Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method

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Heliyon

journal homepage:www.cell.com/heliyon

Research article

Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method

Iftikhar Ahmad

^a

, Hira Ilyas

^a

, Kadir Kutlu

^c

, Vizda Anam

^a,b

, Syed Ibrar Hussain

^a

, Juan Luis García Guirao

^d,e,∗

aDepartmentofMathematics,UniversityofGujrat,Gujrat50700,Pakistan

bBCAM-BasqueCenterforAppliedMathematics,AlamedadeMazarredo14,48009Bilbao,Bizkaia,Basque-Country,Spain

cDepartmentofMathematics,RecepTayyipErdoganUniversity,Rize,Turkey

dDepartmentofAppliedMathematicsandStatistics,TechnicalUniversityofCartagena,Cartagena,Spain

eNonlinearAnalysisandAppliedMathematics(NAAM)-ResearchGroup,DepartmentofMathematics,FacultyofScience,KingAbdulazizUniversity,P.O.Box80203, Jeddah21589,SaudiArabia

A R T I C L E I NF O A B S T R A C T

Keywords:

Sinccollocationmethod(SCM) Hunter-Saxtonequation(HSE) Liquidcrystals(LC) Stabilityanalysis NumericalsolutionofPDEs

Inthis study,numericaltreatmentof liquidcrystal modeldescribedthroughHunter-Saxtonequation(HSE) has been presented by sinc collocation technique through theta weighted scheme due to its enormous applicationsincluding, defects, phase diagrams,self-assembly, rheology, phase transitions, interfaces, and integratedbiologicalapplicationsinmesophasematerialsandprocesses.Sincfunctionsprovidethe procedure forfunctionapproximationoveralltypesofdomainscontainingsingularities,semi-inﬁniteorinﬁnitedomains.

SincfunctionshavebeenusedtoreduceHSEintoanalgebraicsystemofequationsthatmakesthesolutionquite superficial.Thesealgebraicequationshavebeen interpretedasmatrices.Thisprojectedthatsinccollocation techniqueisconsiderablyefficaciousoncomputationalgroundforhigheraccuracyandconvergenceofnumerical solutions.Stabilityanalysisofthe proposedtechniquehasensuredtheaccuracyandreliabilityofthemethod, moreover,asthestabilityparametersatisfiedtheconditiontheproposedsolutionoftheproblemconverges.The solutionoftheHSEispresentedthroughgraphicalfiguresandtablesfordifferentcasesthatareconstructedon variousvaluesof𝜃 andcollocationpoints.Theaccuracyandefficiencyoftheproposedtechniqueisanalyzedon thebasisofabsoluteerrors.

1. Introduction

Liquidcrystals (LCs)is astate between liquidandsolid crystals, for instance, electronic displays,cell membranes, proteins, solutions of soapanddetergentsarewellknownexamplesofLCs.Thehistory of LCsbeginsbackin1888, whenanAustrian physiologist Friedrich Reinitzenstartedexaminingthepropertiesofcholesterol.Therewere severalspeculations aboutthephysico-chemicalproperties ofcholes- terolinthestartbutafterwardsthatclasswascalledoutthecholesteric liquidcrystals[1,2,3].However,neumaticisoneofthenaturallyoc- curringphenomenonwhichallowedthemathematicianstoconvertit intoamodelforfuturepredictions.Therefore,J.K.HunterandRalph Saxtonderivedanasymptoticpartialdiﬀerentialequationwhichgov- ernshyperbolicwavesasamodelofaweaklynonlinearorientational

*

Correspondingauthorat:DepartmentofAppliedMathematicsandStatistics,TechnicalUniversityofCartagena,Cartagena,Spain.

E-mailaddress:[email protected](J.L.G. Guirao).

wavepropagationinamassivenematicliquidcrystaldirectorﬁeld[4].

Itis bi-variational andbi-Hamiltonian structureof integrable partial diﬀerentialequationsknownasHunter-Saxtonequation(HSE). Some assumptionsforHSEweremadeasfollows;

• Therewillbenoﬂuidﬂow,

• Orientation ofmoleculeswill be focusedandshownby director ﬁeldofneumaticLCs(i.e.)𝑣(𝑥,𝑡),

• NokineticenergyduetohigherviscosityofLCs,

• Potentialenergywillconsistuponbend,twistandelasticcoeﬃ- cientofsplay,

Theauthorscreatedawave-markersituationusingtheinitialboundary valueproblem(IBVP)ofHSEpurelyforsplaywaves.

https://doi.org/10.1016/j.heliyon.2021.e07600

Received2December2020;Receivedinrevisedform26January2021;Accepted13July2021

2405-8440/©²⁰²¹^TheÂuthor(s). ^Published^byÊlsevier^Ltd. ^Thisîsânôpenâccessârticleûnder^the^CC^BY-NC-ND^license(http://creativecommons.org/li- censes/by-nc-nd/4.0/).

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ThegeneralHSEform,reportedin[1,5,6],isdeﬁnedas (𝑣_𝑡+𝑣𝑣_𝑥)

𝑥=1

2𝑣²_𝑥, (1)

withconditions

𝑣 (𝑥, 0) = 𝑔 (𝑥) , 𝑣 (0, 𝑡) = 0, 𝑣 (𝐿, 𝑡) = 0, 𝑣_𝑥(𝐿, 𝑡) = 0.

InHunter-Saxtonequation,varioustermspresentsasfollows:

•𝑥 isscaledaspositionthatmovesalonglinearizedwavevelocity 𝑣(𝑥,𝑡),

•𝑡 astimecoordinate,

• When𝑥= 0,thewavemarkerismountedandattheotherendof thechannel thereﬂectionof waveis ignoredbecauseitwasfar awayfromthewavemarker.

Asoverafinitetimescalethisexperimenttakesplacesoattheother end of thechannel nowavemotionproduced.The resultofcurrent workdoesnotsupportthemodelifanyreflectionofwavesproduced backwardorbytwist.Totake accountofthetwistedsplaywaves at theHSEapproximationlevel,a directorfieldwithrotationalinertiais neededasin[7].

{(𝑢𝑡+𝑤𝑢𝑥)

𝑥= 0, −∞ < 𝑥 < ∞, 𝑡 = 0,

𝑤_𝑥𝑥=𝑢²_𝑥, 𝑤_𝑥𝑥> 0, (2)

wherethesecond dependentvariable𝑤(𝑥,𝑡) in Eq. (2) canbe inter- pretedastheamplitudeofsplaywaveoradvectionvelocityof𝑤,that isgeneratedbytwistwaves𝑢(𝑥,𝑡).

HunterandSaxtonused variationalprinciplestoderiveandana- lyze this equation.They studied theinstability in thedirector field of a nematic liquid crystal. Smooth solutions break down in finite times ofasymptoticequation[4].JonathanLenellsprovidedfounda- tionforthegeometricstudyof HSequation,thisexhibitsa geodesic flow.ThesystemofnonlinearDEslikeHunter-Saxton,Camassa-Holm andDegasperis-Procesiwasanalyzedbyvariationalprincipletofindthe weaksolutionsin[8],Volterra-Fredholmintegralequations[9],Boussi- nesq equations [10], Schrödingerequation [11, 12],telegraph PDEs [13,14],Burgersequation[15].Thereareseveralresearchesinthelit- erature,thatareexaminedby thedifferenttechniques[16,17,18,19, 20].A localsolutionoftwo-componentHSEwasprovidedusingKato’s localexistencetheoryin[21],generalsolutionisprovidedthroughlo- caldiscontinuousGalerkin(LDG)andnewdiscontinuousGalerkin(DG) methodin[22].Furthermore,iterativemethodslike,variationaliter- ation method (VIM),modified variational iteration method (VMIM), Adomian decomposition method (ADM), modified Adomian decom- positionmethod(MADM)andHomotopyanalysismethod(HAM)are described [23]. The reciprocal transformations [24], Homotopy de- compositionmethod[25],bivariategeneralizedfractionalorderofthe Chebyshev functions(BGFCF)[26], cubictrigonometric B-Splinecol- locationmethod [27], collocationmethod [28], Harr waveletquasi- linearizationapproach[29],andLipschitzmetric[30],timemarching scheme[31]areappliedtostudythediffusionofneumaticLCs.The generalized Hunter-Saxtonequation is considered using integrability structures[32],NumericalsolutionsofHSEusingLaguerrewaveletand byusingefficientapproachontimedomainsispresentedin [33,34, 35].

ThepropertreatmentsofBCsareone ofthebasicdiﬃcultiesin eval- uatinganynumericalscheme,studybasedonthisconceptiscarriedout in[36,37,38,39,40,41,42].Amongthecomputationalmethods,one oftheimportantclassofmethodswerespectralmethodstosolvelinear andnon-linearpartialdiﬀerentialequations,somehowspectralmethods havefew drawbacksbecausethesemethodsdonotrepresentthephys- icalprocessin spectralspace. Secondly,thesearerestrictedtothose problemswhichhasperiodicboundaryconditions[43].Anothertech- niqueof Lagrange interpolationpolynomials (usedastest functions)

is not somuch useful todeterminetheoscillating solution or prob- lemswithunboundeddomains.A desirableuseof sincfunctionscan dealwiththeseadversitieswhilekeepingtheexperimentalconvergence rate.Asthederivativesofsincfunctionsonboundariesornotdefined, toovercomethisdifficultysinccollocationmethodsalongwithfinite differencemethodisusedtocalculatederivativesnearboundariesfor solvingtheHunter-Saxtonequation[44].

Fromtheabove reportedliteratureitisnoticedthatthesinccol- locationmethod with theta weighted scheme not yet used tosolve Hunter-Saxtonequationarisinginliquidcrystalmodel,thismotivated theauthorstostudyontheproposedmodel.Inthispaper,sinccollo- cationtechniqueisusedtoevaluatethetime-periodicbehaviorofthe Hunter-Saxtonequation,detailed stabilityanalysisof proposed tech- niqueispresentedafterimplementationtotheproblem.Insection1, anintroductionisgiven,section2containsthepreliminaryinformation abouthowtoapplyasincfunctionforpartialdifferentialequationand toapproximatethesolutionandthismethodisappliedontheproposed probleminsection3.Insection4,completestabilityanalysisispre- sented,thennecessaryandsufficientboundsfor𝜃 hasbeenevaluated andconfirmednumericallyinnextsection5withnumericalresultsand graphicalrepresentationofsolution.Inthelastsection5,theconclusion isprovided.

2.Sincbasesfunctions

Generally,Sincfunction[45]isdeﬁnedby

𝑠𝑖𝑛𝑐 (𝑧) =

{1 𝑧 = 0,

sin(𝜋𝑧)

𝜋𝑧 𝑧 ≠ 0, (3)

forall𝑧∈ℂ.

Assumingthestepsize(evenlyspacednodes)ℎ> 0,thetranslated sincfunctiondeﬁnedanddenotedas,

𝑆 (𝑚, ℎ) (𝑥) = 𝑆_𝑚(𝑥) = 𝑆𝑖𝑛𝑐( 𝑥 − 𝑚ℎ ℎ

), 𝑚 ∈ ℤ (4)

𝑠𝑖𝑛𝑐 (𝑚) =

⎧⎪

⎨⎪

⎩

1 𝑥 = 0,

sin (_𝜋

ℎ(𝑥−𝑚ℎ))

𝜋ℎ(𝑥−𝑚ℎ) 𝑥 ≠ 𝑚ℎ. (5)

If𝑣 isafunctiondeﬁnedontherealline,thenforstep-sizeℎ> 0 the Whittakercardinalexpansionfor𝑣 isdeﬁnedby,

𝐶 (𝑚, ℎ) (𝑥) =

∑∞

𝑚=−∞𝑣(𝑚ℎ)𝑆 (𝑚, ℎ) (𝑥) , (6)

whenevertheseriesconverges,𝑣 is approximatedbyusingtheﬁnite numberoftermsintheaboveequation,where𝑥_𝑚=𝑚ℎ andstepsizeℎ isgivenby

ℎ =

√𝜋𝑑

𝛼𝑁, (7)

0< 𝛼 ≤ 1, 𝑑 ≤ 𝜋,

𝑁 issuitablychosen,𝛼 and 𝑑 dependupontheasymptoticbehavior of𝑣.Sincapproximationcanbeconstructedforsemi-infinite,finiteor infiniteintervals.

Definition.Letusconsider𝐷_𝑏betheinfinitestripwithwidth2𝑏 about therealaxis[46]and𝑏> 0 isdefinedas

𝐷_𝑏= {𝑧 ∈ ℂ ∶ |ℑ (𝑧) | < 𝑏} .

Further,for𝜖 belongstounitinterval,let𝐷_𝑏(𝜖) bethe rectangleinthe complexplane

𝐷_𝑏(𝜖) = {

𝑧 ∈ ℂ | ℜ (𝑧) | <1

𝜖, |ℑ (𝑧) | < 𝑏 (1 −𝜖)

}

. (8)

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Let𝐵(𝐷_𝑏)denotethe classoffunctions𝑝 thatareanalyticin𝐷_𝑏,such that

𝑏

∫

−𝑏

|𝑝 (𝑥 + 𝑖𝑦) |𝑑𝑦 ⟶ 0 , 𝑥 ⟶ ±∞, and

𝑁( 𝑝, 𝐷_𝑏)

= lim

𝜖→0

⎛⎜

⎜⎜

⎝ ∫

𝜕𝐷𝑏(𝜖)

|𝑝 (𝑧) |²|𝑑𝑧|

⎞⎟

⎟⎟

⎠

1 2

< ∞,

where𝜕𝐷_𝑏describeboundaryof𝐷_𝑏.

Lemma1.TheHilberttransform[43]ofafunction𝑣 canbeapproximated by

ℍ [𝑣] (𝑥) =1 𝜋𝑃

∞

∫

−∞

𝑓 (𝑥 − 𝑦) 𝑦 𝑑𝑦 ≈ ∑^𝑁

𝑚=−𝑁𝑣_𝑚𝑇_𝑚(𝑥) , (9)

where

𝑇_𝑚(𝑥) =1 − cos[_𝜋

ℎ(𝑥 − 𝑚ℎ)]

𝜋

ℎ(𝑥 − 𝑚ℎ) .

TheentriesofHilberttransformsmatrixatcollocationpoints𝑥_𝑚=𝑚ℎ are ℍ = 𝑇𝑚(

𝑥𝑖)

=

{ 0 𝑚 = 𝑖,

1−(−1)^𝑖−𝑚

𝜋(𝑖−𝑚) 𝑚 ≠ 𝑖. (10)

The generalformulafor nthderivativeoffunction𝑣 atcollocationpoints 𝑥𝑚=𝑚ℎ canbeapproximatedby

𝑣^𝑛( 𝑥_𝑚)

=

∑𝑁 𝑚=−𝑁

𝑣 (𝑚ℎ) d^𝑛 d𝑥^𝑛

[𝑆_𝑚(𝑥)]

, (11)

whereSrepresentsthe sincderivativeandhrepresentsthestepsize.Some generalnotationsare,

𝑆⁽^𝑛)=𝑆_𝑚𝑖^(𝑛), 𝑆_𝑚(𝑥) = [𝑆 (𝑚, ℎ) (𝑥)] |_𝑥=𝑥_𝑖. (12) Forcalculationofderivatives,a generalformulahasbeenwrittenbelowto calculateevenandoddderivatives[43].

Foreven,

𝑆⁽^2𝑠^∗⁾= 1 ℎ²^𝑠^∗

d²^𝑠^∗

d𝑥²^𝑠^∗[𝑆 (𝑚, ℎ) (𝑥)] |_𝑥=𝑥_𝑖

= 1 ℎ²^𝑠^∗

⎧⎪

⎨⎪

⎩

(𝜋)^2𝑠^∗₍₂⁽⁻¹⁾_𝑠∗+1)^𝑠∗, 𝑚 = 𝑖,

(−1)^{(𝑖−𝑚)} (𝑖−𝑚)^2𝑠∗.∑⁽𝑠^∗−1)

𝑙=0 (−1)⁽^{𝑙+1) 2𝑠}^∗^!

(2𝑙+1)!𝜋²^𝑙(𝑖 − 𝑚)²^𝑙, 𝑚 ≠ 𝑖.

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Forodd,

𝑆⁽²^𝑠^∗⁺¹⁾= 1 ℎ⁽²^𝑠^∗⁺¹⁾

d⁽^2𝑠^∗⁺¹⁾

d𝑥⁽²^𝑠^∗⁺¹⁾[𝑆 (𝑚, ℎ) (𝑥)] |_𝑥=𝑥_𝑖

= 1

ℎ⁽²^𝑠^∗⁺¹⁾

⎧⎪

⎨⎪

⎩

0, 𝑚 = 𝑖,

(−1)^{(𝑖−𝑚)} (𝑖−𝑚)⁽^2𝑠∗+1⁾.∑𝑠^∗

𝑙=0(−1)^𝑙

(2𝑠^∗+1)

!

(2𝑙+1)!𝜋²^𝑙(𝑖 − 𝑚)^2𝑙, 𝑚 ≠ 𝑖.

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Allthosederivativeswhichhasbeenusedinthisworkarewritten below.

Inparticularput𝑠= 0,inEq. (13)wehave0thSincderivative 𝑆⁽⁰⁾= [𝑆 (𝑚, ℎ) (𝑥)] |_𝑥=𝑥_𝑖=

{1, 𝑚 = 𝑖,

0, 𝑚 ≠ 𝑖. (15)

For1stSincderivative,put𝑠= 0,inEq. (14)wehave.

𝑆⁽¹⁾=1 ℎ

d

d𝑥[𝑆 (𝑚, ℎ) (𝑥)] |_𝑥=𝑥_𝑖=1 ℎ

{ 0, 𝑚 = 𝑖,

(−1)^{(𝑖−𝑚)}

(𝑖−𝑚) , 𝑚 ≠ 𝑖. (16)

For2ndSincderivative,put𝑠= 1,inEq. (13)wehave.

𝑆⁽²⁾= 1 ℎ²

d²

d𝑥²[𝑆 (𝑚, ℎ) (𝑥)] |_𝑥=𝑥_𝑖= 1 ℎ²

⎧⎪

⎨⎪

⎩

−𝜋²

3 , 𝑚 = 𝑖,

−2(−1)^{(𝑖−𝑚)}

(𝑖−𝑚)² , 𝑚 ≠ 𝑖. (17) PropertiesofSinccollocationmethod

• TheSincCollocationmethodisapplicableoveralltypesofdomains whereothertechniquefail tohandlesingularity, thismethod is applicableonevensemiinfiniteorinfinitedomain,becauseitdid notprovidethe infinitedomainasinfinite.

• TheSCMcanbe successivelyappliedonthehigherorder partial derivativeduetoitspropertyofconvertinghigherorderderivatives intoalgebraicequationsthatareeasytotackle.

• TheSCMcanbeimposed onthatproblemsthathavesingularities withhigherorder partialderivative oron boundary,mostly the singularityisremovedwhenthisisappliedontheproblemifnota suitablepolynomialisappliedtoremovethesingularity.

• Moreover,TheSCMisusedextensivelynotbecauseofexpedient reactionagainsttheproblemhavingsingularitybutalsoduetoin- terestingconvergence.

ThegraphicalabstractofSincCollocationmethodologyandimplemen- tationofSCMonHunterSaxtonequationisillustratedinFig.1.

3.SolvingHunter-Saxtonequation

ConsidertheHunterSaxtonequation.

(𝑣_𝑡+𝑣𝑣_𝑥)

𝑥=1

2𝑣²_𝑥, (18)

𝑣_𝑥𝑡+𝑣𝑣_𝑥𝑥+𝑣²_𝑥−1 2𝑣²_𝑥= 0, 𝑣_𝑥𝑡+𝑣𝑣_𝑥𝑥+1

2𝑣²_𝑥= 0, (19)

withinitialcondition,

𝑣(𝑥, 0) = 1. (20)

Nowapplying𝜃-weightedschemeonEq. (19),where𝛿𝑡 representsthe timestep

(𝑣^𝑘+1_𝑥 −𝑣^𝑘_𝑥 𝛿𝑡

) +𝜃[(

𝑣𝑣𝑥𝑥)𝑘+1

+1 2

(𝑣²_𝑥)𝑘+1] + (1 −𝜃)[(

𝑣𝑣_𝑥𝑥)𝑘+1 2

(𝑣²_𝑥)𝑘]

= 0, 𝑣^𝑘+1_𝑥 −𝑣^𝑘_𝑥+𝜃𝛿𝑡[(

𝑣𝑣_𝑥𝑥)𝑘+1+1 2

(𝑣²_𝑥)𝑘+1] + (1 −𝜃) 𝛿𝑡[(

𝑣𝑣_𝑥𝑥)_𝑘 +1

2 (𝑣²_𝑥)𝑘]

= 0.

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Nowlinearizingsomenonlineartermsforthis,wewritetheTaylorex- pansionof(

𝑣^𝑘+1_𝑥 )2

𝑣^𝑘+1_𝑥 =𝑣_𝑥( 𝑡_𝑘+𝛿𝑡)

(22) 𝑣^𝑘+1_𝑥 =𝑣_𝑥(

𝑡_𝑘) +𝛿𝑡𝑣_𝑥𝑡(

𝑡_𝑘)

+𝑂 (𝛿𝑡)², 𝑣^𝑘+1_𝑥 𝑣^𝑘+1_𝑥 =(

𝑣_𝑥( 𝑡_𝑘)

+𝛿𝑡𝑣_𝑥𝑡( 𝑡_𝑘)

+𝑂 (𝛿𝑡)²) (𝑣_𝑥(

𝑡_𝑘) +𝛿𝑡𝑣_𝑥𝑡(

𝑡_𝑘)

+𝑂 (𝛿𝑡)²)

= (

𝑣^𝑘_𝑥+𝛿𝑡

) +𝑂 (𝛿𝑡)²

)

( 𝑣^𝑘_𝑥+𝛿𝑡

) +𝑂 (𝛿𝑡)²

) ,

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Fig. 1. Graphical Abstract of Sinc collocation method for HSE.

𝑣^𝑘+1_𝑥 𝑣^𝑘+1_𝑥 ≈ 2𝑣^𝑘_𝑥𝑣^𝑘+1_𝑥 −𝑣^𝑘_𝑥𝑣^𝑘_𝑥. (23) Also,weneedtolinearize(

𝑣𝑣_𝑥𝑥)_𝑘+1

sowewriteitsTaylorexpansion.

𝑣^𝑘+1=𝑣( 𝑡_𝑘+𝛿𝑡)

=𝑣( 𝑡_𝑘)

+𝛿𝑡𝑣_𝑡( 𝑡_𝑘)

+𝑂 (𝛿𝑡)², (24)

𝑣^𝑘+1_𝑥𝑥 =𝑣_𝑥𝑥( 𝑡_𝑘+𝛿𝑡)

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=𝑣_𝑥𝑥( 𝑡_𝑘)

+𝛿𝑡𝑣_𝑥𝑥𝑡( 𝑡_𝑘)

+𝑂 (𝛿𝑡)², 𝑣^𝑘+1𝑣^𝑘+1_𝑥𝑥 =(

𝑣( 𝑡_𝑘)

+𝛿𝑡𝑣_𝑡( 𝑡_𝑘)

+𝑂 (𝛿𝑡)²) (𝑣_𝑥𝑥(

𝑡_𝑘) +𝛿𝑡𝑣_𝑥𝑥𝑡(

𝑡_𝑘)

+𝑂 (𝛿𝑡)²)

= (

𝑣^𝑘+𝛿𝑡

(𝑣^𝑘+1−𝑣^𝑘 𝛿𝑡

) +𝑂 (𝛿𝑡)²

)

( 𝑣^𝑘_𝑥𝑥+𝛿𝑡

(𝑣^𝑘+1_𝑥𝑥 −𝑣^𝑘_𝑥𝑥 𝛿𝑡

) +𝑂 (𝛿𝑡)²

) ,

𝑣^𝑘+1𝑣^𝑘+1_𝑥𝑥 ≈𝑣^𝑘+1𝑣^𝑘_𝑥𝑥+𝑣^𝑘𝑣^𝑘+1_𝑥𝑥 −𝑣^𝑘𝑣^𝑘_𝑥𝑥. (26) SubstitutingvaluesfromEqs. (23),(26)intoEq. (21)weget,

𝑣^𝑘+1_𝑥 −𝑣^𝑘_𝑥+𝜃𝛿𝑡[(

𝑣^𝑘+1𝑣^𝑘_𝑥𝑥+𝑣^𝑘𝑣^𝑘+1_𝑥𝑥 −𝑣^𝑘𝑣^𝑘_𝑥𝑥) +1

2

(2𝑣^𝑘_𝑥𝑣^𝑘+1_𝑥 −𝑣^𝑘_𝑥𝑣^𝑘_𝑥)

] (1 −𝜃) 𝛿𝑡[ 𝑣^𝑘𝑣^𝑘_𝑥𝑥+1

2𝑣^𝑘_𝑥𝑣^𝑘_𝑥]

= 0. (27)

Aftersimpliﬁcation

(5)

𝑣^𝑘+1_𝑥 −𝑣^𝑘_𝑥+𝜃𝛿𝑡[

𝑣^𝑘+1𝑣^𝑘_𝑥𝑥+𝑣^𝑘𝑣^𝑘+1_𝑥 +𝑣^𝑘_𝑥𝑣^𝑘+1_𝑥 ] + (1 − 2𝜃) 𝛿𝑡[

𝑣^𝑘𝑣^𝑘_𝑥𝑥+1 2𝑣^𝑘_𝑥𝑣^𝑘_𝑥]

= 0, 𝑣^𝑘+1_𝑥 +𝜃𝛿𝑡[

𝑣^𝑘+1𝑣^𝑘_𝑥𝑥+𝑣^𝑘𝑣^𝑘+1_𝑥 +𝑣^𝑘_𝑥𝑣^𝑘+1_𝑥 ]

=𝑣^𝑘_𝑥− (1 − 2𝜃) 𝛿𝑡[ 𝑣^𝑘𝑣^𝑘_𝑥𝑥+1

2𝑣^𝑘_𝑥𝑣^𝑘_𝑥]

. (28)

Assume,thesolutioncanbeinterpolatedas

𝑣( 𝑥_𝑖, 𝑡_𝑘)

≡ 𝑣^𝑘( 𝑥_𝑖)

≈

∑𝑁 𝑚=1𝑣^𝑘_𝑚𝑆_𝑚(

𝑥_𝑖) .

NowweplugintheassumedsolutionintoaboveEq. (28)

∑𝑁 𝑚=1

𝑣^𝑘+1_𝑚 𝑆⁽¹⁾_𝑚 (𝑥_𝑖) +𝜃𝛿𝑡[∑^𝑁

𝑚=1

𝑣^𝑘+1_𝑚 𝑆_𝑚⁽⁰⁾(𝑥_𝑖)

∑𝑁 𝑛=1

𝑣^𝑘_𝑛𝑆_𝑚⁽²⁾(𝑥_𝑖) +

∑𝑁

𝑚=1𝑣^𝑘_𝑚𝑆⁽⁰⁾_𝑚 (𝑥𝑖)

∑𝑁

𝑛=1𝑣^𝑘+1_𝑛 𝑆⁽¹⁾_𝑚 (𝑥𝑖) +

∑𝑁 𝑚=1

𝑣^𝑘_𝑚𝑆⁽¹⁾_𝑚 (𝑥_𝑖)

∑𝑁 𝑛=1

𝑣^𝑘+1_𝑛 𝑆⁽¹⁾_𝑚 (𝑥_𝑖)]

=

∑𝑁

𝑚=1𝑣^𝑘_𝑚𝑆_𝑚¹(𝑥𝑖) − (1 − 2𝜃)𝛿𝑡[∑^𝑁

𝑚=1𝑣^𝑘_𝑚𝑆_𝑚⁽⁰⁾(𝑥𝑖)

∑𝑁 𝑛=1

𝑣^𝑘_𝑛𝑆_𝑚⁽²⁾(𝑥_𝑖) +1 2

∑𝑁 𝑚=1

𝑣^𝑘_𝑚𝑆_𝑚⁽¹⁾(𝑥_𝑖)

∑𝑁 𝑛=1

𝑣^𝑘_𝑛𝑆_𝑚⁽¹⁾(𝑥_𝑖)] .

(29)

Wewritenon-lineartermsofEq. (29)as, 𝑁1=𝑆⁽²⁾𝑣^𝑘∗𝑆⁽⁰⁾,

𝑁2=𝑆⁽⁰⁾𝑣^𝑘∗𝑆⁽¹⁾, 𝑁3=𝑆⁽¹⁾𝑣^𝑘∗𝑆⁽¹⁾,

(30)

where‘∗’standsforcomponentbycomponentmultiplication [𝑆⁽⁰⁾+𝜃𝛿𝑡(𝑁1+𝑁2+𝑁3)]

𝑣^𝑘+1= [𝑆⁽¹⁾−𝛿𝑡𝑁1− 𝛿𝑡

2𝑁3+ 2𝜃𝛿𝑡𝑁1+𝜃𝛿𝑡𝑁3

]𝑣^𝑘. (31)

Eq. (29)nowbecomesthefollowingmatrixequationbyusingEq. (30) intoit

𝐺𝑣^𝑘+1=𝐻𝑣^𝑘, (32)

where

𝐺 = 𝑆⁽⁰⁾+𝜃𝛿𝑡(𝑁1+𝑁2+𝑁3), (33)

𝐻 = 𝑆⁽¹⁾−𝛿𝑡𝑁1− 𝛿𝑡

2𝑁3+ 2𝜃𝛿𝑡𝑁1+𝜃𝛿𝑡𝑁3, (34) 𝑣^𝑘+1=𝐼⁰∗(

𝐺⁻¹𝐻)

𝑣^𝑘. (35)

4. Stabilityanalysis

InthissectionstabilityanalysisofSinccollocationmethodforsolv- ingHunterSaxtonequation(31)hasbeenpresented.Theerrortermcan bewrittenas

[𝑆⁽⁰⁾+𝜃𝛿𝑡(𝑁1+𝑁2+𝑁3)] 𝑒^𝑘+1

=[

𝑆⁽¹⁾−𝛿𝑡𝑁1− 𝛿𝑡

]𝑒^𝑘, (36)

where𝑆⁽⁰⁾,𝑆⁽¹⁾,𝑁1,𝑁2,and𝑁3aredeﬁnedby,errorisdeﬁnedas:

𝑒^𝑘=|𝑣^𝑘_{𝑒𝑥𝑎𝑐𝑡}−𝑣^𝑘𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒|, (37)

where𝑣^𝑘_{𝑒𝑥𝑎𝑐𝑡}and𝑣^𝑘𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒areexactandapproximatedsolutionattime 𝑡_𝑘.

Eq. (36)canbewrittenas

𝑒^𝑘+1=𝐷𝑒^𝑘, (38)

where

𝐷 ≡ 𝐺⁻¹𝐻 = [𝑆⁽⁰⁾+𝜃𝛿𝑡(𝑁1+𝑁2+𝑁3)]⁻¹

×[𝑆⁽¹⁾−𝛿𝑡𝑁1− 𝛿𝑡

2𝑁3+ 2𝜃𝛿𝑡𝑁1+𝜃𝛿𝑡𝑁3]. (39) Sinccollocationmethod isconsiderednumerically stableif𝜌(𝐷)≤ 1, where𝜌(⋅) denotesthespectralradius.Itwillbestableif

||||

|||

(𝑆⁽¹⁾−𝛿𝑡𝑁1−^𝛿𝑡

) 𝑆⁽⁰⁾+𝜃𝛿𝑡(

𝑁1+𝑁2+𝑁3

) ||

||||

|

≤ 1. (40)

Nowwehaveeigenvaluesof𝑆⁽⁰⁾,𝑆⁽¹⁾,𝑁1,𝑁2,𝑁3as𝜆0,𝜆1,𝜆_𝑁₁,𝜆_𝑁₂and 𝜆_𝑁₃respectively.Fromsection2𝑆⁽⁰⁾isjustanidentitymatrixso𝜆0= 1.

Furthermore

||||

||

𝜆1−𝛿𝑡(𝜆𝑁1+¹

2𝜆𝑁3) + 2𝜃𝛿𝑡(𝜆𝑁1+¹

2𝜆𝑁3) 1 +𝜃𝛿𝑡(𝜆𝑁1+𝜆𝑁2+𝜆𝑁3)

||||

||≤ 1, (41)

{𝑆⁽¹⁾}

𝑚,𝑖= (−1)^𝑚−𝑖

ℎ (𝑚 − 𝑖)= −(−1)^𝑖−𝑚 ℎ (𝑖 − 𝑚)= −{

𝑆⁽¹⁾}

𝑖,𝑚, (42)

where{ 𝑆⁽¹⁾}

𝑚,𝑚= 0.Thisshowsthat𝑆⁽⁰⁾ isskew-symmetricandhave purelyimaginaryvaluesas𝜆1=𝑖∣𝜆1∣

||||

|||

(2𝜃 − 1) 𝜆^𝑅_𝑁₁+ (𝜃 − 0.5) 𝜆^𝑅_𝑁₃ 1 +𝜃(

𝜆^𝑅_𝑁₁+𝜆^𝑅_𝑁₂+𝜆^𝑅_𝑁₃) +𝑖𝜃(

𝜆^𝐼_𝑁₁+𝜆^𝐼_𝑁₂+𝜆^𝐼_𝑁₃)||

||||

| +

𝑖||

||||

|

(𝜆1+ (2𝜃 − 1) 𝜆^𝐼_𝑁₁+ (𝜃 − 0.5) 𝜆^𝐼_𝑁₃) 1 +𝜃(

𝜆^𝐼_𝑁₁+𝜆^𝐼_𝑁₂+𝜆^𝐼_𝑁₃)||

||||

|

≤ 1,

(43)

whichimpliesthat

||||(2𝜃 − 1) 𝜆^𝑅_𝑁₁+ (𝜃 −1

2 )𝜆^𝑅_𝑁₃||

||+ 𝑖||

||(

𝜆 + (2𝜃 − 1) 𝜆^𝐼_𝑁₁+( 𝜃 −1

2 )𝜆^𝐼_𝑁₃)|

|||

≤||

||1 +𝜃(

𝜆^𝐼_𝑁₁+𝜆^𝐼_𝑁₂+𝜆^𝐼_𝑁₃)|

|||, (44) i.e.,

[

(2𝜃 − 1) 𝜆^𝑅_𝑁₁+( 𝜃 −1

2 )𝜆^𝑅_𝑁₃]²

+ [𝜆 + (2𝜃 − 1) 𝜆^𝐼_𝑁₁+(

𝜃 −1 2

)𝜆^𝐼_𝑁₃]2

≤[ 1 +𝜃(

𝜆^𝑅_𝑁₁+𝜆^𝑅_𝑁₂+𝜆^𝑅_𝑁₃)]2

+ [𝜃(

𝜆^𝐼_𝑁₁+𝜆^𝐼_𝑁₂+𝜆^𝐼_𝑁₃)]2

, (45) afteralgebraictreatment,theaboveconditionbecomes

2 (2𝜃 − 1)( 𝜃 −1

2

)𝜆^𝑅_𝑁₁𝜆^𝑅_𝑁₃+ 2 (2𝜃 − 1) 𝜆1𝜆^𝐼_𝑁₁+

(2𝜃 − 1)²𝜆^𝐼_𝑁₁𝜆^𝐼_𝑁₂+ (2𝜃 − 1) 𝜆1𝜆^𝐼_𝑁₃

−𝜃²(

2𝜆^𝑅_𝑁₁𝜆^𝑅_𝑁₂+ 2𝜆^𝑅_𝑁₂𝜆^𝑅_𝑁₃+ 2𝜆^𝑅_𝑁₃𝜆^𝑅_𝑁₁)

− 2𝜃(

𝜆^𝑅_𝑁₁+𝜆^𝑅_𝑁₂+𝜆^𝑅_𝑁₃)

− 𝜃²(

2𝜆^𝐼_𝑁₁𝜆^𝐼_𝑁₂+ 2𝜆^𝐼_𝑁₂𝜆^𝐼_𝑁₃+ 2𝜆^𝐼_𝑁₃𝜆^𝐼_𝑁₁) + ||𝜆¹||²

≤ 1 + (3𝜃 + 1) (1 − 𝜃) |||𝜆^𝑁¹|||

2+ 2𝜃²|||𝜆^𝑁²|||

2

+ (𝜃 −1

8 ) |||𝜆^𝑁³|||

2. (46) Notethatstabilityisassuredif𝜌(𝐺) ≤ 1.Thespectralradiusiscomputed byiterationmatrix,forvaluesofthetabetween0and1.The stability ofthemethodisassuredasFig.2showstheconditionoftheta0≤ 𝜃 ≤ 1 is nearlysuﬃcient. This condition mustholdforall eigen values of thecorresponding matricesfor themethod tobe stable. Noticethat if ²₅≤ 𝜃 ≤ 1, thelefthand sidemaybenegative,dependinguponthe choiceofeigen-values,whereastherighthandsideofEq. (46)isnon- negative.Whichconcludethatthecondition ²

5≤ 𝜃 ≤ 1 isnecessarybut

(6)

Table 1.NumericalSolutionofHSEbySincCollocationmethodforcase2.

x t=0 t=2 t=4 t=6 t=8 t=10 t=12

0.000000 0.000000 0.000000 0.000000 4.78E-18 -1.02E-18 2.26E-19 -3.96E-21 1.111111 0.222222 -0.03945 0.012182 0.004022 0.000678 0.000172 4.35E-05 2.222222 0.444444 0.088753 0.036620 0.009826 0.001588 0.000401 0.000101 3.333333 0.666667 0.096572 0.045222 0.012246 0.001982 0.000501 0.000126 4.444444 0.888889 0.197649 0.063955 0.016628 0.002665 0.000672 0.000169 5.555556 1.111111 0.172924 0.063622 0.016836 0.002711 0.000684 0.000172 6.666667 1.333333 0.278434 0.083603 0.021510 0.003439 0.000867 0.000218 7.777778 1.555556 0.208779 0.072398 0.019117 0.003077 0.000777 0.000196 8.888889 1.777778 0.355958 0.102456 0.026248 0.004192 0.001057 0.000266 10.00000 2.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

Fig. 2. Illustration of convergence dependence on𝜃 for HSE.

notsuﬃcient,forstabilityoftheSinccollocationmethodweinsert𝜃 = 1 inEq. (43)itgives

||||

||| 𝜆^𝑅_𝑁

1+(

1 2

)𝜆^𝑅_𝑁

3+𝑖( 𝜆1+𝜆^𝐼_𝑁

1+(

1 2

)𝜆^𝐼_𝑁

3

) 1 +(

𝜆^𝑅_𝑁₁+𝜆^𝑅_𝑁₂+𝜆^𝑅_𝑁₃) +𝑖(

𝜆^𝐼_𝑁₁+𝜆^𝐼_𝑁₂+𝜆^𝐼_𝑁₃)||

||||

|

≤ 1 , (47)

||||𝜆^𝑅^𝑁¹+1 2𝜆^𝑅_𝑁₃+𝑖(

𝜆1+𝜆^𝐼_𝑁₁+1 2𝜆^𝐼_𝑁₃)|

|||

≤||

||1 +(

𝜆^𝑅_𝑁₁+𝜆^𝑅_𝑁₂+𝜆^𝑅_𝑁₃) +𝑖(

𝜆^𝐼_𝑁₁+𝜆^𝐼_𝑁₂+𝜆^𝐼_𝑁₃)|

|||, (48) usingthetriangleinequality,theaboveequationbecomes

||||𝜆^𝑅^𝑁¹+1 2𝜆^𝑅_𝑁₃+𝑖(

𝜆1+𝜆^𝐼_𝑁₁+1 2𝜆^𝐼_𝑁₃)|

|||

≤ |||1 + 𝜆^𝑅^𝑁¹+𝑖𝜆^𝐼_𝑁₁|||+ |||𝜆^𝑅^𝑁²+𝑖𝜆^𝐼_𝑁₂|||+ |||𝜆^𝑅^𝑁³+𝑖𝜆^𝐼_𝑁₃||| , whichimpliesthat

0≤ |||1 + 𝜆^𝑅^𝑁¹+𝑖𝜆^𝐼_𝑁₁|||+ |||𝜆^𝑅^𝑁²+𝑖𝜆^𝐼_𝑁₂|||+ |||𝜆^𝑅^𝑁³+𝑖𝜆^𝐼_𝑁₃||| .

Fig.2expressesthesuﬃcientconditionfor0≤ 𝜃 ≤ 1.Eq. (43)istrue for allchoice ofeigenvalues. Thuswe obtain𝜃 = 1,is thesuﬃcient conditionfor stabilityof thetechnique.When𝜃 = 0 thetechniqueis restrictivelystablewiththelimitontimestep:

𝛿𝑡 ≤ 1 −Re²(

𝜆1

)−Im²( 𝜆1

)

1 4

(Re²( 𝜆_𝑁₃)

+Im²( 𝜆_𝑁₃))

+Re( 𝜆_𝑁₃𝜆_𝑁₁)

+Im( 𝜆_𝑁₁𝜆1

) . (49)

5. SolutionsforHunter-Saxtonequation

ThesolutionoftheHSEisobtainedbytheSCMwithconsidering diﬀerentvaluesof𝜃 alongwithcollocationpointsandnumberoftime steps,whereMatlabisusedtoreducethelaptime.Thegeneralstepsto obtain thesolutionoftheHSEforalltheproposedcasesareillustrated inAlgorithm1.

Algorithm1Thestepbystepdescriptionofthesolutionobtainingfor HSEthroughSinccollocationmethod.

1:procedure 1(Inputs)

2: Input1: Enterthecollocationpoints={𝑥_𝑚}_𝑚=0 3: Input2: Enterthenumberoftimesteps=𝑛_𝑡 4: Calculate:𝛿𝑥 and𝛿𝑡

5:procedure 2(DerivativesofSincfunction) 6: for𝑖= 1,...,𝑥_𝑚do

7: for𝑗 = 1,...,𝑥_𝑚do 8: if𝑖==𝑗 then 9: 𝑆⁽⁰⁾(𝑖,𝑗)= 1, 10: 𝑆⁽¹⁾(𝑖,𝑗)= 0, 11: 𝑆⁽²⁾(𝑖,𝑗)= −𝜋^2∕3.

12: else

13: 𝑆⁽⁰⁾(𝑖,𝑗)= 0, 14: 𝑆⁽¹⁾(𝑖,𝑗)=⁽⁻¹⁾_{ℎ(𝑗−𝑖)}^𝑗−𝑖, 15: 𝑆⁽²⁾(𝑖,𝑗)=⁻²⁽⁻¹⁾_ℎ2(𝑗−𝑖)^𝑗−𝑖².

16: end

17: end

18: end

19:procedure 3(Iteration) 20: for𝑘= 2,...,𝑛_𝑡do 21: for𝑖= 1,...,𝑥_𝑚do 22: 𝑣^𝑘+1(𝑖,𝑘)=𝐼⁰∗ (𝐺⁻¹𝐻)𝑣^𝑘

23: end

24: end

25: Output: NumericalsolutionoftheHSE

5.1. Case1

ThenumericalsolutionofHunterSaxtonequationispresentedbe- low,using thegraphical figures.The graphical figuresrepresent the wavepropagationin a massivenematic liquidcrystal directorfield.

InCase1, thesolution of HSEis presented.Thenumber ofcolloca- tionpointsanditerationnumbersaretaken10and20,respectively.

Fig.3representsthenumericalsolutionofHSEforvariousvaluesof𝜃,it representsthewavepropagationinamassivenematicliquidcrystaldi- rectorfield,forfourvaluesof𝜃,takentobe,𝜃 = 0.2,0.4,0.6 and𝜃 = 0.8, respectively.Fig.3representsthesolutionofHSEin3-dimension,elab- oratingthewavepropagation,furthermore,thesolutioncontainsthe singularitiesatvariouspointsof 𝑥,𝑡.TheFig.3a-baregraphicalrep- resentation of the 𝜃 = 0.2,0.4 and these are not converges because thestabilityoccursfor 𝜃 > 0.4 whereothertwosubfigures illustrates theconvergent solution of HSE. Approximation has doneby setting 𝛿𝑥=𝐿∕(𝑛𝑥− 1)and𝛿𝑡=𝑇𝑚𝑎𝑥∕(𝑛𝑡− 1)and𝜃 = 1.Fig.4expressesthe wavepropagationinamassivenematicliquidcrystaldirectorfieldat 𝜃 = 1 correspondingtodifferentvaluesofxandt.Fig.5representsthe solutionofHSEat𝑣(𝑡= 0).Fig.6expressesthesolutionofHSEforfour differentvaluesof𝑥,at𝑥= 0,𝑥=𝐿∕4,𝑥=𝐿∕2 and𝑥=𝐿,respectively.

At𝑥= 0,thesolutionofHSEis𝑣(𝑥,𝑡)= 0.At𝑥=𝐿∕4,𝑥=𝐿∕2,𝑥=𝐿, thesolutionisincreasingnearthevalues𝑡= 0.65.

5.2. Case2

InCase2,the graphicalsolution of HSEis presentedfor 𝜃 = 0.6, numberof collocation points anditeration numbersaretaken tobe 10and20,respectively.TheHSEpresentsthewavepropagationina

(7)

Fig. 3. Case 1 wave propagation of HSE for various values of𝜃.

Fig. 4. WavepropagationofHSEforcase1correspondingtovariousvaluesof xandt.

Fig. 5. Wave propagation of HSE for𝑣(𝑡 = 0) of case 1.

massivenematicliquidcrystaldirectorﬁeld.Fig.7expressesthewave propagationinamassivenematicliquidcrystaldirectorﬁeldat𝜃 = 0.6.

Fig.8representsthesolutionofHSEat𝑣(𝑥=𝐿∕2).Fig.9expressesthe solutionofHSEforfourdiﬀerentvaluesof𝑥,at𝑥= 0,𝑥=𝐿∕4,𝑥=𝐿∕2 and𝑥=𝐿,respectively.At𝑥= 0,thesolutionofHSEis𝑣(𝑥,𝑡)= 0.At𝑥= 𝐿∕4,𝑥=𝐿∕2,𝑥=𝐿,thebehavior ofthesolutionofHSEisdecreasing.

ThenumericalsolutionoftheHSEforthiscaseistabulatedinTable1 atdiﬀerenttimesrangesfrom0to12correspondingtovarying𝑥 values from0to10.

Theabsoluteerrors(AEs)oftheobtainedresultswithreferencesolu- tionwhere,referencesolutionisobtainedthroughAdomian’snumerical methodfortheaccuracyandeﬃciencyanalysisoftheproposedtech-

Fig. 6. Solution of HSE for diﬀerent values of𝑥 for case 1.

Fig. 7. WavepropagationofHSEforcase2correspondingtovariousvaluesof xandt.

niquewith𝜃 = 0.9.ThegraphicalillustrationofAEsisdescribedinthe Fig.10.

Table2presentstheabsoluteerrorofHSEwithAdomian’smethod.

A comparisonofproposedtechniqueispresentedinTable3withtheex- istingnumericaltechniques,Haarwaveletquasi-linearizationmethod [29], a collocation method [52] and Chebyshev functions (B-GFCF) collocationmethod[53].Itisconcludedthatourproposedtechnique providesthebetterresultsonthebasisofabsoluteerrors.

(8)

Table 2.AbsoluteErrorwithAdomian’smethod.

t

0.1 0.2 0.3 0.4 0.5

x

0.1 3.92E-01 4.18E-01 6.49E-01 7.11E-01 9.27E-01 0.2 1.80E-01 3.05E-01 4.87E-01 6.22E-01 8.01E-01 0.3 9.75E-02 2.42E-01 4.00E-01 5.47E-01 7.03E-01 0.4 4.47E-02 1.86E-01 3.29E-01 4.71E-01 6.14E-01 0.5 8.75E-04 1.32E-01 2.66E-01 3.99E-01 5.32E-01 0.6 4.34E-02 8.15E-02 2.07E-01 3.31E-01 4.57E-01 0.7 8.33E-02 3.43E-02 1.52E-01 2.70E-01 3.87E-01 0.8 1.20E-01 9.31E-03 1.02E-01 2.13E-01 3.24E-01 0.9 1.55E-01 4.93E-02 5.60E-02 1.61E-01 2.67E-01

1 1.86E-01 8.57E-02 1.43E-02 1.14E-01 2.14E-01

Table 3.ComparativestudyonthebasisofmaxAE.

t Ref. [29] for N=256 Ref. [52] for N=128 Ref. [53] for N=225 SCM for N=10

0.1 4.61 E-06 5.45E-09 7.75E-16 5.86E-18 nt=11

0.01 7.36 E-09 1.30E-07 3.83E-17 1.76E-19 nt=101

0.001 1.00E-11 6.05E-19 1.85E-19 nt=1001

Fig. 8. Case 2 solution of HSE for𝑥 = 𝐿∕2.

Fig. 9. Solutions of HSE at diﬀerent values of𝑥 for case 2.

6. Conclusion

Itisconcluded,onthebasisoftheresultsobtainedintheprevious section thattheproposedtechniquehasbeen provedtobe more re- liable,accurate,stable,andattractive.Initially,cardinalexpansionof Sincbasefunctionshasbeen usedtoapproximatethe spacedimension, thusitallowstheavoidanceoffinitedifferencegridfor2ndand3rdor- dernonlinearpartialdifferentialequations.𝜃-weightedfinitedifference schemehasbeenappliedtoapproximatetimederivative,whichreduces thecomplexityofequations.TheresultsshowedthatSCMsuccessfully

Fig. 10. Absolute errors of the HSE with𝜃 = 0.9.

tackledtheproposedproblem,inthepresenceofthesingularities.Sinc basesfunctionshavebeenusedtominimizethemathematicalcalcu- lations.Oneof thesoundadvantages hasgainedby Sinccollocation methodisabilitytocomposequiteaccurateresultswithMatlabcoding.

TheconvergenceoftheSinccollocationmethodisexaminedby running thealgorithmmanytimeswithsamecollocationpointsthroughMesh pointsthebehaviorisdetermined,mostoftimethediﬀerentmashpoint isprovided whereinothertechniquewecannotﬁndtheexactbehavior betweentheintegerdomainbutfromSinccollocationmethodwecalcu- late.Thestabilityoftheproposedequationisexaminedand graphically illustratedinFig.2thatthesolutionisstablefor𝜃 > 0.4.Thewaveprop- agationvariesasthevaluesof𝜃 correspondingtothe𝑥 and𝑡 values.

Moreover,thevariationofxvaluesalsoeffectsthe wavepropagation valuesasthexincreasestheslopofflowincreasesi-e.,theyhavedi- rectrelationforbothcases.Similarly,atspecificxvalueswithvariation intimevaluesthe solutionoftheHunterSaxtonequationthroughthe proposedmethoddecreases.

InFuture,suchequationswhichmayhaveinfinitedomainsordiffi- culttosolvebyothernumericaltechniquescanbesolvedbySinccol- locationmethod,quitesuccessfully,moreover,thistechniqueprovides moreaccurateandefficientsolutionsforthehigherorderequations[47, 48],alsocanappliedonthesystemofequations[49,50,51].

Declarations

Authorcontributionstatement

I.Ahmad,H.Ilyas,K.Kutlu,V.Anam,S.I.Hussain,J.L.C.Guirao:

Conceivedanddesignedtheexperiments;Performedtheexperiments;

Analyzed and interpreted the data; Contributed reagents, materials, analysistoolsordata;Wrotethepaper.