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On the numerical solution of nonlinear integral equations on non-rectangular domains utilizing thin plate spline collocation method. (English) Zbl 1450.65180

This paper proposes a computational scheme for solving a class of two-dimensional nonlinear Fredholm integral equations of the second kind. Specifically, a meshless method based on the thin plate spline radial basis function for discrete collocation is employed, which estimates the solution without any mesh refinements. The method is particularly attractive when the domain is non-rectangular, because thin plate splines only depend on the pairwise distances between points and not the domain geometry. The authors note that common numerical integration rules for computing the integrals in the scheme are not efficient in this context, due to the limited smoothness of thin plate splines. This obstacle is resolved by introducing a special numerical integration method based on the non-uniform Gauss-Legendre quadrature rule. An algorithm based on the new scheme is presented, along with the error analysis of the scheme. The numerical precision and convergence of the new approach are examined for several two-dimensional nonlinear integral equations on domains of different shapes.

MSC:

65R20 Numerical methods for integral equations
65D12 Numerical radial basis function approximation
45B05 Fredholm integral equations
41A25 Rate of convergence, degree of approximation
41A63 Multidimensional problems
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