119881lowast

119898(2119896

119905 minus 119899)

resp

2(119896+1)2

radic120587119882lowast

119898(2119896

119905 minus 119899) 119905 isin [119899

2119896119899 + 1

2119896] 0 ⩽ 119898 ⩽ 119872

0 ⩽ 119899 ⩽ 2119896

minus 1

0 otherwise

(14)

31 Function Approximation A function 119891(119905) defined over[0 1]may be expanded in terms of Chebyshev wavelets as

119891 (119905) =

infin

sum

119899=0

infin

sum

119898=0

119888119899119898120595119899119898(119905) (15)

where

119888119899119898= (119891 (119905) 120595

119899119898(119905))120596lowast

119894

= int

1

0

120596lowast

119894119891 (119905) 120595

119899119898(119905) 119889119905 (16)

and the weights 119908lowast119894 119894 = 1 2 are given in (12)

Assume that 119891(119905) can be approximated in terms ofChebyshev wavelets as

119891 (119905) ≃

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(119905) (17)

32 Convergence Analysis In this section we state and provea theorem to ascertain that the third- and fourth-kindCheby-shev wavelets expansion of a function 119891(119905) with boundedsecond derivative converges uniformly to 119891(119905)

Theorem 1 Assume that a function 119891(119905) isin 1198712

120596lowast

1

[0 1] 120596lowast1=

radic119905(1 minus 119905) with |11989110158401015840(119905)| ⩽ 119871 can be expanded as an infiniteseries of third-kind Chebyshev wavelets then this series con-verges uniformly to 119891(119905) Explicitly the expansion coefficientsin (16) satisfy the following inequality

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

2radic21205871198711198982

(119899 + 1)52

(1198984 minus 1)

forall119899 ⩾ 0 119898 gt 1 (18)

4 Abstract and Applied Analysis

Proof From (16) it follows that

119888119899119898=2(119896+1)2

radic120587int

(119899+1)2119896

1198992119896

119891 (119905) 119881lowast

119898(2119896

119905 minus 119899) 120596lowast

1(2119896

119905 minus 119899) 119889119905

(19)

If we make use of the substitution 2119896119905minus119899 = cos 120579 in (19) thenwe get

119888119899119898=2(minus119896+1)2

radic120587int

120587

0

119891(cos 120579 + 1198992119896

)

timescos (119898 + (12)) 120579

cos (1205792)radic1 + cos 1205791 minus cos 120579

sin 120579119889120579

=2(minus119896+3)2

radic120587int

120587

0

119891(cos 120579 + 1198992119896

) cos(119898 +1

2) 120579 cos(120579

2)119889120579

=2(minus119896+1)2

radic120587int

120587

0

119891(cos 120579 + 1198992119896

)

times [cos (119898 + 1) 120579 + cos119898120579] 119889120579(20)

which in turn and after performing integration by parts twotimes yields

119888119899119898=

1

251198962radic2120587int

120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579 (21)

where

120574119898(120579) =

1

119898 + 1(sin119898120579119898

minussin (119898 + 2) 120579

119898 + 2)

+1

119898(sin (119898 minus 1) 120579

119898 minus 1minussin (119898 + 1) 120579

119898 + 1)

(22)

Now we have

10038161003816100381610038161198881198991198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1

251198962radic2120587int

120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

=1

251198962radic2120587

10038161003816100381610038161003816100381610038161003816

int

120587

0

11989110158401015840

(cos 120579 + 1198992119896

) 120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

⩽119871

251198962radic2120587int

120587

0

1003816100381610038161003816120574119898 (120579)1003816100381610038161003816 119889120579

⩽119871radic120587

2(5119896+1)2[

1

119898 + 1(1

119898+

1

119898 + 2)

+1

119898(

1

119898 minus 1+

1

119898 + 1)]

=119871radic2120587

251198962[

1

1198982 + 2119898+

1

1198982 minus 1]

lt2119871radic2120587

251198962(

1198982

1198984 minus 1)

(23)

Finally since 119899 ⩽ 2119896 minus 1 we have

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

2radic21205871198711198982

(119899 + 1)52

(1198984 minus 1)

(24)

Remark 2 The estimation in (18) is also valid for the coef-ficients of fourth-kind Chebyshev wavelets expansion Theproof is similar to the proof of Theorem 1

4 Solution of Multipoint BVPs

In this section we present two Chebyshev wavelets col-location methods namely third-kind Chebyshev waveletscollocation method (3CWCM) and fourth-kind Chebyshevwavelets collocationmethod (4CWCM) to numerically solvethe following multipoint boundary value problem (BVP)

119886 (119909) 11991010158401015840

(119909) + 119887 (119909) 1199101015840

(119909) + 119888 (119909) 119910 (119909)

+ 119891 (119909 1199101015840

) + 119892 (119909 119910) = 0 0 ⩽ 119909 ⩽ 1

(25)

1205720119910 (0) + 120572

11199101015840

(0) =

1198980

sum

119894=1

120582119894119910 (120585119894) + 1205750

1205730119910 (1) + 120573

11199101015840

(1) =

1198981

sum

119894=1

120583119894119910 (120578119894) + 1205751

(26)

where 119886(119909) 119887(119909) and 119888(119909) are piecewise continuous on [0 1]119886(0)119886(1) may equal zero 0 lt 120585

119894 120578119894lt 1 120572

119894 120573119894 120582119894 120583119894 and 120575

119894

are constants such that (12057220+1205722

1)(1205732

0+1205732

1) = 0 119891 is a nonlinear

function of 1199101015840 and 119892 is a nonlinear function in 119910Consider an approximate solution to (25) and (26) which

is given in terms of Chebyshev wavelets as

119910119896119872

(119909) =

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(119909) (27)

then the substitution of (27) into (25) enables one to write theresidual of (25) in the form

119877 (119909) =

2119896

minus1

sum

119899=0

119872

sum

119898=2

119888119899119898119886 (119909) 120595

10158401015840

119899119898(119909)

+

2119896

minus1

sum

119899=0

119872

sum

119898=1

119888119899119898119887 (119909) 120595

1015840

119899119898(119909)

+

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898119888 (119909) 120595

119899119898(119909)

+ 119891(119909

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(119909))

+ 119892(119909

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(119909))

(28)

Abstract and Applied Analysis 5

Now the application of the typical collocation method (seeeg [5]) gives

119877 (119909119894) = 0 119894 = 1 2 2

119896

(119872 + 1) minus 2 (29)

where 119909119894are the first (2119896(119872 + 1) minus 2) roots of 119881lowast

2119896(119872+1)

(119909) or119882lowast

2119896(119872+1)

(119909) Moreover the use of the boundary conditions(26) gives

1205720

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(0)

+ 1205721

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(0)

=

1198980

sum

119894=1

2119896

minus1

sum

119899=0

119872

sum

119898=0

120582119894119888119899119898120595119899119898(120585119894) + 1205750

1205730

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(1) + 120573

1

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(1)

=

1198981

sum

119894=1

2119896

minus1

sum

119899=0

119872

sum

119898=0

120583119894119888119899119898120595119899119898(120578119894) + 1205751

(30)

Equations (29) and (30) generate 2119896(119872 + 1) equations inthe unknown expansion coefficients 119888

119899119898 which can be

solved with the aid of the well-known Newtonrsquos iterativemethod Consequently we get the desired approximate solu-tion 119910

119896119872(119909) given by (27)

5 Numerical Examples

In this section the presented algorithms in Section 4 areapplied to solve both of linear and nonlinear multipointBVPs Some examples are considered to illustrate the effi-ciency and applicability of the two proposed algorithms

Example 1 Consider the second-order nonlinear BVP (see[6 28])

11991010158401015840

+3

8119910 +

2

1089(1199101015840

)2

+ 1 = 0 0 lt 119909 lt 1

119910 (0) = 0 119910 (1

3) = 119910 (1)

(31)

The two proposed methods are applied to the problem forthe case corresponding to 119896 = 0 and119872 = 8 The numericalsolutions are shown in Table 1 Due to nonavailability of theexact solution we compare our results with Haar waveletsmethod [6] ADM solution [28] and ODEs Solver fromMathematica which is carried out by using Runge-Kuttamethod This comparison is also shown in Table 1

Example 2 Consider the second-order linear BVP (see [3940])

11991010158401015840

= sinh119909 minus 2 0 lt 119909 lt 1

1199101015840

(0) = 0 119910 (1) = 3119910 (3

5)

(32)

The exact solution of problem (32) is given by

119910 (119909) =1

2(sinh 1 minus 3 sinh 3

5) + sinh119909 minus 1199092 minus 119909 + 11

25

(33)

In Table 2 the maximum absolute error 119864 is listed for119896 = 1 and various values of 119872 while in Table 3 wegive a comparison between the best errors resulted fromthe application of various methods for Example 2 while inFigure 1 we give a comparison between the exact solution of(32) with three approximate solutions

Example 3 Consider the second-order singular nonlinearBVP (see [40 41])

119909 (1 minus 119909) 11991010158401015840

+ 61199101015840

+ 2119910 + 1199102

= 6 cosh 119909 + (2 + 119909 minus 1199092 + sinh119909) sinh119909

0 lt 119909 lt 1

119910 (0) + 119910 (2

3) = sinh(2

3)

119910 (1) +1

2119910 (

4

5) =

1

2sinh(4

5) + sinh 1

(34)

with the exact solution 119910(119909) = sinh119909 In Table 4 the maxi-mum absolute error 119864 is listed for 119896 = 0 and various valuesof 119872 while in Table 5 we give a comparison between thebest errors resulted from the application of various methodsfor Example 3 This table shows that our two algorithms aremore accurate if compared with the two methods developedin [40 41]

Example 4 Consider the second-order nonlinear BVP (see[42])

11991010158401015840

+ (1 + 119909 + 1199093

) 1199102

= 119891 (119909) 0 lt 119909 lt 1

119910 (0) =1

6119910 (

2

9) +

1

3119910 (

7

9) minus 00286634

119910 (1) =1

5119910 (

2

9) +

1

2119910 (

7

9) minus 00401287

(35)

where

119891 (119909) =1

9[ minus 6 cos (119909 minus 1199092) + sin (119909 minus 1199092)

times (minus3(1 minus 2119909)2

+ (1 + 119909 + 1199093

) sin (119909 minus 1199092))]

(36)

6 Abstract and Applied Analysis

00 02 04 06 08 10

0010

0050

0020

0030

0015

0070

x

ln|y(x)|

Exactk = 0M = 4

k = 0M = 3

k = 0M = 2

Figure 1 Different solutions of Example 2

The exact solution of (35) is given by 119910(119909) = (13) sin(119909 minus1199092

) In Table 6 the maximum absolute error 119864 is listed for119896 = 2 and various values of 119872 and in Table 7 we give acomparison between best errors resulted from the applicationof various methods for Example 4 This table shows thatour two algorithms are more accurate if compared with themethod developed in [42]

Example 5 Consider the second-order singular linear BVP

11991010158401015840

+ 119891 (119909) 119910 = 119892 (119909) 0 lt 119909 lt 1

119910 (0) + 16119910 (1

4) = 3

4

radic119890

119910 (1) + 16119910 (3

4) = 3

4

radic1198903

(37)

where

119891 (119909) =

3119909 0 ⩽ 119909 ⩽1

2

21199091

2lt 119909 ⩽ 1

(38)

and 119892(119909) is chosen such that the exact solution of (37) is119910(119909) = 119909(1 minus 119909)119890

119909 In Table 8 the maximum absolute error 119864is listed for 119896 = 1 and various values of119872 while in Figure 2we give a comparison between the exact solution of (37) withthree approximate solutions

Example 6 Consider the following nonlinear second-orderBVP

11991010158401015840

+ (1199101015840

)2

minus 64119910 = 32 0 lt 119909 lt 1 (39)

119910 (0) + 119910 (1

4) = 1 (40)

4119910 (1

2) minus 119910 (1) = 0 (41)

with the exact solution 119910(119909) = 161199092 We solve (39) using

3CWCM for the case corresponding to 119896 = 0 and 119872 = 3

00 02 04 06 08 1000

01

02

03

04

05

x

Exactk = 1M = 3

k = 1M = 2

k = 1M = 1

y(x)

Figure 2 Different solutions of Example 5

to obtain an approximate solution of 119910(119909) If we make use of(27) then the approximate solution 119910

03(119909) can be expanded

in terms of third-kind Chebyshev wavelets as

11991003(119909) = 119888

00radic2

120587+ 11988801radic2

120587(4119909 minus 3)

+ 11988802radic2

120587(161199092

minus 20119909 + 5)

(42)

If we set

V119894= radic

2

1205871198880119894 119894 = 0 1 2 (43)

then (42) reduces to the form

11991003(119909) = V

0+ V1(4119909 minus 3) + V

2(161199092

minus 20119909 + 5) (44)

If we substitute (44) into (39) then the residual of (39) isgiven by

119877 (119909) = 2V2+ [V1+ V2(8119909 minus 5)]

2

minus 4 [V0+ V1(4119909 minus 3)

+ V2(161199092

minus 20119909 + 5)] minus 2

(45)

We enforce the residual to vanish at the first root of 119881lowast3(119909) =

641199093

minus1121199092

+56119909minus7 namely at 1199091= 018825509907063323

to get

610388V22+ 05V2

1minus 349396V

2V1minus 260388V

2

+ 449396V1minus 2V0= 1

(46)

Furthermore the use of the boundary conditions (40) and(41) yields

2V0minus 5V1+ 6V2= 1

3V0minus 5V1minus 5V2= 0

(47)

Abstract and Applied Analysis 7

Table 1 Comparison between different solutions for Example 1

119909 3CWCM 4CWCM Haar method [6] ADMmethod [28] ODEs solver fromMathematica01 006560 006560 006561 00656 00656003 016587 016587 016588 01658 01658705 022369 022369 022369 02236 02236907 023820 023820 023821 02382 02382009 020920 020920 020910 02092 020920

Table 2 The maximum absolute error 119864 for Example 2

119896 119872 10 11 12 13 14 15

0 3CWCM 5173 sdot 10minus9

7184 sdot 10minus11

4418 sdot 10minus12

5018 sdot 10minus14

2442 sdot 10minus15

2220 sdot 10minus16

4CWCM 3324 sdot 10minus10

2533 sdot 10minus11

2811 sdot 10minus12

1665 sdot 10minus14

1110 sdot 10minus15

2220 sdot 10minus16

119872 4 5 6 7 8 9

1 3CWCM 4644 sdot 10minus6

1001 sdot 10minus8

1840 sdot 10minus9

2547 sdot 10minus10

6247 sdot 10minus11

5681 sdot 10minus12

4CWCM 215 sdot 10minus6

5247 sdot 10minus9

7548 sdot 10minus10

1004 sdot 10minus10

2154 sdot 10minus11

4257 sdot 10minus12

Table 3 The best errors for Example 2

Method in [39] Method in [40] 3CWCM 4CWCM900 sdot 10

minus6

400 sdot 10minus5

222 sdot 10minus16

222 sdot 10minus16

Table 4 The maximum absolute error 119864 for Example 3

119872 8 9 10 11 12 133CWCM 4444 sdot 10

minus10

2759 sdot 10minus11

4389 sdot 10minus13

1643 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

4CWCM 4478 sdot 10minus9

2179 sdot 10minus10

1527 sdot 10minus12

2975 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

Table 5 The best errors for Example 3

Method in [40] Method in [41] 3CWCM 4CWCM300 sdot 10

minus8

660 sdot 10minus7

222 sdot 10minus16

222 sdot 10minus16

Table 6 The maximum absolute error 119864 for Example 4

119872 4 5 6 7 8 93CWCM 2881 sdot 10

minus3

3441 sdot 10minus3

1212 sdot 10minus5

1933 sdot 10minus5

1990 sdot 10minus6

3010 sdot 10minus8

4CWCM 1241 sdot 10minus3

8542 sdot 10minus4

7526 sdot 10minus6

1002 sdot 10minus6

6321 sdot 10minus7

2354 sdot 10minus9

Table 7 The best errors for Example 4

Method in [42] 3CWCM 4CWCM800 sdot 10

minus6

3010 sdot 10minus8

235 sdot 10minus9

Table 8 The maximum absolute error 119864 for Example 5

119872 7 8 9 10 11 12 13 143CWCM 71 sdot 10

minus6

32 sdot 10minus7

13 sdot 10minus8

74 sdot 10minus10

13 sdot 10minus11

35 sdot 10minus13

93 sdot 10minus15

81 sdot 10minus16

4CWCM 28 sdot 10minus5

15 sdot 10minus6

63 sdot 10minus8

24 sdot 10minus9

78 sdot 10minus11

23 sdot 10minus12

63 sdot 10minus14

17 sdot 10minus15

8 Abstract and Applied Analysis

The solution of the nonlinear system of (46) and (47) gives

V0= 10 V

1= 5 V

2= 1 (48)

and consequently

11991003(119909) = 10 + 5 (4119909 minus 3) + (16119909

2

minus 20119909 + 5) = 161199092

(49)

which is the exact solution

Remark 3 It is worth noting here that the obtained numericalresults in the previous solved six examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecomparing favorably with the known analytical solutions

6 Concluding Remarks

In this paper two algorithms for obtaining numerical spectralwavelets solutions for second-order multipoint linear andnonlinear boundary value problems are analyzed and dis-cussed Chebyshev polynomials of third and fourth kindsare used One of the advantages of the developed algorithmsis their availability for application on singular boundaryvalue problems Another advantage is that high accurateapproximate solutions are achieved using a few number ofterms of the approximate expansionThe obtained numericalresults are comparing favorably with the analytical ones

Acknowledgment

The authors would like to thank the referee for his valuablecomments and suggestions which improved the paper in itspresent form

References

[1] E H Doha W M Abd-Elhameed and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for the integrated formsof third- and fifth-order elliptic differential equations usinggeneral parameters generalized Jacobi polynomialsrdquo AppliedMathematics and Computation vol 218 no 15 pp 7727ndash77402012

[2] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[3] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013

[4] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[5] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Springer Berlin Germany 2006

[6] Siraj-ul-Islam I Aziz and B Sarler ldquoThe numerical solution ofsecond-order boundary-value problems by collocation methodwith theHaarwaveletsrdquoMathematical andComputerModellingvol 52 no 9-10 pp 1577ndash1590 2010

[7] A H Bhrawy A S Alofi and S I El-Soubhy ldquoAn exten-sion of the Legendre-Galerkin method for solving sixth-orderdifferential equations with variable polynomial coefficientsrdquoMathematical Problems in Engineering vol 2012 Article ID896575 13 pages 2012

[8] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[9] WMAbd-Elhameed ldquoEfficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of (2119899 + 1)th-order linear differential equationsrdquo Journal of the EgyptianMathematical Society vol 17 no 2 pp 189ndash211 2009

[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011

[11] A H Bhrawy A S Alofi and S I El-Soubhy ldquoSpectral shiftedJacobi tau and collocation methods for solving fifth-orderboundary value problemsrdquo Abstract and Applied Analysis vol2011 Article ID 823273 14 pages 2011

[12] M Moshiinsky ldquoSobre los problemas de condiciones a lafrontiera en una dimension de caracteristicas discontinuasrdquoBoletin de La Sociedad Matematica Mexicana vol 7 article 1251950

[13] S P TimoshenkoTheory of Elastic StabilityMcGraw-Hill BookNew York NY USA 2nd edition 1961

[14] R P Agarwal and I Kiguradze ldquoOnmulti-point boundary valueproblems for linear ordinary differential equations with singu-laritiesrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 131ndash151 2004

[15] Z Du ldquoSolvability of functional differential equations withmulti-point boundary value problems at resonancerdquoComputersamp Mathematics with Applications vol 55 no 11 pp 2653ndash26612008

[16] W Feng and J R L Webb ldquoSolvability of 119898-point boundaryvalue problemswith nonlinear growthrdquo Journal ofMathematicalAnalysis and Applications vol 212 no 2 pp 467ndash480 1997

[17] H B Thompson and C Tisdell ldquoThree-point boundary valueproblems for second-order ordinary differential equationsrdquoMathematical and Computer Modelling vol 34 no 3-4 pp 311ndash318 2001

[18] M R Scott and H A Watts ldquoSUPORTmdasha computer code fortwo-point boundary-value problems via orthonormalizationrdquoSandia Labs Report 75-0198 Sandia Laboratories AlbuquerqueNM USA 1975

[19] M R Scott and H A Watts ldquoComputational solution of lineartwo-point boundary value problems via orthonormalizationrdquoSIAM Journal on Numerical Analysis vol 14 no 1 pp 40ndash701977

[20] M R Scott and W H Vandevender ldquoA comparison of severalinvariant imbedding algorithms for the solution of two-pointboundary-value problemsrdquo Applied Mathematics and Compu-tation vol 1 no 3 pp 187ndash218 1975

[21] T YNaComputationalMethods in Engineering BoundaryValueProblems vol 145 Academic Press New York NY USA 1979

Abstract and Applied Analysis 9

[22] K E Bisshopp andDCDrucker ldquoLarge deflection of cantileverbeamsrdquo Quarterly of Applied Mathematics vol 3 pp 272ndash2751945

[23] W Glabisz ldquoThe use of Walsh-wavelet packets in linear bound-ary value problemsrdquo Computers amp Structures vol 82 no 2-3pp 131ndash141 2004

[24] Siraj-ul-Islam M A Noor I A Tirmizi and M A KhanldquoQuadratic non-polynomial spline approach to the solution ofa system of second-order boundary-value problemsrdquo AppliedMathematics and Computation vol 179 no 1 pp 153ndash160 2006

[25] S Robert and J Shipman ldquoSolution of Troeschrsquos two-pointboundary value problems by shooting techniquesrdquo Journal ofComputational Physics vol 10 pp 232ndash241 1972

[26] E Wiebel Confinement of a Plasma Column by RadiationPressure in the Plasma in a Magnetic Field Stanford UniversityPress Stanfor Calif USA 1958

[27] H H Keller and E S Holdrege ldquoRadiation heat transfer forannular fins of trapezoidal profilerdquo International Journal of HighPerformance Computing Applications vol 92 pp 113ndash116 1970

[28] M Tatari andM Dehgan ldquoThe use of the Adomian decomposi-tion method for solving multipoint boundary value problemsrdquoPhysica Scripta vol 73 pp 672ndash676 2006

[29] M Lakestani andM Dehghan ldquoThe solution of a second-ordernonlinear differential equation with Neumann boundary con-ditions using semi-orthogonal B-spline waveletsrdquo InternationalJournal of Computer Mathematics vol 83 no 8-9 pp 685ndash6942006

[30] A Constantmldes Applied Numerical Methods with PersonalComputers McGraw-Hill New York NY USA 1987

[31] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[32] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[33] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[34] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[35] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[36] E H Doha W M Abd- Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[37] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[38] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp Hall New York NY USA 2003

[39] Y Lin J Niu and M Cui ldquoA numerical solution to nonlinearsecond order three-point boundary value problems in thereproducing kernel spacerdquo Applied Mathematics and Computa-tion vol 218 no 14 pp 7362ndash7368 2012

[40] F Geng ldquoSolving singular second order three-point bound-ary value problems using reproducing kernel Hilbert spacemethodrdquo Applied Mathematics and Computation vol 215 no6 pp 2095ndash2102 2009

[41] F Z Geng ldquoA numerical algorithm for nonlinear multi-point boundary value problemsrdquo Journal of Computational andApplied Mathematics vol 236 no 7 pp 1789ndash1794 2012

[42] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

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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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International 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

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Proof From (16) it follows that

119888119899119898=2(119896+1)2

radic120587int

(119899+1)2119896

1198992119896

119891 (119905) 119881lowast

119898(2119896

119905 minus 119899) 120596lowast

1(2119896

119905 minus 119899) 119889119905

(19)

If we make use of the substitution 2119896119905minus119899 = cos 120579 in (19) thenwe get

119888119899119898=2(minus119896+1)2

radic120587int

120587

0

119891(cos 120579 + 1198992119896

)

timescos (119898 + (12)) 120579

cos (1205792)radic1 + cos 1205791 minus cos 120579

sin 120579119889120579

=2(minus119896+3)2

radic120587int

120587

0

119891(cos 120579 + 1198992119896

) cos(119898 +1

2) 120579 cos(120579

2)119889120579

=2(minus119896+1)2

radic120587int

120587

0

119891(cos 120579 + 1198992119896

)

times [cos (119898 + 1) 120579 + cos119898120579] 119889120579(20)

which in turn and after performing integration by parts twotimes yields

119888119899119898=

1

251198962radic2120587int

120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579 (21)

where

120574119898(120579) =

1

119898 + 1(sin119898120579119898

minussin (119898 + 2) 120579

119898 + 2)

+1

119898(sin (119898 minus 1) 120579

119898 minus 1minussin (119898 + 1) 120579

119898 + 1)

(22)

Now we have

10038161003816100381610038161198881198991198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1

251198962radic2120587int

120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

=1

251198962radic2120587

10038161003816100381610038161003816100381610038161003816

int

120587

0

11989110158401015840

(cos 120579 + 1198992119896

) 120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

⩽119871

251198962radic2120587int

120587

0

1003816100381610038161003816120574119898 (120579)1003816100381610038161003816 119889120579

⩽119871radic120587

2(5119896+1)2[

1

119898 + 1(1

119898+

1

119898 + 2)

+1

119898(

1

119898 minus 1+

1

119898 + 1)]

=119871radic2120587

251198962[

1

1198982 + 2119898+

1

1198982 minus 1]

lt2119871radic2120587

251198962(

1198982

1198984 minus 1)

(23)

Finally since 119899 ⩽ 2119896 minus 1 we have

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

2radic21205871198711198982

(119899 + 1)52

(1198984 minus 1)

(24)

Remark 2 The estimation in (18) is also valid for the coef-ficients of fourth-kind Chebyshev wavelets expansion Theproof is similar to the proof of Theorem 1

4 Solution of Multipoint BVPs

In this section we present two Chebyshev wavelets col-location methods namely third-kind Chebyshev waveletscollocation method (3CWCM) and fourth-kind Chebyshevwavelets collocationmethod (4CWCM) to numerically solvethe following multipoint boundary value problem (BVP)

119886 (119909) 11991010158401015840

(119909) + 119887 (119909) 1199101015840

(119909) + 119888 (119909) 119910 (119909)

+ 119891 (119909 1199101015840

) + 119892 (119909 119910) = 0 0 ⩽ 119909 ⩽ 1

(25)

1205720119910 (0) + 120572

11199101015840

(0) =

1198980

sum

119894=1

120582119894119910 (120585119894) + 1205750

1205730119910 (1) + 120573

11199101015840

(1) =

1198981

sum

119894=1

120583119894119910 (120578119894) + 1205751

(26)

where 119886(119909) 119887(119909) and 119888(119909) are piecewise continuous on [0 1]119886(0)119886(1) may equal zero 0 lt 120585

119894 120578119894lt 1 120572

119894 120573119894 120582119894 120583119894 and 120575

119894

are constants such that (12057220+1205722

1)(1205732

0+1205732

1) = 0 119891 is a nonlinear

function of 1199101015840 and 119892 is a nonlinear function in 119910Consider an approximate solution to (25) and (26) which

is given in terms of Chebyshev wavelets as

119910119896119872

(119909) =

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(119909) (27)

then the substitution of (27) into (25) enables one to write theresidual of (25) in the form

119877 (119909) =

2119896

minus1

sum

119899=0

119872

sum

119898=2

119888119899119898119886 (119909) 120595

10158401015840

119899119898(119909)

+

2119896

minus1

sum

119899=0

119872

sum

119898=1

119888119899119898119887 (119909) 120595

1015840

119899119898(119909)

+

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898119888 (119909) 120595

119899119898(119909)

+ 119891(119909

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(119909))

+ 119892(119909

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(119909))

(28)

Abstract and Applied Analysis 5

Now the application of the typical collocation method (seeeg [5]) gives

119877 (119909119894) = 0 119894 = 1 2 2

119896

(119872 + 1) minus 2 (29)

where 119909119894are the first (2119896(119872 + 1) minus 2) roots of 119881lowast

2119896(119872+1)

(119909) or119882lowast

2119896(119872+1)

(119909) Moreover the use of the boundary conditions(26) gives

1205720

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(0)

+ 1205721

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(0)

=

1198980

sum

119894=1

2119896

minus1

sum

119899=0

119872

sum

119898=0

120582119894119888119899119898120595119899119898(120585119894) + 1205750

1205730

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(1) + 120573

1

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(1)

=

1198981

sum

119894=1

2119896

minus1

sum

119899=0

119872

sum

119898=0

120583119894119888119899119898120595119899119898(120578119894) + 1205751

(30)

Equations (29) and (30) generate 2119896(119872 + 1) equations inthe unknown expansion coefficients 119888

119899119898 which can be

solved with the aid of the well-known Newtonrsquos iterativemethod Consequently we get the desired approximate solu-tion 119910

119896119872(119909) given by (27)

5 Numerical Examples

In this section the presented algorithms in Section 4 areapplied to solve both of linear and nonlinear multipointBVPs Some examples are considered to illustrate the effi-ciency and applicability of the two proposed algorithms

Example 1 Consider the second-order nonlinear BVP (see[6 28])

11991010158401015840

+3

8119910 +

2

1089(1199101015840

)2

+ 1 = 0 0 lt 119909 lt 1

119910 (0) = 0 119910 (1

3) = 119910 (1)

(31)

The two proposed methods are applied to the problem forthe case corresponding to 119896 = 0 and119872 = 8 The numericalsolutions are shown in Table 1 Due to nonavailability of theexact solution we compare our results with Haar waveletsmethod [6] ADM solution [28] and ODEs Solver fromMathematica which is carried out by using Runge-Kuttamethod This comparison is also shown in Table 1

Example 2 Consider the second-order linear BVP (see [3940])

11991010158401015840

= sinh119909 minus 2 0 lt 119909 lt 1

1199101015840

(0) = 0 119910 (1) = 3119910 (3

5)

(32)

The exact solution of problem (32) is given by

119910 (119909) =1

2(sinh 1 minus 3 sinh 3

5) + sinh119909 minus 1199092 minus 119909 + 11

25

(33)

In Table 2 the maximum absolute error 119864 is listed for119896 = 1 and various values of 119872 while in Table 3 wegive a comparison between the best errors resulted fromthe application of various methods for Example 2 while inFigure 1 we give a comparison between the exact solution of(32) with three approximate solutions

Example 3 Consider the second-order singular nonlinearBVP (see [40 41])

119909 (1 minus 119909) 11991010158401015840

+ 61199101015840

+ 2119910 + 1199102

= 6 cosh 119909 + (2 + 119909 minus 1199092 + sinh119909) sinh119909

0 lt 119909 lt 1

119910 (0) + 119910 (2

3) = sinh(2

3)

119910 (1) +1

2119910 (

4

5) =

1

2sinh(4

5) + sinh 1

(34)

with the exact solution 119910(119909) = sinh119909 In Table 4 the maxi-mum absolute error 119864 is listed for 119896 = 0 and various valuesof 119872 while in Table 5 we give a comparison between thebest errors resulted from the application of various methodsfor Example 3 This table shows that our two algorithms aremore accurate if compared with the two methods developedin [40 41]

Example 4 Consider the second-order nonlinear BVP (see[42])

11991010158401015840

+ (1 + 119909 + 1199093

) 1199102

= 119891 (119909) 0 lt 119909 lt 1

119910 (0) =1

6119910 (

2

9) +

1

3119910 (

7

9) minus 00286634

119910 (1) =1

5119910 (

2

9) +

1

2119910 (

7

9) minus 00401287

(35)

where

119891 (119909) =1

9[ minus 6 cos (119909 minus 1199092) + sin (119909 minus 1199092)

times (minus3(1 minus 2119909)2

+ (1 + 119909 + 1199093

) sin (119909 minus 1199092))]

(36)

6 Abstract and Applied Analysis

00 02 04 06 08 10

0010

0050

0020

0030

0015

0070

x

ln|y(x)|

Exactk = 0M = 4

k = 0M = 3

k = 0M = 2

Figure 1 Different solutions of Example 2

The exact solution of (35) is given by 119910(119909) = (13) sin(119909 minus1199092

) In Table 6 the maximum absolute error 119864 is listed for119896 = 2 and various values of 119872 and in Table 7 we give acomparison between best errors resulted from the applicationof various methods for Example 4 This table shows thatour two algorithms are more accurate if compared with themethod developed in [42]

Example 5 Consider the second-order singular linear BVP

11991010158401015840

+ 119891 (119909) 119910 = 119892 (119909) 0 lt 119909 lt 1

119910 (0) + 16119910 (1

4) = 3

4

radic119890

119910 (1) + 16119910 (3

4) = 3

4

radic1198903

(37)

where

119891 (119909) =

3119909 0 ⩽ 119909 ⩽1

2

21199091

2lt 119909 ⩽ 1

(38)

and 119892(119909) is chosen such that the exact solution of (37) is119910(119909) = 119909(1 minus 119909)119890

119909 In Table 8 the maximum absolute error 119864is listed for 119896 = 1 and various values of119872 while in Figure 2we give a comparison between the exact solution of (37) withthree approximate solutions

Example 6 Consider the following nonlinear second-orderBVP

11991010158401015840

+ (1199101015840

)2

minus 64119910 = 32 0 lt 119909 lt 1 (39)

119910 (0) + 119910 (1

4) = 1 (40)

4119910 (1

2) minus 119910 (1) = 0 (41)

with the exact solution 119910(119909) = 161199092 We solve (39) using

3CWCM for the case corresponding to 119896 = 0 and 119872 = 3

00 02 04 06 08 1000

01

02

03

04

05

x

Exactk = 1M = 3

k = 1M = 2

k = 1M = 1

y(x)

Figure 2 Different solutions of Example 5

to obtain an approximate solution of 119910(119909) If we make use of(27) then the approximate solution 119910

03(119909) can be expanded

in terms of third-kind Chebyshev wavelets as

11991003(119909) = 119888

00radic2

120587+ 11988801radic2

120587(4119909 minus 3)

+ 11988802radic2

120587(161199092

minus 20119909 + 5)

(42)

If we set

V119894= radic

2

1205871198880119894 119894 = 0 1 2 (43)

then (42) reduces to the form

11991003(119909) = V

0+ V1(4119909 minus 3) + V

2(161199092

minus 20119909 + 5) (44)

If we substitute (44) into (39) then the residual of (39) isgiven by

119877 (119909) = 2V2+ [V1+ V2(8119909 minus 5)]

2

minus 4 [V0+ V1(4119909 minus 3)

+ V2(161199092

minus 20119909 + 5)] minus 2

(45)

We enforce the residual to vanish at the first root of 119881lowast3(119909) =

641199093

minus1121199092

+56119909minus7 namely at 1199091= 018825509907063323

to get

610388V22+ 05V2

1minus 349396V

2V1minus 260388V

2

+ 449396V1minus 2V0= 1

(46)

Furthermore the use of the boundary conditions (40) and(41) yields

2V0minus 5V1+ 6V2= 1

3V0minus 5V1minus 5V2= 0

(47)

Abstract and Applied Analysis 7

Table 1 Comparison between different solutions for Example 1

119909 3CWCM 4CWCM Haar method [6] ADMmethod [28] ODEs solver fromMathematica01 006560 006560 006561 00656 00656003 016587 016587 016588 01658 01658705 022369 022369 022369 02236 02236907 023820 023820 023821 02382 02382009 020920 020920 020910 02092 020920

Table 2 The maximum absolute error 119864 for Example 2

119896 119872 10 11 12 13 14 15

0 3CWCM 5173 sdot 10minus9

7184 sdot 10minus11

4418 sdot 10minus12

5018 sdot 10minus14

2442 sdot 10minus15

2220 sdot 10minus16

4CWCM 3324 sdot 10minus10

2533 sdot 10minus11

2811 sdot 10minus12

1665 sdot 10minus14

1110 sdot 10minus15

2220 sdot 10minus16

119872 4 5 6 7 8 9

1 3CWCM 4644 sdot 10minus6

1001 sdot 10minus8

1840 sdot 10minus9

2547 sdot 10minus10

6247 sdot 10minus11

5681 sdot 10minus12

4CWCM 215 sdot 10minus6

5247 sdot 10minus9

7548 sdot 10minus10

1004 sdot 10minus10

2154 sdot 10minus11

4257 sdot 10minus12

Table 3 The best errors for Example 2

Method in [39] Method in [40] 3CWCM 4CWCM900 sdot 10

minus6

400 sdot 10minus5

222 sdot 10minus16

222 sdot 10minus16

Table 4 The maximum absolute error 119864 for Example 3

119872 8 9 10 11 12 133CWCM 4444 sdot 10

minus10

2759 sdot 10minus11

4389 sdot 10minus13

1643 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

4CWCM 4478 sdot 10minus9

2179 sdot 10minus10

1527 sdot 10minus12

2975 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

Table 5 The best errors for Example 3

Method in [40] Method in [41] 3CWCM 4CWCM300 sdot 10

minus8

660 sdot 10minus7

222 sdot 10minus16

222 sdot 10minus16

Table 6 The maximum absolute error 119864 for Example 4

119872 4 5 6 7 8 93CWCM 2881 sdot 10

minus3

3441 sdot 10minus3

1212 sdot 10minus5

1933 sdot 10minus5

1990 sdot 10minus6

3010 sdot 10minus8

4CWCM 1241 sdot 10minus3

8542 sdot 10minus4

7526 sdot 10minus6

1002 sdot 10minus6

6321 sdot 10minus7

2354 sdot 10minus9

Table 7 The best errors for Example 4

Method in [42] 3CWCM 4CWCM800 sdot 10

minus6

3010 sdot 10minus8

235 sdot 10minus9

Table 8 The maximum absolute error 119864 for Example 5

119872 7 8 9 10 11 12 13 143CWCM 71 sdot 10

minus6

32 sdot 10minus7

13 sdot 10minus8

74 sdot 10minus10

13 sdot 10minus11

35 sdot 10minus13

93 sdot 10minus15

81 sdot 10minus16

4CWCM 28 sdot 10minus5

15 sdot 10minus6

63 sdot 10minus8

24 sdot 10minus9

78 sdot 10minus11

23 sdot 10minus12

63 sdot 10minus14

17 sdot 10minus15

8 Abstract and Applied Analysis

The solution of the nonlinear system of (46) and (47) gives

V0= 10 V

1= 5 V

2= 1 (48)

and consequently

11991003(119909) = 10 + 5 (4119909 minus 3) + (16119909

2

minus 20119909 + 5) = 161199092

(49)

which is the exact solution

Remark 3 It is worth noting here that the obtained numericalresults in the previous solved six examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecomparing favorably with the known analytical solutions

6 Concluding Remarks

In this paper two algorithms for obtaining numerical spectralwavelets solutions for second-order multipoint linear andnonlinear boundary value problems are analyzed and dis-cussed Chebyshev polynomials of third and fourth kindsare used One of the advantages of the developed algorithmsis their availability for application on singular boundaryvalue problems Another advantage is that high accurateapproximate solutions are achieved using a few number ofterms of the approximate expansionThe obtained numericalresults are comparing favorably with the analytical ones

Acknowledgment

The authors would like to thank the referee for his valuablecomments and suggestions which improved the paper in itspresent form

References

[1] E H Doha W M Abd-Elhameed and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for the integrated formsof third- and fifth-order elliptic differential equations usinggeneral parameters generalized Jacobi polynomialsrdquo AppliedMathematics and Computation vol 218 no 15 pp 7727ndash77402012

[2] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[3] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013

[4] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[5] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Springer Berlin Germany 2006

[6] Siraj-ul-Islam I Aziz and B Sarler ldquoThe numerical solution ofsecond-order boundary-value problems by collocation methodwith theHaarwaveletsrdquoMathematical andComputerModellingvol 52 no 9-10 pp 1577ndash1590 2010

[7] A H Bhrawy A S Alofi and S I El-Soubhy ldquoAn exten-sion of the Legendre-Galerkin method for solving sixth-orderdifferential equations with variable polynomial coefficientsrdquoMathematical Problems in Engineering vol 2012 Article ID896575 13 pages 2012

[8] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[9] WMAbd-Elhameed ldquoEfficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of (2119899 + 1)th-order linear differential equationsrdquo Journal of the EgyptianMathematical Society vol 17 no 2 pp 189ndash211 2009

[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011

[11] A H Bhrawy A S Alofi and S I El-Soubhy ldquoSpectral shiftedJacobi tau and collocation methods for solving fifth-orderboundary value problemsrdquo Abstract and Applied Analysis vol2011 Article ID 823273 14 pages 2011

[12] M Moshiinsky ldquoSobre los problemas de condiciones a lafrontiera en una dimension de caracteristicas discontinuasrdquoBoletin de La Sociedad Matematica Mexicana vol 7 article 1251950

[13] S P TimoshenkoTheory of Elastic StabilityMcGraw-Hill BookNew York NY USA 2nd edition 1961

[14] R P Agarwal and I Kiguradze ldquoOnmulti-point boundary valueproblems for linear ordinary differential equations with singu-laritiesrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 131ndash151 2004

[15] Z Du ldquoSolvability of functional differential equations withmulti-point boundary value problems at resonancerdquoComputersamp Mathematics with Applications vol 55 no 11 pp 2653ndash26612008

[16] W Feng and J R L Webb ldquoSolvability of 119898-point boundaryvalue problemswith nonlinear growthrdquo Journal ofMathematicalAnalysis and Applications vol 212 no 2 pp 467ndash480 1997

[17] H B Thompson and C Tisdell ldquoThree-point boundary valueproblems for second-order ordinary differential equationsrdquoMathematical and Computer Modelling vol 34 no 3-4 pp 311ndash318 2001

[18] M R Scott and H A Watts ldquoSUPORTmdasha computer code fortwo-point boundary-value problems via orthonormalizationrdquoSandia Labs Report 75-0198 Sandia Laboratories AlbuquerqueNM USA 1975

[19] M R Scott and H A Watts ldquoComputational solution of lineartwo-point boundary value problems via orthonormalizationrdquoSIAM Journal on Numerical Analysis vol 14 no 1 pp 40ndash701977

[20] M R Scott and W H Vandevender ldquoA comparison of severalinvariant imbedding algorithms for the solution of two-pointboundary-value problemsrdquo Applied Mathematics and Compu-tation vol 1 no 3 pp 187ndash218 1975

[21] T YNaComputationalMethods in Engineering BoundaryValueProblems vol 145 Academic Press New York NY USA 1979

Abstract and Applied Analysis 9

[22] K E Bisshopp andDCDrucker ldquoLarge deflection of cantileverbeamsrdquo Quarterly of Applied Mathematics vol 3 pp 272ndash2751945

[23] W Glabisz ldquoThe use of Walsh-wavelet packets in linear bound-ary value problemsrdquo Computers amp Structures vol 82 no 2-3pp 131ndash141 2004

[24] Siraj-ul-Islam M A Noor I A Tirmizi and M A KhanldquoQuadratic non-polynomial spline approach to the solution ofa system of second-order boundary-value problemsrdquo AppliedMathematics and Computation vol 179 no 1 pp 153ndash160 2006

[25] S Robert and J Shipman ldquoSolution of Troeschrsquos two-pointboundary value problems by shooting techniquesrdquo Journal ofComputational Physics vol 10 pp 232ndash241 1972

[26] E Wiebel Confinement of a Plasma Column by RadiationPressure in the Plasma in a Magnetic Field Stanford UniversityPress Stanfor Calif USA 1958

[27] H H Keller and E S Holdrege ldquoRadiation heat transfer forannular fins of trapezoidal profilerdquo International Journal of HighPerformance Computing Applications vol 92 pp 113ndash116 1970

[28] M Tatari andM Dehgan ldquoThe use of the Adomian decomposi-tion method for solving multipoint boundary value problemsrdquoPhysica Scripta vol 73 pp 672ndash676 2006

[29] M Lakestani andM Dehghan ldquoThe solution of a second-ordernonlinear differential equation with Neumann boundary con-ditions using semi-orthogonal B-spline waveletsrdquo InternationalJournal of Computer Mathematics vol 83 no 8-9 pp 685ndash6942006

[30] A Constantmldes Applied Numerical Methods with PersonalComputers McGraw-Hill New York NY USA 1987

[31] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[32] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[33] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[34] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[35] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[36] E H Doha W M Abd- Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[37] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[38] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp Hall New York NY USA 2003

[39] Y Lin J Niu and M Cui ldquoA numerical solution to nonlinearsecond order three-point boundary value problems in thereproducing kernel spacerdquo Applied Mathematics and Computa-tion vol 218 no 14 pp 7362ndash7368 2012

[40] F Geng ldquoSolving singular second order three-point bound-ary value problems using reproducing kernel Hilbert spacemethodrdquo Applied Mathematics and Computation vol 215 no6 pp 2095ndash2102 2009

[41] F Z Geng ldquoA numerical algorithm for nonlinear multi-point boundary value problemsrdquo Journal of Computational andApplied Mathematics vol 236 no 7 pp 1789ndash1794 2012

[42] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

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

Abstract and Applied Analysis 5

Now the application of the typical collocation method (seeeg [5]) gives

119877 (119909119894) = 0 119894 = 1 2 2

119896

(119872 + 1) minus 2 (29)

where 119909119894are the first (2119896(119872 + 1) minus 2) roots of 119881lowast

2119896(119872+1)

(119909) or119882lowast

2119896(119872+1)

(119909) Moreover the use of the boundary conditions(26) gives

1205720

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(0)

+ 1205721

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(0)

=

1198980

sum

119894=1

2119896

minus1

sum

119899=0

119872

sum

119898=0

120582119894119888119899119898120595119899119898(120585119894) + 1205750

1205730

2119896

minus1

sum

119899=0

119872

sum

119898=0

119888119899119898120595119899119898(1) + 120573

1

2119896

minus1

sum

119899=0

119872

sum

119898=1

1198881198991198981205951015840

119899119898(1)

=

1198981

sum

119894=1

2119896

minus1

sum

119899=0

119872

sum

119898=0

120583119894119888119899119898120595119899119898(120578119894) + 1205751

(30)

Equations (29) and (30) generate 2119896(119872 + 1) equations inthe unknown expansion coefficients 119888

119899119898 which can be

solved with the aid of the well-known Newtonrsquos iterativemethod Consequently we get the desired approximate solu-tion 119910

119896119872(119909) given by (27)

5 Numerical Examples

In this section the presented algorithms in Section 4 areapplied to solve both of linear and nonlinear multipointBVPs Some examples are considered to illustrate the effi-ciency and applicability of the two proposed algorithms

Example 1 Consider the second-order nonlinear BVP (see[6 28])

11991010158401015840

+3

8119910 +

2

1089(1199101015840

)2

+ 1 = 0 0 lt 119909 lt 1

119910 (0) = 0 119910 (1

3) = 119910 (1)

(31)

The two proposed methods are applied to the problem forthe case corresponding to 119896 = 0 and119872 = 8 The numericalsolutions are shown in Table 1 Due to nonavailability of theexact solution we compare our results with Haar waveletsmethod [6] ADM solution [28] and ODEs Solver fromMathematica which is carried out by using Runge-Kuttamethod This comparison is also shown in Table 1

Example 2 Consider the second-order linear BVP (see [3940])

11991010158401015840

= sinh119909 minus 2 0 lt 119909 lt 1

1199101015840

(0) = 0 119910 (1) = 3119910 (3

5)

(32)

The exact solution of problem (32) is given by

119910 (119909) =1

2(sinh 1 minus 3 sinh 3

5) + sinh119909 minus 1199092 minus 119909 + 11

25

(33)

In Table 2 the maximum absolute error 119864 is listed for119896 = 1 and various values of 119872 while in Table 3 wegive a comparison between the best errors resulted fromthe application of various methods for Example 2 while inFigure 1 we give a comparison between the exact solution of(32) with three approximate solutions

Example 3 Consider the second-order singular nonlinearBVP (see [40 41])

119909 (1 minus 119909) 11991010158401015840

+ 61199101015840

+ 2119910 + 1199102

= 6 cosh 119909 + (2 + 119909 minus 1199092 + sinh119909) sinh119909

0 lt 119909 lt 1

119910 (0) + 119910 (2

3) = sinh(2

3)

119910 (1) +1

2119910 (

4

5) =

1

2sinh(4

5) + sinh 1

(34)

with the exact solution 119910(119909) = sinh119909 In Table 4 the maxi-mum absolute error 119864 is listed for 119896 = 0 and various valuesof 119872 while in Table 5 we give a comparison between thebest errors resulted from the application of various methodsfor Example 3 This table shows that our two algorithms aremore accurate if compared with the two methods developedin [40 41]

Example 4 Consider the second-order nonlinear BVP (see[42])

11991010158401015840

+ (1 + 119909 + 1199093

) 1199102

= 119891 (119909) 0 lt 119909 lt 1

119910 (0) =1

6119910 (

2

9) +

1

3119910 (

7

9) minus 00286634

119910 (1) =1

5119910 (

2

9) +

1

2119910 (

7

9) minus 00401287

(35)

where

119891 (119909) =1

9[ minus 6 cos (119909 minus 1199092) + sin (119909 minus 1199092)

times (minus3(1 minus 2119909)2

+ (1 + 119909 + 1199093

) sin (119909 minus 1199092))]

(36)

6 Abstract and Applied Analysis

00 02 04 06 08 10

0010

0050

0020

0030

0015

0070

x

ln|y(x)|

Exactk = 0M = 4

k = 0M = 3

k = 0M = 2

Figure 1 Different solutions of Example 2

The exact solution of (35) is given by 119910(119909) = (13) sin(119909 minus1199092

) In Table 6 the maximum absolute error 119864 is listed for119896 = 2 and various values of 119872 and in Table 7 we give acomparison between best errors resulted from the applicationof various methods for Example 4 This table shows thatour two algorithms are more accurate if compared with themethod developed in [42]

Example 5 Consider the second-order singular linear BVP

11991010158401015840

+ 119891 (119909) 119910 = 119892 (119909) 0 lt 119909 lt 1

119910 (0) + 16119910 (1

4) = 3

4

radic119890

119910 (1) + 16119910 (3

4) = 3

4

radic1198903

(37)

where

119891 (119909) =

3119909 0 ⩽ 119909 ⩽1

2

21199091

2lt 119909 ⩽ 1

(38)

and 119892(119909) is chosen such that the exact solution of (37) is119910(119909) = 119909(1 minus 119909)119890

119909 In Table 8 the maximum absolute error 119864is listed for 119896 = 1 and various values of119872 while in Figure 2we give a comparison between the exact solution of (37) withthree approximate solutions

Example 6 Consider the following nonlinear second-orderBVP

11991010158401015840

+ (1199101015840

)2

minus 64119910 = 32 0 lt 119909 lt 1 (39)

119910 (0) + 119910 (1

4) = 1 (40)

4119910 (1

2) minus 119910 (1) = 0 (41)

with the exact solution 119910(119909) = 161199092 We solve (39) using

3CWCM for the case corresponding to 119896 = 0 and 119872 = 3

00 02 04 06 08 1000

01

02

03

04

05

x

Exactk = 1M = 3

k = 1M = 2

k = 1M = 1

y(x)

Figure 2 Different solutions of Example 5

to obtain an approximate solution of 119910(119909) If we make use of(27) then the approximate solution 119910

03(119909) can be expanded

in terms of third-kind Chebyshev wavelets as

11991003(119909) = 119888

00radic2

120587+ 11988801radic2

120587(4119909 minus 3)

+ 11988802radic2

120587(161199092

minus 20119909 + 5)

(42)

If we set

V119894= radic

2

1205871198880119894 119894 = 0 1 2 (43)

then (42) reduces to the form

11991003(119909) = V

0+ V1(4119909 minus 3) + V

2(161199092

minus 20119909 + 5) (44)

If we substitute (44) into (39) then the residual of (39) isgiven by

119877 (119909) = 2V2+ [V1+ V2(8119909 minus 5)]

2

minus 4 [V0+ V1(4119909 minus 3)

+ V2(161199092

minus 20119909 + 5)] minus 2

(45)

We enforce the residual to vanish at the first root of 119881lowast3(119909) =

641199093

minus1121199092

+56119909minus7 namely at 1199091= 018825509907063323

to get

610388V22+ 05V2

1minus 349396V

2V1minus 260388V

2

+ 449396V1minus 2V0= 1

(46)

Furthermore the use of the boundary conditions (40) and(41) yields

2V0minus 5V1+ 6V2= 1

3V0minus 5V1minus 5V2= 0

(47)

Abstract and Applied Analysis 7

Table 1 Comparison between different solutions for Example 1

119909 3CWCM 4CWCM Haar method [6] ADMmethod [28] ODEs solver fromMathematica01 006560 006560 006561 00656 00656003 016587 016587 016588 01658 01658705 022369 022369 022369 02236 02236907 023820 023820 023821 02382 02382009 020920 020920 020910 02092 020920

Table 2 The maximum absolute error 119864 for Example 2

119896 119872 10 11 12 13 14 15

0 3CWCM 5173 sdot 10minus9

7184 sdot 10minus11

4418 sdot 10minus12

5018 sdot 10minus14

2442 sdot 10minus15

2220 sdot 10minus16

4CWCM 3324 sdot 10minus10

2533 sdot 10minus11

2811 sdot 10minus12

1665 sdot 10minus14

1110 sdot 10minus15

2220 sdot 10minus16

119872 4 5 6 7 8 9

1 3CWCM 4644 sdot 10minus6

1001 sdot 10minus8

1840 sdot 10minus9

2547 sdot 10minus10

6247 sdot 10minus11

5681 sdot 10minus12

4CWCM 215 sdot 10minus6

5247 sdot 10minus9

7548 sdot 10minus10

1004 sdot 10minus10

2154 sdot 10minus11

4257 sdot 10minus12

Table 3 The best errors for Example 2

Method in [39] Method in [40] 3CWCM 4CWCM900 sdot 10

minus6

400 sdot 10minus5

222 sdot 10minus16

222 sdot 10minus16

Table 4 The maximum absolute error 119864 for Example 3

119872 8 9 10 11 12 133CWCM 4444 sdot 10

minus10

2759 sdot 10minus11

4389 sdot 10minus13

1643 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

4CWCM 4478 sdot 10minus9

2179 sdot 10minus10

1527 sdot 10minus12

2975 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

Table 5 The best errors for Example 3

Method in [40] Method in [41] 3CWCM 4CWCM300 sdot 10

minus8

660 sdot 10minus7

222 sdot 10minus16

222 sdot 10minus16

Table 6 The maximum absolute error 119864 for Example 4

119872 4 5 6 7 8 93CWCM 2881 sdot 10

minus3

3441 sdot 10minus3

1212 sdot 10minus5

1933 sdot 10minus5

1990 sdot 10minus6

3010 sdot 10minus8

4CWCM 1241 sdot 10minus3

8542 sdot 10minus4

7526 sdot 10minus6

1002 sdot 10minus6

6321 sdot 10minus7

2354 sdot 10minus9

Table 7 The best errors for Example 4

Method in [42] 3CWCM 4CWCM800 sdot 10

minus6

3010 sdot 10minus8

235 sdot 10minus9

Table 8 The maximum absolute error 119864 for Example 5

119872 7 8 9 10 11 12 13 143CWCM 71 sdot 10

minus6

32 sdot 10minus7

13 sdot 10minus8

74 sdot 10minus10

13 sdot 10minus11

35 sdot 10minus13

93 sdot 10minus15

81 sdot 10minus16

4CWCM 28 sdot 10minus5

15 sdot 10minus6

63 sdot 10minus8

24 sdot 10minus9

78 sdot 10minus11

23 sdot 10minus12

63 sdot 10minus14

17 sdot 10minus15

8 Abstract and Applied Analysis

The solution of the nonlinear system of (46) and (47) gives

V0= 10 V

1= 5 V

2= 1 (48)

and consequently

11991003(119909) = 10 + 5 (4119909 minus 3) + (16119909

2

minus 20119909 + 5) = 161199092

(49)

which is the exact solution

Remark 3 It is worth noting here that the obtained numericalresults in the previous solved six examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecomparing favorably with the known analytical solutions

6 Concluding Remarks

In this paper two algorithms for obtaining numerical spectralwavelets solutions for second-order multipoint linear andnonlinear boundary value problems are analyzed and dis-cussed Chebyshev polynomials of third and fourth kindsare used One of the advantages of the developed algorithmsis their availability for application on singular boundaryvalue problems Another advantage is that high accurateapproximate solutions are achieved using a few number ofterms of the approximate expansionThe obtained numericalresults are comparing favorably with the analytical ones

Acknowledgment

The authors would like to thank the referee for his valuablecomments and suggestions which improved the paper in itspresent form

References

[1] E H Doha W M Abd-Elhameed and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for the integrated formsof third- and fifth-order elliptic differential equations usinggeneral parameters generalized Jacobi polynomialsrdquo AppliedMathematics and Computation vol 218 no 15 pp 7727ndash77402012

[2] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[3] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013

[4] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[5] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Springer Berlin Germany 2006

[6] Siraj-ul-Islam I Aziz and B Sarler ldquoThe numerical solution ofsecond-order boundary-value problems by collocation methodwith theHaarwaveletsrdquoMathematical andComputerModellingvol 52 no 9-10 pp 1577ndash1590 2010

[7] A H Bhrawy A S Alofi and S I El-Soubhy ldquoAn exten-sion of the Legendre-Galerkin method for solving sixth-orderdifferential equations with variable polynomial coefficientsrdquoMathematical Problems in Engineering vol 2012 Article ID896575 13 pages 2012

[8] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[9] WMAbd-Elhameed ldquoEfficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of (2119899 + 1)th-order linear differential equationsrdquo Journal of the EgyptianMathematical Society vol 17 no 2 pp 189ndash211 2009

[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011

[11] A H Bhrawy A S Alofi and S I El-Soubhy ldquoSpectral shiftedJacobi tau and collocation methods for solving fifth-orderboundary value problemsrdquo Abstract and Applied Analysis vol2011 Article ID 823273 14 pages 2011

[12] M Moshiinsky ldquoSobre los problemas de condiciones a lafrontiera en una dimension de caracteristicas discontinuasrdquoBoletin de La Sociedad Matematica Mexicana vol 7 article 1251950

[13] S P TimoshenkoTheory of Elastic StabilityMcGraw-Hill BookNew York NY USA 2nd edition 1961

[14] R P Agarwal and I Kiguradze ldquoOnmulti-point boundary valueproblems for linear ordinary differential equations with singu-laritiesrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 131ndash151 2004

[15] Z Du ldquoSolvability of functional differential equations withmulti-point boundary value problems at resonancerdquoComputersamp Mathematics with Applications vol 55 no 11 pp 2653ndash26612008

[16] W Feng and J R L Webb ldquoSolvability of 119898-point boundaryvalue problemswith nonlinear growthrdquo Journal ofMathematicalAnalysis and Applications vol 212 no 2 pp 467ndash480 1997

[17] H B Thompson and C Tisdell ldquoThree-point boundary valueproblems for second-order ordinary differential equationsrdquoMathematical and Computer Modelling vol 34 no 3-4 pp 311ndash318 2001

[18] M R Scott and H A Watts ldquoSUPORTmdasha computer code fortwo-point boundary-value problems via orthonormalizationrdquoSandia Labs Report 75-0198 Sandia Laboratories AlbuquerqueNM USA 1975

[19] M R Scott and H A Watts ldquoComputational solution of lineartwo-point boundary value problems via orthonormalizationrdquoSIAM Journal on Numerical Analysis vol 14 no 1 pp 40ndash701977

[20] M R Scott and W H Vandevender ldquoA comparison of severalinvariant imbedding algorithms for the solution of two-pointboundary-value problemsrdquo Applied Mathematics and Compu-tation vol 1 no 3 pp 187ndash218 1975

[21] T YNaComputationalMethods in Engineering BoundaryValueProblems vol 145 Academic Press New York NY USA 1979

Abstract and Applied Analysis 9

[22] K E Bisshopp andDCDrucker ldquoLarge deflection of cantileverbeamsrdquo Quarterly of Applied Mathematics vol 3 pp 272ndash2751945

[23] W Glabisz ldquoThe use of Walsh-wavelet packets in linear bound-ary value problemsrdquo Computers amp Structures vol 82 no 2-3pp 131ndash141 2004

[24] Siraj-ul-Islam M A Noor I A Tirmizi and M A KhanldquoQuadratic non-polynomial spline approach to the solution ofa system of second-order boundary-value problemsrdquo AppliedMathematics and Computation vol 179 no 1 pp 153ndash160 2006

[25] S Robert and J Shipman ldquoSolution of Troeschrsquos two-pointboundary value problems by shooting techniquesrdquo Journal ofComputational Physics vol 10 pp 232ndash241 1972

[26] E Wiebel Confinement of a Plasma Column by RadiationPressure in the Plasma in a Magnetic Field Stanford UniversityPress Stanfor Calif USA 1958

[27] H H Keller and E S Holdrege ldquoRadiation heat transfer forannular fins of trapezoidal profilerdquo International Journal of HighPerformance Computing Applications vol 92 pp 113ndash116 1970

[28] M Tatari andM Dehgan ldquoThe use of the Adomian decomposi-tion method for solving multipoint boundary value problemsrdquoPhysica Scripta vol 73 pp 672ndash676 2006

[29] M Lakestani andM Dehghan ldquoThe solution of a second-ordernonlinear differential equation with Neumann boundary con-ditions using semi-orthogonal B-spline waveletsrdquo InternationalJournal of Computer Mathematics vol 83 no 8-9 pp 685ndash6942006

[30] A Constantmldes Applied Numerical Methods with PersonalComputers McGraw-Hill New York NY USA 1987

[31] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[32] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[33] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[34] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[35] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[36] E H Doha W M Abd- Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[37] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[38] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp Hall New York NY USA 2003

[39] Y Lin J Niu and M Cui ldquoA numerical solution to nonlinearsecond order three-point boundary value problems in thereproducing kernel spacerdquo Applied Mathematics and Computa-tion vol 218 no 14 pp 7362ndash7368 2012

[40] F Geng ldquoSolving singular second order three-point bound-ary value problems using reproducing kernel Hilbert spacemethodrdquo Applied Mathematics and Computation vol 215 no6 pp 2095ndash2102 2009

[41] F Z Geng ldquoA numerical algorithm for nonlinear multi-point boundary value problemsrdquo Journal of Computational andApplied Mathematics vol 236 no 7 pp 1789ndash1794 2012

[42] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

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

6 Abstract and Applied Analysis

00 02 04 06 08 10

0010

0050

0020

0030

0015

0070

x

ln|y(x)|

Exactk = 0M = 4

k = 0M = 3

k = 0M = 2

Figure 1 Different solutions of Example 2

The exact solution of (35) is given by 119910(119909) = (13) sin(119909 minus1199092

) In Table 6 the maximum absolute error 119864 is listed for119896 = 2 and various values of 119872 and in Table 7 we give acomparison between best errors resulted from the applicationof various methods for Example 4 This table shows thatour two algorithms are more accurate if compared with themethod developed in [42]

Example 5 Consider the second-order singular linear BVP

11991010158401015840

+ 119891 (119909) 119910 = 119892 (119909) 0 lt 119909 lt 1

119910 (0) + 16119910 (1

4) = 3

4

radic119890

119910 (1) + 16119910 (3

4) = 3

4

radic1198903

(37)

where

119891 (119909) =

3119909 0 ⩽ 119909 ⩽1

2

21199091

2lt 119909 ⩽ 1

(38)

and 119892(119909) is chosen such that the exact solution of (37) is119910(119909) = 119909(1 minus 119909)119890

119909 In Table 8 the maximum absolute error 119864is listed for 119896 = 1 and various values of119872 while in Figure 2we give a comparison between the exact solution of (37) withthree approximate solutions

Example 6 Consider the following nonlinear second-orderBVP

11991010158401015840

+ (1199101015840

)2

minus 64119910 = 32 0 lt 119909 lt 1 (39)

119910 (0) + 119910 (1

4) = 1 (40)

4119910 (1

2) minus 119910 (1) = 0 (41)

with the exact solution 119910(119909) = 161199092 We solve (39) using

3CWCM for the case corresponding to 119896 = 0 and 119872 = 3

00 02 04 06 08 1000

01

02

03

04

05

x

Exactk = 1M = 3

k = 1M = 2

k = 1M = 1

y(x)

Figure 2 Different solutions of Example 5

to obtain an approximate solution of 119910(119909) If we make use of(27) then the approximate solution 119910

03(119909) can be expanded

in terms of third-kind Chebyshev wavelets as

11991003(119909) = 119888

00radic2

120587+ 11988801radic2

120587(4119909 minus 3)

+ 11988802radic2

120587(161199092

minus 20119909 + 5)

(42)

If we set

V119894= radic

2

1205871198880119894 119894 = 0 1 2 (43)

then (42) reduces to the form

11991003(119909) = V

0+ V1(4119909 minus 3) + V

2(161199092

minus 20119909 + 5) (44)

If we substitute (44) into (39) then the residual of (39) isgiven by

119877 (119909) = 2V2+ [V1+ V2(8119909 minus 5)]

2

minus 4 [V0+ V1(4119909 minus 3)

+ V2(161199092

minus 20119909 + 5)] minus 2

(45)

We enforce the residual to vanish at the first root of 119881lowast3(119909) =

641199093

minus1121199092

+56119909minus7 namely at 1199091= 018825509907063323

to get

610388V22+ 05V2

1minus 349396V

2V1minus 260388V

2

+ 449396V1minus 2V0= 1

(46)

Furthermore the use of the boundary conditions (40) and(41) yields

2V0minus 5V1+ 6V2= 1

3V0minus 5V1minus 5V2= 0

(47)

Abstract and Applied Analysis 7

Table 1 Comparison between different solutions for Example 1

119909 3CWCM 4CWCM Haar method [6] ADMmethod [28] ODEs solver fromMathematica01 006560 006560 006561 00656 00656003 016587 016587 016588 01658 01658705 022369 022369 022369 02236 02236907 023820 023820 023821 02382 02382009 020920 020920 020910 02092 020920

Table 2 The maximum absolute error 119864 for Example 2

119896 119872 10 11 12 13 14 15

0 3CWCM 5173 sdot 10minus9

7184 sdot 10minus11

4418 sdot 10minus12

5018 sdot 10minus14

2442 sdot 10minus15

2220 sdot 10minus16

4CWCM 3324 sdot 10minus10

2533 sdot 10minus11

2811 sdot 10minus12

1665 sdot 10minus14

1110 sdot 10minus15

2220 sdot 10minus16

119872 4 5 6 7 8 9

1 3CWCM 4644 sdot 10minus6

1001 sdot 10minus8

1840 sdot 10minus9

2547 sdot 10minus10

6247 sdot 10minus11

5681 sdot 10minus12

4CWCM 215 sdot 10minus6

5247 sdot 10minus9

7548 sdot 10minus10

1004 sdot 10minus10

2154 sdot 10minus11

4257 sdot 10minus12

Table 3 The best errors for Example 2

Method in [39] Method in [40] 3CWCM 4CWCM900 sdot 10

minus6

400 sdot 10minus5

222 sdot 10minus16

222 sdot 10minus16

Table 4 The maximum absolute error 119864 for Example 3

119872 8 9 10 11 12 133CWCM 4444 sdot 10

minus10

2759 sdot 10minus11

4389 sdot 10minus13

1643 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

4CWCM 4478 sdot 10minus9

2179 sdot 10minus10

1527 sdot 10minus12

2975 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

Table 5 The best errors for Example 3

Method in [40] Method in [41] 3CWCM 4CWCM300 sdot 10

minus8

660 sdot 10minus7

222 sdot 10minus16

222 sdot 10minus16

Table 6 The maximum absolute error 119864 for Example 4

119872 4 5 6 7 8 93CWCM 2881 sdot 10

minus3

3441 sdot 10minus3

1212 sdot 10minus5

1933 sdot 10minus5

1990 sdot 10minus6

3010 sdot 10minus8

4CWCM 1241 sdot 10minus3

8542 sdot 10minus4

7526 sdot 10minus6

1002 sdot 10minus6

6321 sdot 10minus7

2354 sdot 10minus9

Table 7 The best errors for Example 4

Method in [42] 3CWCM 4CWCM800 sdot 10

minus6

3010 sdot 10minus8

235 sdot 10minus9

Table 8 The maximum absolute error 119864 for Example 5

119872 7 8 9 10 11 12 13 143CWCM 71 sdot 10

minus6

32 sdot 10minus7

13 sdot 10minus8

74 sdot 10minus10

13 sdot 10minus11

35 sdot 10minus13

93 sdot 10minus15

81 sdot 10minus16

4CWCM 28 sdot 10minus5

15 sdot 10minus6

63 sdot 10minus8

24 sdot 10minus9

78 sdot 10minus11

23 sdot 10minus12

63 sdot 10minus14

17 sdot 10minus15

8 Abstract and Applied Analysis

The solution of the nonlinear system of (46) and (47) gives

V0= 10 V

1= 5 V

2= 1 (48)

and consequently

11991003(119909) = 10 + 5 (4119909 minus 3) + (16119909

2

minus 20119909 + 5) = 161199092

(49)

which is the exact solution

Remark 3 It is worth noting here that the obtained numericalresults in the previous solved six examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecomparing favorably with the known analytical solutions

6 Concluding Remarks

In this paper two algorithms for obtaining numerical spectralwavelets solutions for second-order multipoint linear andnonlinear boundary value problems are analyzed and dis-cussed Chebyshev polynomials of third and fourth kindsare used One of the advantages of the developed algorithmsis their availability for application on singular boundaryvalue problems Another advantage is that high accurateapproximate solutions are achieved using a few number ofterms of the approximate expansionThe obtained numericalresults are comparing favorably with the analytical ones

Acknowledgment

The authors would like to thank the referee for his valuablecomments and suggestions which improved the paper in itspresent form

References

[1] E H Doha W M Abd-Elhameed and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for the integrated formsof third- and fifth-order elliptic differential equations usinggeneral parameters generalized Jacobi polynomialsrdquo AppliedMathematics and Computation vol 218 no 15 pp 7727ndash77402012

[2] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[3] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013

[4] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[5] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Springer Berlin Germany 2006

[6] Siraj-ul-Islam I Aziz and B Sarler ldquoThe numerical solution ofsecond-order boundary-value problems by collocation methodwith theHaarwaveletsrdquoMathematical andComputerModellingvol 52 no 9-10 pp 1577ndash1590 2010

[7] A H Bhrawy A S Alofi and S I El-Soubhy ldquoAn exten-sion of the Legendre-Galerkin method for solving sixth-orderdifferential equations with variable polynomial coefficientsrdquoMathematical Problems in Engineering vol 2012 Article ID896575 13 pages 2012

[8] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[9] WMAbd-Elhameed ldquoEfficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of (2119899 + 1)th-order linear differential equationsrdquo Journal of the EgyptianMathematical Society vol 17 no 2 pp 189ndash211 2009

[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011

[11] A H Bhrawy A S Alofi and S I El-Soubhy ldquoSpectral shiftedJacobi tau and collocation methods for solving fifth-orderboundary value problemsrdquo Abstract and Applied Analysis vol2011 Article ID 823273 14 pages 2011

[12] M Moshiinsky ldquoSobre los problemas de condiciones a lafrontiera en una dimension de caracteristicas discontinuasrdquoBoletin de La Sociedad Matematica Mexicana vol 7 article 1251950

[13] S P TimoshenkoTheory of Elastic StabilityMcGraw-Hill BookNew York NY USA 2nd edition 1961

[14] R P Agarwal and I Kiguradze ldquoOnmulti-point boundary valueproblems for linear ordinary differential equations with singu-laritiesrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 131ndash151 2004

[15] Z Du ldquoSolvability of functional differential equations withmulti-point boundary value problems at resonancerdquoComputersamp Mathematics with Applications vol 55 no 11 pp 2653ndash26612008

[16] W Feng and J R L Webb ldquoSolvability of 119898-point boundaryvalue problemswith nonlinear growthrdquo Journal ofMathematicalAnalysis and Applications vol 212 no 2 pp 467ndash480 1997

[17] H B Thompson and C Tisdell ldquoThree-point boundary valueproblems for second-order ordinary differential equationsrdquoMathematical and Computer Modelling vol 34 no 3-4 pp 311ndash318 2001

[18] M R Scott and H A Watts ldquoSUPORTmdasha computer code fortwo-point boundary-value problems via orthonormalizationrdquoSandia Labs Report 75-0198 Sandia Laboratories AlbuquerqueNM USA 1975

[19] M R Scott and H A Watts ldquoComputational solution of lineartwo-point boundary value problems via orthonormalizationrdquoSIAM Journal on Numerical Analysis vol 14 no 1 pp 40ndash701977

[20] M R Scott and W H Vandevender ldquoA comparison of severalinvariant imbedding algorithms for the solution of two-pointboundary-value problemsrdquo Applied Mathematics and Compu-tation vol 1 no 3 pp 187ndash218 1975

[21] T YNaComputationalMethods in Engineering BoundaryValueProblems vol 145 Academic Press New York NY USA 1979

Abstract and Applied Analysis 9

[22] K E Bisshopp andDCDrucker ldquoLarge deflection of cantileverbeamsrdquo Quarterly of Applied Mathematics vol 3 pp 272ndash2751945

[23] W Glabisz ldquoThe use of Walsh-wavelet packets in linear bound-ary value problemsrdquo Computers amp Structures vol 82 no 2-3pp 131ndash141 2004

[24] Siraj-ul-Islam M A Noor I A Tirmizi and M A KhanldquoQuadratic non-polynomial spline approach to the solution ofa system of second-order boundary-value problemsrdquo AppliedMathematics and Computation vol 179 no 1 pp 153ndash160 2006

[25] S Robert and J Shipman ldquoSolution of Troeschrsquos two-pointboundary value problems by shooting techniquesrdquo Journal ofComputational Physics vol 10 pp 232ndash241 1972

[26] E Wiebel Confinement of a Plasma Column by RadiationPressure in the Plasma in a Magnetic Field Stanford UniversityPress Stanfor Calif USA 1958

[27] H H Keller and E S Holdrege ldquoRadiation heat transfer forannular fins of trapezoidal profilerdquo International Journal of HighPerformance Computing Applications vol 92 pp 113ndash116 1970

[28] M Tatari andM Dehgan ldquoThe use of the Adomian decomposi-tion method for solving multipoint boundary value problemsrdquoPhysica Scripta vol 73 pp 672ndash676 2006

[29] M Lakestani andM Dehghan ldquoThe solution of a second-ordernonlinear differential equation with Neumann boundary con-ditions using semi-orthogonal B-spline waveletsrdquo InternationalJournal of Computer Mathematics vol 83 no 8-9 pp 685ndash6942006

[30] A Constantmldes Applied Numerical Methods with PersonalComputers McGraw-Hill New York NY USA 1987

[31] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[32] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[33] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[34] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[35] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[36] E H Doha W M Abd- Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[37] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[38] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp Hall New York NY USA 2003

[39] Y Lin J Niu and M Cui ldquoA numerical solution to nonlinearsecond order three-point boundary value problems in thereproducing kernel spacerdquo Applied Mathematics and Computa-tion vol 218 no 14 pp 7362ndash7368 2012

[40] F Geng ldquoSolving singular second order three-point bound-ary value problems using reproducing kernel Hilbert spacemethodrdquo Applied Mathematics and Computation vol 215 no6 pp 2095ndash2102 2009

[41] F Z Geng ldquoA numerical algorithm for nonlinear multi-point boundary value problemsrdquo Journal of Computational andApplied Mathematics vol 236 no 7 pp 1789ndash1794 2012

[42] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

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

Abstract and Applied Analysis 7

Table 1 Comparison between different solutions for Example 1

119909 3CWCM 4CWCM Haar method [6] ADMmethod [28] ODEs solver fromMathematica01 006560 006560 006561 00656 00656003 016587 016587 016588 01658 01658705 022369 022369 022369 02236 02236907 023820 023820 023821 02382 02382009 020920 020920 020910 02092 020920

Table 2 The maximum absolute error 119864 for Example 2

119896 119872 10 11 12 13 14 15

0 3CWCM 5173 sdot 10minus9

7184 sdot 10minus11

4418 sdot 10minus12

5018 sdot 10minus14

2442 sdot 10minus15

2220 sdot 10minus16

4CWCM 3324 sdot 10minus10

2533 sdot 10minus11

2811 sdot 10minus12

1665 sdot 10minus14

1110 sdot 10minus15

2220 sdot 10minus16

119872 4 5 6 7 8 9

1 3CWCM 4644 sdot 10minus6

1001 sdot 10minus8

1840 sdot 10minus9

2547 sdot 10minus10

6247 sdot 10minus11

5681 sdot 10minus12

4CWCM 215 sdot 10minus6

5247 sdot 10minus9

7548 sdot 10minus10

1004 sdot 10minus10

2154 sdot 10minus11

4257 sdot 10minus12

Table 3 The best errors for Example 2

Method in [39] Method in [40] 3CWCM 4CWCM900 sdot 10

minus6

400 sdot 10minus5

222 sdot 10minus16

222 sdot 10minus16

Table 4 The maximum absolute error 119864 for Example 3

119872 8 9 10 11 12 133CWCM 4444 sdot 10

minus10

2759 sdot 10minus11

4389 sdot 10minus13

1643 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

4CWCM 4478 sdot 10minus9

2179 sdot 10minus10

1527 sdot 10minus12

2975 sdot 10minus14

4441 sdot 10minus16

2220 sdot 10minus16

Table 5 The best errors for Example 3

Method in [40] Method in [41] 3CWCM 4CWCM300 sdot 10

minus8

660 sdot 10minus7

222 sdot 10minus16

222 sdot 10minus16

Table 6 The maximum absolute error 119864 for Example 4

119872 4 5 6 7 8 93CWCM 2881 sdot 10

minus3

3441 sdot 10minus3

1212 sdot 10minus5

1933 sdot 10minus5

1990 sdot 10minus6

3010 sdot 10minus8

4CWCM 1241 sdot 10minus3

8542 sdot 10minus4

7526 sdot 10minus6

1002 sdot 10minus6

6321 sdot 10minus7

2354 sdot 10minus9

Table 7 The best errors for Example 4

Method in [42] 3CWCM 4CWCM800 sdot 10

minus6

3010 sdot 10minus8

235 sdot 10minus9

Table 8 The maximum absolute error 119864 for Example 5

119872 7 8 9 10 11 12 13 143CWCM 71 sdot 10

minus6

32 sdot 10minus7

13 sdot 10minus8

74 sdot 10minus10

13 sdot 10minus11

35 sdot 10minus13

93 sdot 10minus15

81 sdot 10minus16

4CWCM 28 sdot 10minus5

15 sdot 10minus6

63 sdot 10minus8

24 sdot 10minus9

78 sdot 10minus11

23 sdot 10minus12

63 sdot 10minus14

17 sdot 10minus15

8 Abstract and Applied Analysis

The solution of the nonlinear system of (46) and (47) gives

V0= 10 V

1= 5 V

2= 1 (48)

and consequently

11991003(119909) = 10 + 5 (4119909 minus 3) + (16119909

2

minus 20119909 + 5) = 161199092

(49)

which is the exact solution

Remark 3 It is worth noting here that the obtained numericalresults in the previous solved six examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecomparing favorably with the known analytical solutions

6 Concluding Remarks

In this paper two algorithms for obtaining numerical spectralwavelets solutions for second-order multipoint linear andnonlinear boundary value problems are analyzed and dis-cussed Chebyshev polynomials of third and fourth kindsare used One of the advantages of the developed algorithmsis their availability for application on singular boundaryvalue problems Another advantage is that high accurateapproximate solutions are achieved using a few number ofterms of the approximate expansionThe obtained numericalresults are comparing favorably with the analytical ones

Acknowledgment

The authors would like to thank the referee for his valuablecomments and suggestions which improved the paper in itspresent form

References

[1] E H Doha W M Abd-Elhameed and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for the integrated formsof third- and fifth-order elliptic differential equations usinggeneral parameters generalized Jacobi polynomialsrdquo AppliedMathematics and Computation vol 218 no 15 pp 7727ndash77402012

[2] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[3] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013

[4] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[5] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Springer Berlin Germany 2006

[6] Siraj-ul-Islam I Aziz and B Sarler ldquoThe numerical solution ofsecond-order boundary-value problems by collocation methodwith theHaarwaveletsrdquoMathematical andComputerModellingvol 52 no 9-10 pp 1577ndash1590 2010

[7] A H Bhrawy A S Alofi and S I El-Soubhy ldquoAn exten-sion of the Legendre-Galerkin method for solving sixth-orderdifferential equations with variable polynomial coefficientsrdquoMathematical Problems in Engineering vol 2012 Article ID896575 13 pages 2012

[8] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[9] WMAbd-Elhameed ldquoEfficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of (2119899 + 1)th-order linear differential equationsrdquo Journal of the EgyptianMathematical Society vol 17 no 2 pp 189ndash211 2009

[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011

[11] A H Bhrawy A S Alofi and S I El-Soubhy ldquoSpectral shiftedJacobi tau and collocation methods for solving fifth-orderboundary value problemsrdquo Abstract and Applied Analysis vol2011 Article ID 823273 14 pages 2011

[12] M Moshiinsky ldquoSobre los problemas de condiciones a lafrontiera en una dimension de caracteristicas discontinuasrdquoBoletin de La Sociedad Matematica Mexicana vol 7 article 1251950

[13] S P TimoshenkoTheory of Elastic StabilityMcGraw-Hill BookNew York NY USA 2nd edition 1961

[14] R P Agarwal and I Kiguradze ldquoOnmulti-point boundary valueproblems for linear ordinary differential equations with singu-laritiesrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 131ndash151 2004

[15] Z Du ldquoSolvability of functional differential equations withmulti-point boundary value problems at resonancerdquoComputersamp Mathematics with Applications vol 55 no 11 pp 2653ndash26612008

[16] W Feng and J R L Webb ldquoSolvability of 119898-point boundaryvalue problemswith nonlinear growthrdquo Journal ofMathematicalAnalysis and Applications vol 212 no 2 pp 467ndash480 1997

[17] H B Thompson and C Tisdell ldquoThree-point boundary valueproblems for second-order ordinary differential equationsrdquoMathematical and Computer Modelling vol 34 no 3-4 pp 311ndash318 2001

[18] M R Scott and H A Watts ldquoSUPORTmdasha computer code fortwo-point boundary-value problems via orthonormalizationrdquoSandia Labs Report 75-0198 Sandia Laboratories AlbuquerqueNM USA 1975

[19] M R Scott and H A Watts ldquoComputational solution of lineartwo-point boundary value problems via orthonormalizationrdquoSIAM Journal on Numerical Analysis vol 14 no 1 pp 40ndash701977

[20] M R Scott and W H Vandevender ldquoA comparison of severalinvariant imbedding algorithms for the solution of two-pointboundary-value problemsrdquo Applied Mathematics and Compu-tation vol 1 no 3 pp 187ndash218 1975

[21] T YNaComputationalMethods in Engineering BoundaryValueProblems vol 145 Academic Press New York NY USA 1979

Abstract and Applied Analysis 9

[22] K E Bisshopp andDCDrucker ldquoLarge deflection of cantileverbeamsrdquo Quarterly of Applied Mathematics vol 3 pp 272ndash2751945

[23] W Glabisz ldquoThe use of Walsh-wavelet packets in linear bound-ary value problemsrdquo Computers amp Structures vol 82 no 2-3pp 131ndash141 2004

[24] Siraj-ul-Islam M A Noor I A Tirmizi and M A KhanldquoQuadratic non-polynomial spline approach to the solution ofa system of second-order boundary-value problemsrdquo AppliedMathematics and Computation vol 179 no 1 pp 153ndash160 2006

[25] S Robert and J Shipman ldquoSolution of Troeschrsquos two-pointboundary value problems by shooting techniquesrdquo Journal ofComputational Physics vol 10 pp 232ndash241 1972

[26] E Wiebel Confinement of a Plasma Column by RadiationPressure in the Plasma in a Magnetic Field Stanford UniversityPress Stanfor Calif USA 1958

[27] H H Keller and E S Holdrege ldquoRadiation heat transfer forannular fins of trapezoidal profilerdquo International Journal of HighPerformance Computing Applications vol 92 pp 113ndash116 1970

[28] M Tatari andM Dehgan ldquoThe use of the Adomian decomposi-tion method for solving multipoint boundary value problemsrdquoPhysica Scripta vol 73 pp 672ndash676 2006

[29] M Lakestani andM Dehghan ldquoThe solution of a second-ordernonlinear differential equation with Neumann boundary con-ditions using semi-orthogonal B-spline waveletsrdquo InternationalJournal of Computer Mathematics vol 83 no 8-9 pp 685ndash6942006

[30] A Constantmldes Applied Numerical Methods with PersonalComputers McGraw-Hill New York NY USA 1987

[31] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[32] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[33] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[34] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[35] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[36] E H Doha W M Abd- Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[37] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[38] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp Hall New York NY USA 2003

[39] Y Lin J Niu and M Cui ldquoA numerical solution to nonlinearsecond order three-point boundary value problems in thereproducing kernel spacerdquo Applied Mathematics and Computa-tion vol 218 no 14 pp 7362ndash7368 2012

[40] F Geng ldquoSolving singular second order three-point bound-ary value problems using reproducing kernel Hilbert spacemethodrdquo Applied Mathematics and Computation vol 215 no6 pp 2095ndash2102 2009

[41] F Z Geng ldquoA numerical algorithm for nonlinear multi-point boundary value problemsrdquo Journal of Computational andApplied Mathematics vol 236 no 7 pp 1789ndash1794 2012

[42] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

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

8 Abstract and Applied Analysis

The solution of the nonlinear system of (46) and (47) gives

V0= 10 V

1= 5 V

2= 1 (48)

and consequently

11991003(119909) = 10 + 5 (4119909 minus 3) + (16119909

2

minus 20119909 + 5) = 161199092

(49)

which is the exact solution

Remark 3 It is worth noting here that the obtained numericalresults in the previous solved six examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecomparing favorably with the known analytical solutions

6 Concluding Remarks

In this paper two algorithms for obtaining numerical spectralwavelets solutions for second-order multipoint linear andnonlinear boundary value problems are analyzed and dis-cussed Chebyshev polynomials of third and fourth kindsare used One of the advantages of the developed algorithmsis their availability for application on singular boundaryvalue problems Another advantage is that high accurateapproximate solutions are achieved using a few number ofterms of the approximate expansionThe obtained numericalresults are comparing favorably with the analytical ones

Acknowledgment

The authors would like to thank the referee for his valuablecomments and suggestions which improved the paper in itspresent form

References

[1] E H Doha W M Abd-Elhameed and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for the integrated formsof third- and fifth-order elliptic differential equations usinggeneral parameters generalized Jacobi polynomialsrdquo AppliedMathematics and Computation vol 218 no 15 pp 7727ndash77402012

[2] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[3] E H Doha W M Abd-Elhameed and A H Bhrawy ldquoNewspectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobipolynomialsrdquo Collectanea Mathematica vol 64 no 3 pp 373ndash394 2013

[4] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[5] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Springer Berlin Germany 2006

[6] Siraj-ul-Islam I Aziz and B Sarler ldquoThe numerical solution ofsecond-order boundary-value problems by collocation methodwith theHaarwaveletsrdquoMathematical andComputerModellingvol 52 no 9-10 pp 1577ndash1590 2010

[7] A H Bhrawy A S Alofi and S I El-Soubhy ldquoAn exten-sion of the Legendre-Galerkin method for solving sixth-orderdifferential equations with variable polynomial coefficientsrdquoMathematical Problems in Engineering vol 2012 Article ID896575 13 pages 2012

[8] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[9] WMAbd-Elhameed ldquoEfficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of (2119899 + 1)th-order linear differential equationsrdquo Journal of the EgyptianMathematical Society vol 17 no 2 pp 189ndash211 2009

[10] A H Bhrawy and W M Abd-Elhameed ldquoNew algorithm forthe numerical solutions of nonlinear third-order differentialequations using Jacobi-Gauss collocation methodrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 837218 14pages 2011

[11] A H Bhrawy A S Alofi and S I El-Soubhy ldquoSpectral shiftedJacobi tau and collocation methods for solving fifth-orderboundary value problemsrdquo Abstract and Applied Analysis vol2011 Article ID 823273 14 pages 2011

[12] M Moshiinsky ldquoSobre los problemas de condiciones a lafrontiera en una dimension de caracteristicas discontinuasrdquoBoletin de La Sociedad Matematica Mexicana vol 7 article 1251950

[13] S P TimoshenkoTheory of Elastic StabilityMcGraw-Hill BookNew York NY USA 2nd edition 1961

[14] R P Agarwal and I Kiguradze ldquoOnmulti-point boundary valueproblems for linear ordinary differential equations with singu-laritiesrdquo Journal of Mathematical Analysis and Applications vol297 no 1 pp 131ndash151 2004

[15] Z Du ldquoSolvability of functional differential equations withmulti-point boundary value problems at resonancerdquoComputersamp Mathematics with Applications vol 55 no 11 pp 2653ndash26612008

[16] W Feng and J R L Webb ldquoSolvability of 119898-point boundaryvalue problemswith nonlinear growthrdquo Journal ofMathematicalAnalysis and Applications vol 212 no 2 pp 467ndash480 1997

[17] H B Thompson and C Tisdell ldquoThree-point boundary valueproblems for second-order ordinary differential equationsrdquoMathematical and Computer Modelling vol 34 no 3-4 pp 311ndash318 2001

[18] M R Scott and H A Watts ldquoSUPORTmdasha computer code fortwo-point boundary-value problems via orthonormalizationrdquoSandia Labs Report 75-0198 Sandia Laboratories AlbuquerqueNM USA 1975

[19] M R Scott and H A Watts ldquoComputational solution of lineartwo-point boundary value problems via orthonormalizationrdquoSIAM Journal on Numerical Analysis vol 14 no 1 pp 40ndash701977

[20] M R Scott and W H Vandevender ldquoA comparison of severalinvariant imbedding algorithms for the solution of two-pointboundary-value problemsrdquo Applied Mathematics and Compu-tation vol 1 no 3 pp 187ndash218 1975

[21] T YNaComputationalMethods in Engineering BoundaryValueProblems vol 145 Academic Press New York NY USA 1979

Abstract and Applied Analysis 9

[22] K E Bisshopp andDCDrucker ldquoLarge deflection of cantileverbeamsrdquo Quarterly of Applied Mathematics vol 3 pp 272ndash2751945

[23] W Glabisz ldquoThe use of Walsh-wavelet packets in linear bound-ary value problemsrdquo Computers amp Structures vol 82 no 2-3pp 131ndash141 2004

[24] Siraj-ul-Islam M A Noor I A Tirmizi and M A KhanldquoQuadratic non-polynomial spline approach to the solution ofa system of second-order boundary-value problemsrdquo AppliedMathematics and Computation vol 179 no 1 pp 153ndash160 2006

[25] S Robert and J Shipman ldquoSolution of Troeschrsquos two-pointboundary value problems by shooting techniquesrdquo Journal ofComputational Physics vol 10 pp 232ndash241 1972

[26] E Wiebel Confinement of a Plasma Column by RadiationPressure in the Plasma in a Magnetic Field Stanford UniversityPress Stanfor Calif USA 1958

[27] H H Keller and E S Holdrege ldquoRadiation heat transfer forannular fins of trapezoidal profilerdquo International Journal of HighPerformance Computing Applications vol 92 pp 113ndash116 1970

[28] M Tatari andM Dehgan ldquoThe use of the Adomian decomposi-tion method for solving multipoint boundary value problemsrdquoPhysica Scripta vol 73 pp 672ndash676 2006

[29] M Lakestani andM Dehghan ldquoThe solution of a second-ordernonlinear differential equation with Neumann boundary con-ditions using semi-orthogonal B-spline waveletsrdquo InternationalJournal of Computer Mathematics vol 83 no 8-9 pp 685ndash6942006

[30] A Constantmldes Applied Numerical Methods with PersonalComputers McGraw-Hill New York NY USA 1987

[31] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[32] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[33] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[34] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[35] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[36] E H Doha W M Abd- Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[37] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[38] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp Hall New York NY USA 2003

[39] Y Lin J Niu and M Cui ldquoA numerical solution to nonlinearsecond order three-point boundary value problems in thereproducing kernel spacerdquo Applied Mathematics and Computa-tion vol 218 no 14 pp 7362ndash7368 2012

[40] F Geng ldquoSolving singular second order three-point bound-ary value problems using reproducing kernel Hilbert spacemethodrdquo Applied Mathematics and Computation vol 215 no6 pp 2095ndash2102 2009

[41] F Z Geng ldquoA numerical algorithm for nonlinear multi-point boundary value problemsrdquo Journal of Computational andApplied Mathematics vol 236 no 7 pp 1789ndash1794 2012

[42] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

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

Abstract and Applied Analysis 9

[22] K E Bisshopp andDCDrucker ldquoLarge deflection of cantileverbeamsrdquo Quarterly of Applied Mathematics vol 3 pp 272ndash2751945

[23] W Glabisz ldquoThe use of Walsh-wavelet packets in linear bound-ary value problemsrdquo Computers amp Structures vol 82 no 2-3pp 131ndash141 2004

[24] Siraj-ul-Islam M A Noor I A Tirmizi and M A KhanldquoQuadratic non-polynomial spline approach to the solution ofa system of second-order boundary-value problemsrdquo AppliedMathematics and Computation vol 179 no 1 pp 153ndash160 2006

[25] S Robert and J Shipman ldquoSolution of Troeschrsquos two-pointboundary value problems by shooting techniquesrdquo Journal ofComputational Physics vol 10 pp 232ndash241 1972

[26] E Wiebel Confinement of a Plasma Column by RadiationPressure in the Plasma in a Magnetic Field Stanford UniversityPress Stanfor Calif USA 1958

[27] H H Keller and E S Holdrege ldquoRadiation heat transfer forannular fins of trapezoidal profilerdquo International Journal of HighPerformance Computing Applications vol 92 pp 113ndash116 1970

[28] M Tatari andM Dehgan ldquoThe use of the Adomian decomposi-tion method for solving multipoint boundary value problemsrdquoPhysica Scripta vol 73 pp 672ndash676 2006

[29] M Lakestani andM Dehghan ldquoThe solution of a second-ordernonlinear differential equation with Neumann boundary con-ditions using semi-orthogonal B-spline waveletsrdquo InternationalJournal of Computer Mathematics vol 83 no 8-9 pp 685ndash6942006

[30] A Constantmldes Applied Numerical Methods with PersonalComputers McGraw-Hill New York NY USA 1987

[31] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[32] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[33] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[34] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[35] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[36] E H Doha W M Abd- Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[37] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[38] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp Hall New York NY USA 2003

[39] Y Lin J Niu and M Cui ldquoA numerical solution to nonlinearsecond order three-point boundary value problems in thereproducing kernel spacerdquo Applied Mathematics and Computa-tion vol 218 no 14 pp 7362ndash7368 2012

[40] F Geng ldquoSolving singular second order three-point bound-ary value problems using reproducing kernel Hilbert spacemethodrdquo Applied Mathematics and Computation vol 215 no6 pp 2095ndash2102 2009

[41] F Z Geng ldquoA numerical algorithm for nonlinear multi-point boundary value problemsrdquo Journal of Computational andApplied Mathematics vol 236 no 7 pp 1789ndash1794 2012

[42] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

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

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