修正的Helmholtz方程柯西问题的四阶修正法
本论文在半无界区域上考虑修正的Helmholtz方程柯西问题。x=0处的柯西数据已经给出,要求0
www.1mpi.com
Fourth-orderModi edMethodForthe
CauchyproblemoftheModi ed
HelmholtzEquation
R.Shi,T.Wei andH.H.Qin
SchoolofMathematicsandStatistics,LanzhouUniversity,
Lanzhou,730000,P.R.China
April15,2008
Abstract—Inthispaper,theCauchyproblemforthemodi edHelmholtz
equationinanin nitestripplane0<x≤1,y∈Risconsidered.TheCauchydataontheendx=0isgivenandthesolutionisthensoughtfortheinterval0<x≤1.Thisproblemishighlyill-posedandthesolution(ifitexists)doesnotdependcontinuouslyonthegivendata.Inthispaper,weproposeafourth-ordermodi edmethodtosolvetheCauchyproblem.Convergenceestimatesarepresentedunderthesuitablechoicesofregularizationparametersandtheaprioriassumptionontheboundsoftheexactsolution.Numericalimplementationisconsideredandasimpleexampleisgiven.
Keywords—Cauchyproblemforthemodi edHelmholtzequation,Ill-posed
problem,Fourth-ordermodi edmethod
1.Introduction
Themodi edHelmholtzequationarisesinmanyareas,especiallyinpracticalphysicalapplications,suchasimplicitmarchingschemesfortheheatequation,Debye-HuckeltheoryandthelinearizationofthePoisson-Boltzmannequationandsoon,see[1,2,5,8,9].Thedirectproblems,i.e.Dirichlet,Neumannormixedboundaryvalueproblemsforthemodi edHelmholtzequationhavebeenstudiedextensivelyinthepastcentury.However,insomepracticalproblems,theboundarydataonthewholeboundarycannotbeobtained.Weonlyknow
The
workdescribedinthispaperwassupportedbytheNSFofChina(10571079,
10671085)
Correspondingauthor.
1
