New ultraspherical wavelets collocation method for solving 2nth-order initial and boundary value problems

Egyptian

Mathematical

Society

Journal

of

the

Egyptian

Mathematical

Society

www.etms-eg.org www.elsevier.com/locate/joems

Original

Article

New

ultraspherical

wavelets

collocation

method

for

solving

2nth-order

initial

and

boundary

value

problems

E.H.

Doha

,

W.M.

Abd-Elhameed

,

Y.H.

Youssri

b_{DepartmentofMathematics,FacultyofScience,UniversityofJeddah,Jeddah,SaudiArabia}

Received1August2014;revised5May2015;accepted20May2015 Availableonline16June2015

Keywords Ultraspherical polynomials; Wavelets; Bernstein-typeinequality; Collocationmethod; Even-order differential equations; Burger’sequation

Abstract Inthispaper,anewspectralalgorithmbasedonemployingultrasphericalwaveletsalong withthespectralcollocationmethodisdeveloped.Theproposedalgorithmisutilizedtosolvelinear andnonlineareven-orderinitialandboundaryvalueproblems.Thisalgorithmissupportedby study-ingtheconvergenceanalysisof theusedultrasphericalwaveletsexpansion.Theprincipleideafor obtainingtheproposedspectralnumericalsolutionsfortheabove-mentionedproblemsisactually basedonusingwaveletscollocationmethodtoreducethelinearornonlineardifferentialequations withtheirinitialorboundaryconditionsintosystemsoflinearornonlinearalgebraicequationsin theunknownexpansioncoefficients.SomespecificimportantproblemssuchasLane–Emdenand Burger’stypeequationscanbesolvedefficientlywiththesuggestedalgorithm.Somenumerical ex-amplesaregivenforthesakeoftestingtheefficiencyandtheapplicabilityoftheproposedalgorithm.

MathematicsSubjectClassification: 65M70;65N35;35C10;42C10

Copyright2015,EgyptianMathematicalSociety.ProductionandhostingbyElsevierB.V. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

∗ _{Correspondingauthor.}

E-mail addresses: [email protected] (E.H. Doha), [email protected] (W.M. Abd-Elhameed), [email protected] (Y.H.Youssri).

PeerreviewunderresponsibilityofEgyptianMathematicalSociety.

ProductionandhostingbyElsevier

1. Introduction

Manypracticalandphysicalproblemsinfieldsofscienceand engineeringareformulatedasboundaryorinitialvalue prob-lems. The nonlinear boundary value problemsare crucial in various applications as they arise frequently in many areas of science andengineering. Forexample, the deflection of a

S1110-256X(15)00032-2Copyright2015,EgyptianMathematicalSociety.ProductionandhostingbyElsevierB.V.Thisisanopenaccessarticle undertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

uniformly loadedrectangular platesupportedoverthe entire surfacebyan elastic foundationandrigidly supportedalong theedges,satisfiesafourth-orderdifferentialequationseefor example[6,8,12–15,20,28–30,41].

Spectralmethods play prominentroles in solving various kindsofdifferentialequations.Itisknownthattherearethree mostwidelyusedspectralmethods,theyarethetau, colloca-tion,andGalerkinmethods.Collocationmethodshavebecome increasinglypopularforsolvingdifferentialequations,in partic-ular,theyareveryusefulinprovidinghighlyaccuratesolutions tononlineardifferentialequations(see,forexample[1,6,8,12– 15,19,20,28–30,41,49]).

Higheven-orderdifferentialequationshavebeenextensively discussedbyalargenumberofauthorsduetotheirgreat im-portancein variousapplicationsinmany fields.Forexample, inthesequenceofpapers[18,16,17],theauthorstreatedsuch equationsbytheGalerkinmethod. Theyconstructedsuitable basis functions whichsatisfythe boundaryconditions ofthe givendifferential equation.Forthispurpose, theyused com-pactcombinationsofvariousorthogonalpolynomials.The sug-gestedalgorithmsinthesearticlesaresuitableforhandling one-and two-dimensional linear high even-order boundary value problems.

Inthispaper,weaimtogiveanewalgorithmforhandling bothoflinearandnonlineareven-orderinitial andboundary valueproblemsbasedonemployingultrasphericalwavelets col-locationmethod(UWCM).

Waveletscoveragreatareainmathematicalmodels,andthey havebeenusedto handlea widerangeofmedical and engi-neeringdisciplines;inparticular,waveletsareveryappropriate insignalanalysis,imagesegmentations,timefrequency analy-sisandfastalgorithmsforeasyimplementation.Waveletsgive accuraterepresentationofavarietyoffunctionsandoperators. Moreover,waveletsconnectbetweenfastnumericalalgorithms, (see[11,38,32]).

Legendrewaveletshavebeenpreviouslyemployedfor solv-ingvarious differentialandintegralequations (see for exam-ple,[35,34,36,46–48]).Also,firstandsecondkindsChebyshev waveletshavebeenusedforsolvingsomeintegerandfractional orders differential equations (see for example, [7,56,31,25]). Recently, Abd-Elhameed et al. in [2,3], have introduced newChebyshev waveletsalgorithmsfor solvingsecond-order boundaryvalueproblems.Up tonow,andtothebestofour knowledge,theuseofultrasphericalwaveletsinvariousspectral numericalapplicationsistracelessintheliterature.Thisgivesus amotivationtointroduceandusetheultrasphericalwaveletsin variousapplications.

Theoutlinesofthispaperareorganizedasfollows.In Sec-tion2,wegivesomerelevantpropertiesofultraspherical poly-nomialsandtheirshiftedforms.Moreover,inthissection, ul-traspherical wavelets are constructed. In Section 3, the con-vergence of the ultraspherical wavelets expansion is proved. Section 4 is concerned with presenting and implementing a numerical algorithm for solving even-order linear and non-linearinitial orboundary value problems based on employ-ingultrasphericalwaveletstogether withthespectral colloca-tionmethod.Section5isconcernedwithconsideringsome nu-mericalexamplesaimingtoillustratetheefficiencyand appli-cabilityofthedevelopedalgorithm.Conclusions aregivenin Section6.

2. Somepropertiesofultrasphericalpolynomialsand ultrasphericalwavelets

This sectionisconcerned withpresenting anoverview on ul-traspherical polynomials andtheirshiftedpolynomials. Also, ultrasphericalwaveletsareconstructedinthissection.

2.1. Somepropertiesofultrasphericalpolynomials

The ultraspherical polynomials are a classof symmetric Ja-cobi polynomials.Theyareasequenceoforthogonal polyno-mialsdefinedontheinterval(−1,1)withrespecttotheweight functionw(x)=(1−x2₎α−12_{associatedwiththerealparameter}

(α >−1

2).Explicitly,theysatisfytheorthogonalityproperty 1 −1( 1₋x2₎α−12_C(α) m (x)Cn(α)(x)dx= _π21−2α₍n+2α) n!(n+α)((α))2, m=n, 0_, m ₌ n. (1)

Also, they arethe eigenfunctionsof the following singular Sturm– Liouvilleequation

(1₋x2_)D2_φ

m(x)−(2α+1)xDφm(x)+m(m+2α)φm(x)=0, D_≡ d

dx.

Formorepropertiesofultrasphericalpolynomials,onecanbereferred to[5].

Thefollowingintegralformula(see,[5])isneeded

Cn(α)(x)w(x)dx=− 2α (1−x2₎α+12 n(n+2α) C (α+1) n−1 (x), n≥1. (2)

Also,thefollowingtheoremisusefulinderivingtheconvergence theoremfortheexpansionoftheultrasphericalwavelets.

Theorem1([22]Bernstein-typeinequality). Thefollowing in-equalityholdsforultrasphericalpolynomials:

(sin_{θ )}α_|C_n(α)₍cos_{θ )|<} 2 1−α_n+3α 2 (α)n+1+α₂, 0≤θ≤π, 0<α <1. (3)

2.2. Shiftedultrasphericalpolynomials

Theshiftedultrasphericalpolynomialsaredefinedon[0,1]by

∼ C

(α)

n (x)=Cn(α)(2x−1).Allresultsofultraspherical

polynomi-alscanbeeasilytransformedtogivethecorrespondingresults for their shifted polynomials. The orthogonality relation for

∼ C

(α)

n (x)withrespecttotheweightfunctionω˜ =(x−x2) α−1 2 _is givenby 1 0 ( x−x2)α− 1 2_C∼(α) n (x) ∼ C (α) m (x)dx= ⎧ ⎨ ⎩ π21−4α_(n+2α) n!₍n_+α)((α))2, m=n, 0, m = n.

Formorepropertiesofultrasphericalpolynomialssee[40]. 2.3. Constructionofultrasphericalwavelets

Waveletsconstituteofafamilyoffunctionsconstructedfrom dilation and translationof singlefunction called the mother

wavelet.IfthedilationparameterAandthetranslation parame-terBvarycontinuously,thenthefollowingfamilyofcontinuous waveletsisobtained: ψA,B(t)=|A|−1/2ψ t−B A , A,B∈R, A = 0. (4) Ultrasphericalwaveletsψnm(α)(t)=ψ(k,n,m,α,t)havefive

ar-guments:k,ncanbeassumedtobeanypositiveinteger,mis thedegreefortheultrasphericalpolynomial,αistheknown ul-trasphericalparameterandtisthenormalizedtime.Theyare definedontheinterval[0,1]by

ψ(α) nm(t)= ⎧ ⎨ ⎩2 k+1 2 ξ_m,αC(α) m 2k+1t−2n−1, t∈[n 2k, n+1 2k ], 0, otherwise, (5) where0≤m≤M,0≤n≤2k_−1,and ξm,α=2α(α) m!(m+α) 2π (m+2α). (6)

Remark1. Itisworthnotingherethatψ(12)

nm (t)isidenticalto

Legendrewavelets (see,[35]), ψ(0)

nm(t)is identicaltofirst kind

Chebyshevwavelets(see,[7])andψ(1)

nm(t)isidenticaltosecond

kindChebyshevwavelets(see,[56]).

3. Functionapproximationandconvergenceanalysis

Afunction f(t)definedon[0,1]maybeexpandedintermsof ultrasphericalwaveletsas f(t)= ∞ n=0 ∞ m=0 cnmψnm(α)(t), where cnm= f(t),ψnm(α)(t) w= 1 0 (t−t2₎α−12 _f(t)ψ(α) nm(t)dt. (7)

Also,thisfunctioncanbeapproximatedbythetruncatedfinite series f(t)≈ 2k−1 n=0 M−1 m=0 cnmψnm(α)(t).

Now,westateandprovethefollowingimportanttheoremwhich ascertainsthattheultrasphericalwaveletsexpansionofa func-tionf(x)withboundedsecondderivative,convergesuniformly tof(x).

Theorem 2. A function f(x)∈L2 ˜

ω[0,1], ω˜ = (x−x2₎α−12, ₀<α <_{1 can be expanded as an infinite}

se-ries of ultraspherical wavelets, which converges uniformly to f(x),provided|f(x)|≤L.Explicitly,theexpansioncoefficients in (7)satisfytheinequality

|cnm|< 4L₍1_+α)2_(m+1+3α 2) √ m!₍m_+α) (m+α₂)√(m+2α)(m−2)4(n+1)52 , ∀n_≥0_,m_>2. (8)

Proof. Fromrelations(5)and(7),onecanwrite

cnm=2 k+1 2 ξ_m,α n+1 2k n 2k f(t)Cm(α) 2k+1_{t−2n−1ω(2}k_t−n)dt.(9)

Integrationoftherighthandsideof(9)byparts,andmaking useofrelation(2),yield

cnm= 25−2kα ξ_m,α m(m+2α) n+1 2k n 2k f(t)Cm(α−+11) 2k+1_t−2n−1(2k_t−n) ×(1−2kt+n)ω(2kt−n)dt. (10)

Anotherintegrationbypartsandmakinguseofthe substitu-tion:2k+1_{t−2n−1=cosθ,enableonetowrite}

cnm = √ 2(α)2ξm,α 22α+5k 2 (m−1)₂(m+2α−1)₂ × π 0 f 1+2n+cosθ 2k+1 Cm(α−+22)(cosθ )(sinθ ) 2α+4_dθ . (11) Now,assumingthatm>2,takingintoaccounttheassumption |f(t)|≤L,andwiththeaidofTheorem1,weobtain

|cnm|≤ √ 2|(α)2ξm,α| 22α+52k(m−1)2(m+2α−1)2 × π 0 f 1+2n+cosθ 2k+1 |C_m(α₋+₂2)(cosθ )|(sinθ )2α+4dθ ≤ √ 2L_|α|(1_+α)ξm,α| 22α+5k 2 (m₋1₎2(m+2α−1)2 × π 0 | C_m(α₋+₂2)₍cos_{θ )|(}sin_{θ )}2α+4d_θ < √ 2πL|α|(1+α)|ξm,α| m+1+3₂α3+₂α 23α+5k 2 (m₋1₎2(m+2α−1)2(α+2) m₊α₂2₊α₂ .

Knowingthatα >0,n<2k_{−1,andwiththeaidofrelation(6),weget}

|cnm|< 2L|α|(1+α)3+₂αm+1+3₂α√m!(m+α) 4α₍2₊α₂₎m₊α₂√₍m₊2_α)(m₋2₎4_(n+1)5₂ < 4L(1+α) 2_m+1+3α 2 _√ m!₍m_+α) m+α₂√(m+2α)(m−2)4(n+1)52 .

Thiscompletestheproofofthetheorem.

Note1. Itshouldbenotedherethat,forlargevaluesofmandn, andmakinguseofthewellknownStirling’sformula(see,[33]), itcanbeeasilyshownthat|cnm|isofO(n−

5 2m−2).

Note 2. It isworthyto note herethat theresultobtained in

Theorem2maybe improvedbutwith someextraconditions onthefunction f(x),tobeconcise,iffisdifferentiablep-times forsomep>1and|f(p)_{(x)|≤L,wecanshowthat|c}

nm|isof O(n−p−1

4. Solutionofhigheven-orderdifferentialequations

Inthissection,andbasedontheultrasphericalwavelets expan-sionalongwiththespectralcollocationmethod,wereducethe 2νth-orderinitialandboundaryvalueproblemswiththeir ini-tialorboundaryconditionstosystemsoflinearornonlinear algebraicequationswhichcanbeefficientlysolved.

Considerthe2νth-orderdifferentialequations

u(2ν)_{(x)+φx,u(x),u(x),...,u}(2ν−1)_{(x)=0, x∈(0,1),}

(12) subjecttotheinitialconditions

u(i)_(0)=a

i, i=0,1,...,2ν−1, (13)

ortheDirichletboundaryconditions

u(i)_(0)=a

i, u(i)(1)=bi, i=0,1,...,ν−1. (14)

Theapproximatesolutionof(12)canbeexpandedintermsof ultrasphericalwaveletsas:

uk,M,α(x)= 2k₋₁ n=0 M m=0 cnmψnm(α)(x). (15)

Substitutionof(15)into(12),enablesonetowritetheresidual of(12)as R(x)= 2k₋₁ n=0 M m=2ν cnmD2νψnm(α)+φ ⎛ ⎝_x,2 k₋₁ n=0 M m=0 cnmψnm(α), × 2k−1 n=0 M m=1 cnmDψnm(α),..., 2k−1 n=0 M m=2ν−1 cnmD2ν−1ψnm(α) ⎞ ⎠.(16) Theapplicationofthecollocationmethod(seeforexample,[9]) impliesthat

R(xi)=0, i=1,2,...,

2k_{(M+1)−2ν, (17)}

wherexiarethefirst(2k(M+1)−2)rootsofC₂(α)k₍M+1).

More-over,theuseofboundaryconditions(13)gives

2k₋₁ n=0 M m=0 cnmD(i)ψnm(α)(0)=ai, i=0,1,...,2ν−1, (18)

whiletheuseof(14)gives

2k₋₁ n=0 M m=0 cnmD(i)ψnm(α)(0)=ai, 2k₋₁ n=0 M m=0 cnmD(i)ψnm(α)(1)=bi, i=0,1,...,ν−1. (19) Eqs. (17) with (18) or (19) generate a set of 2k_(M+1)

intheunknownexpansioncoefficients,{cnm:0≤n≤2k−1;

0≤m≤M}. This system of equations can be solved with the aidof aNewton’s iterativemethodwith theinitial guess

cnm=10−m,toobtaintheunknowncomponentsofthevector C,andhencetherequiredspectralwaveletssolutionuk,M,α(x)

given by (15) can be obtained. In fact, solving such sys-tem of nonlinear equations and its convergence is not an easy task, and is to be considered as one disadvantage of using Newton’s method. The interested reader is referred to the useful references [26,42,39] for the theory of Newton’s method – (Linearization anditeration ofnonlinear operator equations, the convergence of Newton’s method, Uniqueness of the solution) – for solvinggeneralized systemsof nonlin-ear algebraic equations in Banach spaces. In reality, New-ton’s method provides a powerful tool for the theoretical as well as the numerical investigation of nonlinear operator equations.

5. Numericalresultsanddiscussions

Example 1. Consider the singular Lane–Emden initial value problem(IVP)(see,[37]):

u+1 xu

_+u5_{=0, x∈(0,1], u(0)=1, u(0)=0,}

(20) withtheexactsolutionu(x)=(1+x₃2)−

1 2

.InTable1,the max-imum absoluteerror Eis listedfork=0andvariousvalues of M;andinTable2,wegiveacomparison between the so-lutionresultedfromtheapplicationofUWCMforthecase cor-respondstok=0,M=11andα=−0.49,withthenumerical solutionobtainedin[37]incaseofk=0,M=11.

Example 2. Consider the singular boundary value problem (BVP)(see,[37]):

Table2 ComparisonbetweenUWCMandthesolutionin[37] forExample1.

x Methodin[37] UWCM Exactsolution

0.1 0.99503720 0.99503719 0.99503719 0.2 0.98058067 0.98058067 0.98058067 0.3 0.95782628 0.95782628 0.95782628 0.4 0.92847670 0.92847669 0.92847669 0.5 0.89442718 0.89442719 0.89442719 0.6 0.85749292 0.85749292 0.85749292 0.7 0.81923192 0.81923192 0.81923192 0.8 0.78086880 0.78086880 0.78086880 0.9 0.74329414 0.74329414 0.74329414 1 0.70710678 0.70710678 0.70710678

Table1 MaximumabsoluteerrorEforExample1.

α M E M E M E M E M E 1 5 4.01×10−6 8 1.79×10−8 11 6.49×10−11 14 1.44×10−13 17 4.44×10−16 1 2 3.77×10− 6 _5.09×10−9 _1.32×10−11 _3.26×10−14 _4.44×10−16 0 3.03_×10−6 _1.28×10−8 _2.90×10−11 _4.94×10−14 _3.33×10−16 −0.49 2.71×10−6 1.07×10−8 3.21×10−11 3.80×10−14 2.22×10−16

Table3 MaximumabsoluteerrorEforExample2. α M E M E M E M E M E 1 3 5.97×10−6 5 1.99×10−8 7 3.84×10−11 9 1.19×10−12 11 9.10×10−15 1 2 2.23×10− 6 _6.02×10−9 _1.06×10−11 _2.86×10−13 _2.55×10−15 0 1.63×10−7 2.85×10−10 3.00×10−13 8.43×10−15 8.88×10−16 −0_.49 5.82_×10−7 _1.10×10−9 _1.45×10−12 _3.64×10−14 _9.99×10−16 u+0.5 x u _=eu_(0.5−eu_{), x∈(0,1], u(0)=ln2,} u(1)=0, (21)

withtheexactsolutionu(x)=ln( 2

x2₊1).InTable3,the

maxi-mumabsoluteerrorEislistedfork=1andvariousvaluesof

M,whileinTable4wegiveacomparisonbetweenthe approx-imatesolutionresultedfromtheapplicationofUWCMforthe casecorrespondstok=0,M=11andα=1withthe numeri-calsolutionobtainedin[37]incaseofk=0,M=11.

Remark2. WementionherethatcomparisonsinTables2and

4illustratethatwiththesamenumberofretainedmodes,the UWCMsolutionstoExamples1and2aremoreaccuratethan thoseobtainedbyNasabetal.in[37],andcomparingfavorably withtheanalyticalsolutionsofthesetwoexamples.

Table4 ComparisonbetweenUWCMandthesolutionin[37] forExample2.

x Methodin[37] UWCM Exactsolution

0.1 0.68319682 0.68319684 0.68319684 0.2 0.65392655 0.65392646 0.65392646 0.3 0.65392655 0.60696948 0.60696948 0.4 0.54472710 0.54472717 0.54472717 0.5 0.47000366 0.47000362 0.47000362 0.6 0.38566250 0.38566248 0.38566248 0.7 0.29437106 0.29437106 0.29437106 0.8 0.19845088 0.19845093 0.19845093 0.9 0.09982033 0.09982033 0.09982033 1 0.00000000 0.00000000 0.00000000

Example 3 (Burgers’ Equation). Consider the singular BVP (see,[44]): 1 Re u ₊λ xu ₋ λ x2u =uu+ f(x), x∈(0,1], u(0)=0, u(1)=cosh1, (22) where f(x) is chosen such thatthe exactsolution of(22) is

u(x)=x2_{coshx. Approximate solutions of this problem are}

computedforthecasescorrespondtothetwochoices λ=1,

Re=10 and λ=2,Re=50. The absolute errors for such casesaredisplayedinTable5fork=0andforvariousvaluesof

α,whileinTable6wegiveacomparisonbetweentheerrors ob-tainedusingtheB-Splinecollocation(BSC)solutionin[44]and UWCM,whileFig.1illustratesacomparisonbetweentheexact solutionandsomenumericalsolutions.

Remark3. WementionherethatTable6illustratesthatwe ob-tainedmoreaccuratesolutionsthanthoseobtainedin[44]in spiteofBSCdividestheinterval into80equalparts(n=80) which increasethecomplexity ofthe algorithm,while inthe presentalgorithm(k=0).

Example 4. Considerthefourth-order nonlinear beam equa-tion(see,[44,10,4,51,52]):

Table6 ThebestmaximumabsoluteerrorsofExample3.

λ=1,Re=10 λ=2,Re=50 BSC[44](n=80) 1.967×10−15 1.085×10−15 UWCM(k=0) 1.330×10−16 1.460×10−16

Table5 MaximumabsoluteerrorEforExample3.

λ Re α M E M E M E M E 1 10 1 9 1.01×10−7 11 1.46×10−10 13 1.20×10−13 15 1.33×10−15 1 2 9.02×10− 8 _1.17×10−10 _8.94×10−14 _7.32×10−15 0 7.64_×10−8 8.83_×10−11 5.55_×10−13 2.79_×10−14 −0.49 5.90×10−8 4.92×10−11 8.98×10−11 4.53×10−12 2 50 1 9 2.69×10−8 11 1.64×10−11 13 2.88×10−15 15 1.46×10−16 1 2 2.15×10− 8 _9.89×10−12 _1.22×10−15 _2.22×10−16 0 1.60_×10−8 _3.59×10−12 _3.92×10−15 _3.72×10−15 −0.49 9.23×10−9 1.08×10−12 2.05×10−12 3.05×10−12 1 103 1 9 1.23×10−4 11 2.35×10−7 13 5.22×10−11 15 4.68×10−12 1 2 5.22×10− 4 _5.27×10−8 _2.57×10−11 _1.25×10−12 0 8.88_×10−4 _7.24×10−8 _9.27×10−11 _5.71×10−11 −0.49 2.22×10−5 2.35×10−8 3.59×10−8 6.24×10−8

Figure1 DifferentsolutionsofExample3.

u(4)=6e−4u− 12

(1+x)4, x∈[0,1], (23)

u(0)=0, u(1)=ln2, u(0)=−1, u(1)=−1

4, (24) withtheexactsolutionu(x)=ln(1+x).InTable7,the maxi-mumabsoluteerrorEislistedfork=1andvariousvaluesof

M,andinTable8wegiveacomparisonbetweenthebesterrors resultedfromdifferentnumericalsolutions.

Example 5. Consider the fourth-order linear equation (see,

[24,43,45,54,55,27]):

u(4)_{+xu=0, x∈[0,1], (25)}

u(0)=1, u(1)=0, u(0)=0, u(1)=−4e, (26)

withtheexactsolutionu(x)=x(1−x)ex_{.InTable9,the}

max-imumabsoluteerrorEislistedfork=1andvariousvaluesof

M,whileinTable10 we giveacomparison between thebest errorsobtainedbydifferenterrors.

Remark4. WementionherethatTables8and10illustratethat UWCMismoreaccuratethanotheralgorithmsusedby differ-entauthors(see,[4,10,24,27,43–45,51,52,54,55]).

Example6. ConsiderthesingularBVP(see,[37]):

u+e1x_u+_u=_6x+_x3+_3x2_ex1, _x∈(₀,_1],

u(0)=0, u(1)=1, (27) withtheexactsolutionu(x)=x3_{.Eq.(27)issolvedbyapplying}

thealgorithmdescribedinSection4forthecasecorresponds toα=k=0andM=4,toobtaintheapproximatesolution

u0,4,0(x).Expandingu0,4,0(x)intermsofthefistkindChebyshev

wavelets,weget

u0,4,0(x)=c0,0ξ0+c0,1ξ1(2x−1)+c0,2ξ2(8x2−8x+1)

+c0,3ξ3(32x3−48x2+18x−1)

=c0+c1(2x−1)+c2(8x2−8x+1)

+c3(32x3−48x2+18x−1), (28)

whereci=ξic0,i, i=0,1,2,3.Now,theresidualofEq.(27)is

givenby R(x)=32c3x3+8c2x2−48c3x2−8c2x+210c3x +c1(2x−1)+c0+17c2−97c3−x3−6x +e1x_6c 3 16x2_−16x+3+8c 2(2x−1)+2c1−3x2 . (29)

Table7 MaximumabsoluteerrorEforExample4.

α M E M E M E M E M E 1 6 2.46_×10−5 _{8 1.54×10}−7 _{10 5.56×10}−10 _{12 2.62×10}−12 _{14 3.44×10}−15 1 2 2.52×10− 5 _1.46×10−7 _5.06×10−10 _2.45×10−12 _1.30×10−13 0 2.56×10−5 1.37×10−7 4.28×10−10 2.49×10−12 1.32×10−12 −0_.49 2.59_×10−5 _1.25×10−7 _3.57×10−10 _2.18×10−12 _2.35×10−13

Table8 ComparisonbetweenthebesterrorsforExample4.

Methodin[44] Methodin[10] Methodin[4] Methodin[51] Methodin[52] UWCM

1.25×10−12 6.30×10−11 5.40×10−8 2.70×10−12 6.70×10−12 3.44×10−15

Table9 maximumabsoluteerrorEforExample5.

α M E M E M E M E M E 1 7 1.26×10−3 9 1.05×10−5 11 3.75×10−8 13 7.12×10−11 15 7.43×10−14 1 2 1.35×10− 3 _1.10×10−5 _3.81×10−8 _6.99×10−11 _8.59×10−14 0 1.44_×10−3 _1.15×10−5 _3.86×10−8 _6.81×10−11 _7.26×10−14 −0.49 1.56×10−3 1.21×10−5 3.89×10−8 6.57×10−11 6.92×10−14

Table10 ComparisonbetweenthebesterrorsforExample5.

Method [24] [43] [45] [54] [55] [27] UWCM

E 4.98×10−11 5.01×10−11 1.67×10−9 5.40×10−5 2.70×10−5 6.38×10−13 6.92×10−14

WesatisfyEq.(29)atthefirsttworootsoftheshiftedChebyshev polynomialT₄∗(x),namelyatx1,2= 14(2±

2−√2). Further-more,theuseoftheboundaryconditionsyields,

c0−c1+c2−c3=0, (30)

c0+c1+c2+c3=1, (31)

andsolutionoftheresultedsystemofequationsgives,

c0= 5 16, c1= 15 32, c2= 3 16, c3= 1 32, andconsequently u0,4,0(x)= 5 16 + 15 32(2x−1)+ 3 16(8x 2_−8x+1) + 1 32(32x 3_−48x2_+18x−1)=x3_{, (32)}

whichistheexactsolution.

Example 7. Consider the following linear eighth-order BVP (see,[50,45,21,23,53]):

y(8)−y=−8ex; 0<x<1, y(0)=1,

y(1)(0)=0, y(2)(0)=−1, y(3)(0)=−2,

y(4)(0)=−3, y(5)(0)=−4, y(1)(1)=−e,

y(2)_{(1)=−2e, (33)}

withtheexactsolutiony(x)=(1−x)ex_{.InTable11,the}

max-imumabsoluteerrorEislistedforvariousvaluesofkandM, whileinTable12,acomparisonbetweenthebesterrorresulted fromtheapplicationofUWCMforthecaseλ= 1

2 – to

numer-icallysolve(33) – with thenumerical algorithmsobtained in

[50,45,21,23,53]isdisplayed.Thiscomparison showsthatour

suggestednumericalmethodismoreaccuratethanthose devel-opedin[50,45,21,23,53].

Example 8. Consider the following nonlinear second-order boundaryvalueproblem:

y(x)+y(x)2−y(x)=sinh2x, y(0)=1,y(1)=cosh1(34), withtheexactsolutiony(x)=coshx.

InTable13,welistthemaximumabsoluteerrorE,for vari-ousvaluesofkandM.

Example9. Considerthefollowingsecond-orderlinearsingular initialvalueproblem(see[35]):

y(x)+ f(x)y(x)+y(x)=g(x), y(0)=0,y(0)=1, (35) where f(x)= ⎧ ⎪ ⎨ ⎪ ⎩ −4x+4 3, 0≤x< 1 4, x−1 3, 1 4≤x≤1,

and g(x) is chosen such that the exact solution of (35) is

y(x)= ex_sinx

1+x2. We apply UWCM with λ=1,k=2. In

Table14,welistthemaximumabsoluteerrorforvarious val-uesofM,whileinFig.2,theexactsolutioniscomparedwith

Table13 MaximumabsoluteerrorEforExample8. k M E k M E

0 3 5.69_×10−4 _{1 2 1.12×10}−6

4 1.35×10−5 3 1.97×10−8

Table11 MaximumabsoluteerrorEforExample7.

k=0 M 11 12 13 14 15 16

E 3.04_×10−9 _2.65×10−10 _5.81×10−12 _3.65×10−13 _8.95×10−14 _5.66×10−16 k=1 M 8 9 10 11 12 13

E 5.47×10−6 5.87×10−8 5.44×10−10 5.22×10−125.77×10−14 8.47×10−16

Table12 ComparisonbetweenthebesterrorsforExample7.

x UWCM [50] [45] [21] [23] [53]

0.25 4.44×10−16 2.55×10−15 3.03×10−10 2.16×10−9 4.58×10−9 3.89×10−10

0.5 2.66×10−16 9.65×10−14 7.73×10−9 1.16×10−7 9.84×10−9 1.16×10−7

0.75 1.22_×10−16 _5.63×10−13 _3.12×10−8 _1.05×10−6 _1.09×10−5 _1.05×10−6

Table14 MaximumabsoluteerrorEforExample9. M E M E

2 2.98×10−4 6 1.85×10−10

4 1.70_×10−5 _{8 9.12×10}−11

Figure2 NumericalandexactsolutionsofExample9.

Table15 ComparisonbetweendifferenterrorsforExample9. x 0.2 0.4 0.6 0.8 Methodin [35] 4.30_×10−12 _1.24×10−11 _2.97×10−12 _4.35×10−12 Present method 6.47×10−13 1.21×10−12 1.74×10−12 2.32×10−12

thethreenumericalsolutionscorrespondtok=2,M=2,k=

2,M=4andk=2,M=6,alsoinTable15wegivea compar-isonbetweentheerrorobtainedin[35]andthepresentmethod,

k=2,M=8.

Note 3. It is worthy to note here that the results obtained inExample 9areincompleteagreementwith theresults ob-tainedbyAbd-Elhameedetal.[3],asexpectedthesecondkind Chebyshevwaveletsisadirectspecialcaseoftheultraspherical wavelets.

6. Conclusions

Inthispaper,anewalgorithm forobtainingnumerical spec-tralsolutions for2νth-order linearandnonlineardifferential equationsisdevelopedandimplemented. Actuallythe deriva-tionofthisalgorithmisbasedonconstructingtheshifted ul-trasphericalwaveletsandapplyingspectralcollocationmethod. The developed algorithm has the advantage that, high ac-curate approximate solutions areachieved using afew num-berofultrasphericalwavelets.Theobtainedapproximate solu-tions areveryclose tothe analyticalones. Onedisadvantage ofthe proposedmethodis howto handlethenonlinear sys-temusingNewton’smethod,thishasbeendiscusseddeeplyin Section4.

Acknowledgment

Theauthorswouldliketothanktherefereesforcarefulreading andveryhelpfulcommentstoimprovethispaper.

References

[1] W.M. Abd- Elhameed, E.H. Doha, Y.H. Youssri, Ef-ficient spectral-Petrov–Galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials, Quaest. Math. 36 (2013) 15–38.

[2] W.M.Abd-Elhameed,E.H.Doha,Y.H.Youssri,Newwavelets col-location method forsolving second-order multiPoint boundary valueproblemsusingChebyshevpolynomialsofthirdandfourth kinds,Abstr.Appl.Anal.2013(2013)542839.

[3] W.M. Abd-Elhameed, E.H. Doha, Y.H. Youssri, New spectral secondkindchebyshevwaveletsalgorithmforsolvinglinearand nonlinear second-order differential equations involving singu-lar and Bratutype equations, Abstr. Appl. Anal. 2013 (2013) 71756.

[4] R.P.Agarwall,G.Akrivis,Boundaryvalueproblemsoccurringin platedeflectiontheory,J.Comput.Appl.Math.8(1982)145–154. [5] G.E.Andrews,R.Askey,R.Roy,SpecialFunctions,Cambridge

UniversityPress,Cambridge,1999.

[6] C.Basdevant,M.Deville,P.Haldenwang,J.M.Lacroix,J. Ouz-zani,R.Peyret,P.Orlandi,A.T.Patera,Spectralandfinite differ-encesolutionsoftheBurgersequation,Comp.Fluids14(1986) 23–41.

[7] E.Babolian,F.Fattah zadeh,Numericalsolutionofdifferential equationsbyusingChebyshevwaveletoperationalmatrixof inte-gration,Appl.Math.Comput.188(2007)417–426.

[8] J.P. Boyd, One point pseudospectral collocation for the oned-imensional Bratu equation, Appl. Math. Comput. 217 (2011) 5553–5565.

[9] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral MethodsonFluidDynamics,Springer-Verlag,1988.

[10] M.M.Chawla,C.P.Katti,Finitedifferencemethodsfortwo-point boundaryvalueproblemsinvolvinghighorderdifferential equa-tions,BITNumer.Math.19(1979)27–33.

[11] A. Constantmldes, Applied Numerical Methods with Personal Computers,McGraw-Hill,NewYork,1987.

[12] M.Dehghan,A.Nikpour,Numericalsolution ofthesystemof second-orderboundaryvalueproblemsusingthelocalradialbasis functionsbaseddifferentialquadraturecollocationmethod,Appl. Math.Modell.37(18)(2013)8578–8599.

[13] M.Dehghan,S.Aryanmehr,M.R.Eslahchi,Atechniqueforthe numericalsolutionofinitial-valueproblemsbasedon aclassof Birkhoff-type interpolation method, J. Comp.Appl. Math.244 (2013)125–139.

[14] M.Dehghan,F.Shakeri,Asemi-numericaltechniqueforsolving themulti-pointboundaryvalueproblemsandengineering applica-tions,Int.J.Numer.Method.H.21(7)(2011)794–809.

[15] M.Dehghan,M.Tatari,Findingapproximatesolutionsforaclass ofthird-ordernon-linearboundaryvalueproblemsviathe decom-positionmethodofAdomian,Int.J.Comp.Math.87(6)(2010) 1256–1263.

[16] E.H.Doha,W.M.Abd-Elhameed,Efficientspectral-Galerkin al-gorithms for direct solution of second-order equations using ultraspherical polynomials, SIAM J. Sci. Comput. 24: (2002) 548–571.

[17] E.H. Doha, W.M. Abd-Elhameed, M.A. Bassuony, New algo-rithmsforsolvinghigheven-orderdifferentialequationsusingthird andfourthChebyshev–Galerkinmethods,J.Comput.Phys.236: (2013)563–579.

[18] E.H.Doha,W.M.Abd-Elhameed,A.H.Bhrawy,Newspectral– Galerkinalgorithmsfordirectsolutionofhigheven-order differ-entialequationsusingsymmetricgeneralizedJacobipolynomials, Collect.Math.64(2013)373–394.

[19]E.H. Doha, W.M. Abd- Elhameed, Y.H. Youssri, Second kind Chebyshev operational matrix algorithm for solving differen-tial equations ofLane–Emdentype, NewAstron. 23–24(2013) 113–117.

[20]M.R.Eslahchi,M.Dehghan,S.Ahmadi,ThegeneralJacobi ma-trixmethodforsolvingsomenonlinearordinarydifferential equa-tions,Appl.Math.Modell.36(8)(2012)3387–3398.

[21]A. Golbabai,M. Javidi,Application ofhomotopyperturbation methodforsolvingeighth-orderboundaryvalueproblems,Appl. Math.Comp.191(2)(2007)334–346.

[22]C.Giordano,A.Laforgia,OntheBernstein-typeinequalitiesfor ultrasphericalpolynomials, J. Comput.Appl. Math.153 (2003) 243–248.

[23]J.H. He, The variationaliteration method foreighth-order ini-tial-boundaryvalueproblems,Phys.Scripta76(6)(2007)680–682. [24]M.K.Jain,S.R.K.Iyengar,J.S.V.Saldanha,Numericalsolutionof afourth-orderordinarydifferentialequation,J.Eng.Math.11(4) (1977)373–380.

[25]M.Kajani,A.Tavassoli,H.Vencheh,M.Ghasemi,TheChebyshev waveletsoperationalmatrixofintegrationandproductoperation matrix,Int.J.Comput.Math.86(7)(2009)1118–1125.

[26]L. Kantorovitch, The method of successive approximation for functionalequations,ActaMath.71(1)(1939)63–97.

[27]A.Khan,P.Khandelwal,Non-polynomialsexticsplineapproach forthesolutionoffourth-orderboundaryvalueproblems,Appl. Math.Comput.218(2011)3320–3329.

[28]M.Lakestani,M.Dehghan,FourtechniquesbasedontheB-spline expansion andthecollocation approachforthenumerical solu-tionoftheLane–Emdenequation,Math.Meth.Appl.Sci.36(16) (2013)2243–2253.

[29]M.Lakestani,B.N.Saray,M.Dehghan,Numericalsolutionfor the weakly singularFredholm integro-differential equations us-ingLegendremultiwavelets,J.Comp.Appl.Math.235(11)(2011) 3291–3303.

[30]M.Lakestani,M.Dehghan,Thesolutionofasecond-order nonlin-eardifferentialequationwithNeumannboundaryconditions us-ingsemi-orthogonalB-splinewavelets,IntJ.Comp.Math.83(8-9) (2006)685–694.

[31]Y. Li, Solvinganonlinearfractional differentialequation using Chebyshevwavelets,Commun.NonlinearSci.Numer.Simul.15 (9)(2010)2284–2292.

[32]G.Liu,X.Feng,M.Li,Relationshipofd-dimensionalcontinuous multi-scalewaveletshrinkage withintegro-differentialequations, ChaosSolitonFract.40(2009)1118–1126.

[33]Y.-C.Li,AnoteonanidentityofthegammafunctionandStirling’s formula,RealAnal.Exchang.32(1)(2007)267–272.

[34]F. Mohammadi, M.M. Hosseini, Legendre waveletmethod for solvinglinearstiff systems,J.Adv.Res.Diff.Eq.2(1)(2010)47–57. [35]F.Mohammadi,M.M.Hosseini,AnewLegendrewavelet opera-tionalmatrixofderivativeanditsapplicationsinsolvingthe sin-gular ordinary differentialequations,J. Frank. Inst.348 (2011) 1787–1796.

[36]F. Mohammadi, M.M. Hosseini, S.T. Mohyud-din, Legendre waveletGalerkin methodforsolvingordinarydifferential equa-tions withnonanalyticsolution,Int. J.Syst. Sci. 42(4)(2011) 579–585.

[37] A.K.Nasab,A.Kilicman,E.Babolian,Z.P.Atabakan,Wavelet analysismethodforsolvinglinearandnonlinearsingular bound-aryvalueproblems,Appl.Math.Model.(2013)5876–5886. [38] D.E.Newland,AnIntroductiontoRandomVibrations,Spectral

and Wavelet Analysis, LongmanScientific and Technical, New York,1993.

[39] J.M. Ortega, W.C. Rheinboldt, Iterative Solution ofNonlinear EquationsinSeveral Variables,vol.30,SiamPublishingHouse, 1970.

[40] E.D.Rainville,SpecialFunctions,NewYork,1960.

[41] K. Parand, M. Dehghan, A. Pirkhedri, The Sinc-collocation methodforsolvingtheThomas–Fermiequation,J.Comp.Appl. Math.237(1)(2013)244–252.

[42] L.B.Rall,AnoteontheconvergenceofNewton’smethod,SIAM J.Numer.Anal.11(1)(1974)34–36.

[43] M.A.Ramadan,I.F.Lashien,W.K.Zahra,Quintic non-polyno-mialsplinesolutionsforfourth-order two-pointboundaryvalue problem, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 1105–1114.

[44] J.Rashidinia,M.Ghasemia,B-splinecollocationforsolutionof two-pointboundaryvalueproblems,J.Comput.Appl.Math.235 (2011)2325–2342.

[45] J. Rashidinia, Direct methods for the solution of a linear fourth-ordertwo-pointboundaryvalueproblem,Int.J.Eng.Sci. 13(2002)37–48.

[46] M.Razzaghi,S.Yousefi,Legendrewaveletsoperationalmatrixof integration,Int.J.Syst.Sci.32(4)(2001)495–502.

[47] M.Razzaghi,S.Yousefi,Legendrewaveletsmethodforthesolution ofnonlinearproblemsinthecalculusofvariations,Math.Comput. Model.34(2001)45–54.

[48] M. Razzaghi, S. Yousefi, Legendre wavelets method for con-strained optimal control problems, Math. Meth. Appl. Sci. 25 (2002)529–539.

[49] B.Saka,I.Dag,QuarticB-splinecollocationmethodtothe nu-mericalsolutionsoftheBurgers’equation,ChaosSolitonFract. 32(2007)1125–1137.

[50] S.S. Siddiqi, M. Iftikhar, Numerical solution of higher order boundaryvalue,Problems,Abst.Appl.Anal.(2013)427521(12 pages).

[51] E.H.Twizell,Asixth-orderextrapolationmethodforspecial non-linearfourth-orderboundaryvalueproblems,Comp.Meth.Appl. Mech.Eng.62(1987)293–303.

[52] E.H.Twizell,S.I.A.Tirmizi,Multiderivativemethodsfornonlinear beamproblems,Commun.Appl.Numer.Meth.4(1988)43–50. [53] M.Torvattanabun,S.Koonprasert,Variationaliterationmethod

forsolvingeighth-orderboundaryvalueproblems,ThaiJ.Math.8 (4)(2010)121–129.

[54] R.A.Usmani,S.A.Warsi,Smoothsplinesolutionsforboundary valueproblemsinplatedeflectiontheory,Comput.Math.Appl.6 (1980)205–211.

[55] R.A.Usmani,Theuseofquarticsplinesinthenumericalsolution ofafourth-orderboundaryvalueproblem,J.Comput.Appl.Math. 44(1992)187–199.

[56] L. Zhu, Q.B.Fan, Solvingfractional nonlinear Fredholm inte-gro-differentialequationsbythesecondkindChebyshevwavelet, Commun.NonlinearSci.Numer.Simulat.17(6)(2012)2333–2341.