一類橢圓型偏微分方程反問題的無網(wǎng)格方法.pdf

1、70Sj7!洳≥：≥～碩士學(xué)位⑧論文論文題目：一類橢圓型偏微分方程反問題的無網(wǎng)格方法姓名金邦梯學(xué)科(郇Ip)計算數(shù)學(xué)指導(dǎo)教師鄭耀教授所在學(xué)院理學(xué)院提交日期2005年3月摘要MeshlessMethodsforSomeInverseProblemsAssoeiatedwithEllipticPartialDifferentialEquationsAbstractInverseproblemsariseinmanyengineeringan

2、dscientificcontexts，andtheyarenowplayingacentralrolethereInpracticalapplications，theproblemunderinvestigationisoftengovernedbypartialdifferentialequationsHencethesolutionofpracticalinverseproblemsoftenleadstosolvinginver

3、seproblemsassociatedwithpartialdifferentialequationsThisthesisfocusesontheclassofinverseproblemsthatcanbeformulatedasCauchyproblemswithellipticpartialdifferentialequationswhicharenotoriousfortheir川posednessInparticularwe

4、willinvestigatetheCauchyproblemassociatedwiththeLaplaceequationforthesteady—statebeatconduction，nondestructivetesting洲D(zhuǎn)T)andgeophysics，theHelmholtzequationsforacousticanalysis，electromagneticfield，vibrationofmembranesand

5、otherstructures，andheatconductioninafin，andtheNaviersysteminlinearelasticityInthethesis，severalnovelnumericalschemesbasedonmeshlessmethodsaleproposedforthesolutionoftheinverseproblemsassociatedwithellipticpartialdifferen

6、tialequationsThesemeshlessmethodscanbeformulatedinthegeneralframeworkofradialbasisfunctioncollocationtechniquesPreliminarynumericalresultsfortheLaplaceequationandNaviersystemarepresented，andtrickyissuesassociatedwiththes

7、emethods，suchastheill—conditioningwiththeinterpolationmatrixandeffectofshapeparameterareinvestigatedItisfoundthatusingregularizationmethods，suchasthetruncatedsingularvaluedecompositionandTitdaonovregularization，tosolveth

8、eresultingsystemoflinearequationsnotonlymitigatesthe川conditioningwiththematrixequation，butalsoalleviatesthedifficultywithdetermininganappropriateshapeparameterintheradialbasisfunctionsItisconcludedthattheradialbasisfunct

9、ioncollocationtechniquescoupledwithregularizationmethodscouldbecompetitivealternativestoexistingmethodsfortheseproblems[[owever，forspecificequations，suchastheLaplaceequation，theHelmholtzequationandtheNaviersystem，wecanim

10、provethecomputationalefficiencysignificantlybyemployingdifferentialoperatorgearedradialbasisfunctions、Tobemorespecific，wemayemploythefundamentalandgeneralsolutionstothedifferentialoperatorsastheradialbasisfunctions，which