{"title":"Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation","authors":"B. Barnes, Anthony Y. Aidoo, J. Ackora-Prah","doi":"10.1155/2022/4628634","DOIUrl":null,"url":null,"abstract":"The ill-posed Helmholtz equation with inhomogeneous boundary deflection in a Hilbert space is regularized using the divergence regularization method (DRM). The DRM includes a positive integer scaler that homogenizes the inhomogeneous boundary deflection in the Helmholtz equation’s Cauchy issue. This guarantees the existence and uniqueness of the equation’s solution. To reestablish the stability of the regularized Helmholtz equation and regularized Cauchy boundary conditions, the DRM uses its regularization term \n \n \n \n 1\n +\n \n \n α\n \n \n 2\n m\n \n \n \n \n \n \n e\n \n \n m\n \n \n \n , where \n \n α\n >\n 0\n \n is the regularization parameter. As a result, DRM restores all three Hadamard requirements for well-posedness.","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":"9 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Abstract and Applied Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2022/4628634","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 1
Abstract
The ill-posed Helmholtz equation with inhomogeneous boundary deflection in a Hilbert space is regularized using the divergence regularization method (DRM). The DRM includes a positive integer scaler that homogenizes the inhomogeneous boundary deflection in the Helmholtz equation’s Cauchy issue. This guarantees the existence and uniqueness of the equation’s solution. To reestablish the stability of the regularized Helmholtz equation and regularized Cauchy boundary conditions, the DRM uses its regularization term
1
+
α
2
m
e
m
, where
α
>
0
is the regularization parameter. As a result, DRM restores all three Hadamard requirements for well-posedness.
利用散度正则化方法对Hilbert空间中具有非均匀边界偏转的不适定Helmholtz方程进行了正则化。DRM包括一个正整数定标器，用于均匀化亥姆霍兹方程柯西问题中的非均匀边界偏转。这保证了方程解的存在性和唯一性。为了重新建立正则化亥姆霍兹方程和正则化柯西边界条件的稳定性，DRM使用其正则化项1+α2 m e m，其中α>0是正则化参数。因此，DRM恢复了Hadamard对适定性的所有三个要求。
期刊介绍：
Abstract and Applied Analysis is a mathematical journal devoted exclusively to the publication of high-quality research papers in the fields of abstract and applied analysis. Emphasis is placed on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimization theory, and control theory. Abstract and Applied Analysis supports the publication of original material involving the complete solution of significant problems in the above disciplines. Abstract and Applied Analysis also encourages the publication of timely and thorough survey articles on current trends in the theory and applications of analysis.
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