预览加载中，请您耐心等待几秒...

Haar小波方法求解分数阶Poisson方程的数值解.docx

预览
1/2
2/2

在线预览结束，喜欢就下载吧，查找使用更方便

下载提示 文本预览

如果您无法下载资料，请参考说明：

1、部分资料下载需要金币，请确保您的账户上有足够的金币

2、已购买过的文档，再次下载不重复扣费

3、资料包下载后请先用软件解压，在使用对应软件打开

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