Haar小波方法求解分数阶Poisson方程的数值解-豆柴文库

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Haar小波方法求解分数阶Poisson方程的数值解
ThefractionalorderPoissonequationisageneralizationoftheclassicalPoissonequationtoincludefractionalderivatives.Ithasbeenstudiedextensivelyinrecentyearsduetoitspotentialapplicationsinvariousfieldssuchassignalprocessing,imagedenoising,andbiomedicalengineering.Inthispaper,weaimtosolvethefractionalorderPoissonequationusingtheHaarwaveletmethod.
ThefractionalorderPoissonequationisdefinedas:
(-1)^m∂^(2m)u(x)+(-1)^nD^(2n)u(x)=f(x),x∈Ω
u(x)=g(x),x∈∂Ω
whereΩrepresentsthedomainoftheproblem,u(x)istheunknownfunctiontobedetermined,f(x)isagivensourceterm,andg(x)isagivenboundarycondition.Theoperators∂^(k)andD^(m)representtheclassicalderivativeoforderkandthefractionalderivativeoforderm,respectively.
ToapproximatethesolutionofthefractionalorderPoissonequation,weutilizetheHaarwaveletmethod.TheHaarwaveletisapiecewiseconstantfunctiondefinedonacompactsupportinterval.Ithastheadvantageofbeingnumericallyefficientandeasytoimplement.
First,weexpandtheunknownfunctionu(x)andthesourcetermf(x)intermsoftheHaarwaveletbasisfunctions:
u(x)=∑(j,k)u(j,k)φ(j,k)(x)
f(x)=∑(j,k)f(j,k)φ(j,k)(x)
whereu(j,k)andf(j,k)aretheexpansioncoefficients,andφ(j,k)(x)representstheHaarwaveletbasisfunctionatleveljandpositionk.
SubstitutingtheaboveexpansionsintothefractionalorderPoissonequation,weobtainthefollowingsystemofequations:
∑(j',k')(2^(j'α)Δ(2^j'x-k')u(j',k'))φ(j,k)(x)+
∑(j',k')(2^(j'β)Δ(2^j'x-k')(-1)^(α-β)
∑(j',k')(2^(j'β)Δ(2^j'x-k')u(j',k')
=∑(j',k')f(j',k')φ(j,k)(x)forallφ(j,k)(x)
whereαandβarethefractionalordersofthederivatives,andΔ()istheKroneckerdeltafunction.
Tosolvethesystemofequations,wetruncatetheinfinitesumandretainonlyafinitenumberofbasisfunctions.Theresultingsystemofequationscanbewritteninmatrixformas:
Au=f
whereAisasquarematrixformedbythecoefficientsofthebasisfunctions,uisthevectorofexpansioncoefficients,andfisthevectorofexpansioncoefficientsofthesourceterm.
WecansolvethismatrixequationnumericallyusingvarioustechniquessuchasGaussianeliminationoriterativemethods.Thesemethodsca