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谱-Galerkin方法求解分数阶偏积分微分方程(英文).docx

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谱-Galerkin方法求解分数阶偏积分微分方程(英文) Title:Spectral-GalerkinMethodforSolvingFractionalPartialIntegral-DifferentialEquations Abstract: Fractionalcalculushasgainedsignificantattentioninrecentyearsduetoitsabilitytomodelcomplexsystemsmoreaccurately.Fractionalpartialintegral-differentialequations(FPIDEs)aregeneralizationsofclassicalPDEsthatinvolvebothfractionalderivativesandintegrals.Inthispaper,wepresenttheSpectral-GalerkinMethod(SGM)asanefficienttoolfornumericallysolvingFPIDEs.TheSGMcombinestheGalerkinframeworkwithbasisfunctionsdefinedonaspectraldomain,providingareliableandaccurateapproachtohandlingthecomplexityoffractionalcalculus. 1.Introduction: Thestudyoffractionalcalculushasexpandedtherealmofmathematicalmodelingandanalysis,leadingtomoreaccuratedescriptionsofphysicalphenomenathatclassicalcalculusfailstocapture.FPIDEsariseinvariousareassuchasphysics,engineering,finance,andbiology,makingtheirnumericalsolutionasubjectofconsiderableinterest.TraditionalnumericalmethodsoftenstruggletoefficientlysolveFPIDEsduetothenon-localcharacteristicsintroducedbyfractionalderivativesandintegrals.TheSpectral-GalerkinMethodoffersapromisingalternativebyleveragingthepropertiesofspectralapproximationsandGalerkinframework. 2.TheoreticalBackground: Thissectiondiscussesthefundamentalconceptsoffractionalcalculus,includingRiemann-LiouvilleandCaputofractionalderivatives,andfractionalintegrals.WealsopresentthebasicpropertiesandbehaviorsofFPIDEs,highlightingthechallengestheyposefortraditionalnumericalmethods. 3.SpectralApproximation: TheSpectral-GalerkinMethodreliesonapproximatingthesolutionsofFPIDEsusingbasisfunctionsdefinedonaspectraldomain.Wediscussthechoiceofspectralbasisfunctionsandtheirproperties,suchasorthogonalpolynomials(e.g.,Fourierseries,Legendrepolynomials)orwavelets,andtheirabilitytoefficientlyhandlenon-localityandsingularityproblemstypicallyencounteredinFPIDEs. 4.GalerkinFramework: TheGalerkinframeworkisawidelyusedtechniqueforthenumericalsolutionofPDEs,anditformsthefoundationoftheSpectral-Ga