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非线性Fredholm-Volterra积分方程的Legendre谱配置法研究.docx

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非线性Fredholm-Volterra积分方程的Legendre谱配置法研究 Title:StudyingtheLegendreSpectralCollocationMethodforNonlinearFredholm-VolterraIntegralEquations Abstract: TheFredholm-Volterraintegralequationsplayafundamentalroleinvariousareasofscienceandengineering,astheydescribetherelationbetweenanunknownfunctionanditsintegraloveraspecificinterval.However,solvingsuchequationsanalyticallyisoftenchallenging,especiallywhentheequationsarenonlinear.Totacklethischallenge,variousnumericalmethodshavebeendeveloped.Inthispaper,wefocusontheapplicationoftheLegendrespectralcollocationmethodforsolvingnonlinearFredholm-Volterraintegralequations.Wediscussthetheoreticalbackgroundofthemethod,itsimplementationsteps,andpresentnumericalexperimentstodemonstrateitsefficiencyandaccuracy. 1.Introduction: Fredholm-Volterraintegralequationsareessentialinnumerousphysicalphenomena,suchasfractionalcalculus,electricalcircuits,andfluidmechanics.Theseequationsprovideapowerfultoolforstudyingintegraloperators'uniquepropertiesandtheirsolutions.However,whendealingwithnonlinearintegralequations,traditionalanalyticalmethodsbecomeinadequate.Therefore,thedevelopmentofefficientnumericalmethodsisnecessary. 2.TheLegendreSpectralCollocationMethod: TheLegendrespectralcollocationmethodisanaccurateandefficientnumericaltechniqueforsolvingintegralequations.ItutilizestheLegendre-Gauss-Lobatto(LGL)pointsascollocationpoints.TheLGLpointsarechosentoensuretheaccuracyofthemethodbysolvingtheintegralequationsatspecificnodes.Byapproximatingtheunknownfunctionusingatruncatedseriesexpansion,asystemofalgebraicequationsisobtained,whichcanbesolvedusingalinearornonlinearsolver.Thismethodensureshighaccuracyandfastconvergence. 3.ImplementationSteps: TheimplementationoftheLegendrespectralcollocationmethodfornonlinearFredholm-Volterraintegralequationsinvolvesthefollowingsteps: a.Transformingthenonlinearintegralequationintoalinearformusingsuitabletransformationtechniques. b.Selectingappropriatebasisfunctionsforconstructingthetruncatedseriesexpansion. c.Choosi