Regularized MFS-Based Boundary Identification in Two-Dimensional Helmholtz-Type Equations

摘要: Abstract: We study the stable numerical identification of an unknown portionof boundary on which a given condition is provided and additionalCauchy data are remaining known portion two-dimensional domain for problems governed by either Helmholtz or modifiedHelmholtz equation. This inverse geometric problem solved using methodof fundamental solutions (MFS) in conjunction with Tikhonov regularizationmethod. The optimal value regularization parameter chosen according toHansen’s L-curve criterion. stability, convergence, accuracy efficiency ofthe proposed method investigated considering several examples.Keywords: Helmholtz-Type Equations; Inverse Geometric Problem; Method ofFundamental Solutions (MFS); Regularization.1 IntroductionIn direct mechanics, goal to determine response systemwhen governing system partial differential equations, initial bound-ary conditions primary and/or secondary fields, material properties andthe geometry occupied under investigation allknown. existence uniqueness solution such have beenwell established. If, however, at least one above partially entirelymissing, this yields problem. It well that ingeneral unstable, sense small measurement errors input maysignificantly amplify solution, see e.g. Hadamard (1923).An important class mechanics represented ge-ometric can be divided into following categories: (i) shape