第49卷第4期2023年8月文章编号：1673-5196(2023)04-0151-06带Neumann边界条件的Helmholtz方程柯西问题的一种新的正则化方法余亚辉*1，李振平1.2（1.洛阳理工学院数学与物理教学部，河南洛阳471023；2.西北师范大学数学与统计学院，甘肃兰州730070）摘要：考虑矩形域上带Neumann边界条件的Helmholtz方程的柯西问题，该问题是一类严重不适定的偏微分方程反问题，即它的解不连续依赖于输入数据.基于经典的Tikhonov正则化方法利用自设计过滤化子修改核函数的思想，提出一种新的正则化求解方法，给出该问题基于分离变量的近似解，对正则化参数的先验和后验两种选取规则下精确解与近似解进行误差分析，得到满足收敛性和稳定性的Holder型误差估计.关键词：Helmholtz方程柯西问题；Neumann边界条件；不适定问题；正则化；后验参数选取中图分类号：0175AnewregularizationmethodfortheCauchyproblemoftheHelmholtz(1.DepartmentofMathematicsandPhysics,LuoyangInstituteofScienceandTechnology,Luoyang471023,China;2.CollegeofMathe-maticsandStatistics,NorthwestNormalUniversity,Lanzhou730070,China)Abstract:ACauchyproblemfortheHelmholtzequationwithNeumannboundaryconditionsinarectan-gledomainisdiscussedinthispaper.Thisproblemisaseriousill-posedinverseproblemofpartialdiffer-entialequations,thatis,thesolutiondoesnotdependcontinuouslyontheinputdata.Basedontheclassi-calTikhonovregularizationmethodusingtheideaofmodifyingthekernelfunctionbyself-designedfilterssubsets,anewregularizationmethodispresentedforapproximatingthesolutiontothisproblembasedonthemethodofseparatingvariables.Afterperformingerroranalysisontheexactandapproximatesolutionsunderbothaprioriandaposterioriselectionrulesoftheregularizationparameters,theHolder-typeerrorestimatessatisfyingconvergenceandstabilityareobtained.Keywords:CauchyproblemforHelmholtzequation;Neumannboundarycondition;ill-posedproblem;regularization;aposterioriparameterchoicerule本文考虑矩形域上带Neumann边界条件的Helmholtz方程的柯西问题[1[u(α,y)+k²u(α,y)=00
