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The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation. (English) Zbl 1088.35079

Summary: The application of the method of fundamental solutions to inverse problems associated with the two-dimensional biharmonic equation is investigated. The resulting system of linear algebraic equations is ill-conditioned and, therefore, its solution is regularized by employing the 0th-order Tikhonov functional, while the choice of the regularization parameter is based on the \(L\)-curve method. Numerical results are presented for both smooth and piecewise smooth geometries. The convergence and the 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:

35R25 Ill-posed problems for PDEs
35J40 Boundary value problems for higher-order elliptic equations
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
35R30 Inverse problems for PDEs
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35A08 Fundamental solutions to PDEs
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