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A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations. (English) Zbl 1070.65115
Summary: The application of the method of fundamental solutions to the Cauchy problem associated with three-dimensional Helmholtz-type equations is investigated. The resulting system of linear algebraic equations is ill-conditioned and therefore its solution is regularized by employing the zeroth-order Tikhonov functional, while the choice of the regularization parameter is based on the \(L\)-curve method. Numerical results are presented for under-, equally- and over-determined Cauchy problems in a piecewise smooth geometry. The convergence, accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are analysed.

MSC:
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35R30 Inverse problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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