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The method of fundamental solutions for non-linear thermal explosions. (English) Zbl 0839.65143
On the background of a spontaneous ignition problem, the author considers a technique for solving a nonlinear Poisson equation in general two-dimensional domains. This consists of a Picard iteration to handle the nonlinearity, of a splitting of the remaining linear Poisson equation into an inhomogeneous part (which is solved disregarding boundary conditions by interpolation with thin plate splines) and a Laplace equation for the correction of the boundary values (which is approximated by collocating a sum of fundamental solutions).
This approach requires the solution of two medium-size dense linear systems (for fixed interpolation points in the interior and fixed collocation points on the boundary), is said to be faster and more accurate than the boundary element method and the finite element method (FEM). Results of 3 tests are displaced confirming that accuracy is comparable with that of FEM (however, missing are CPU times; several hundred iterations are reported to have been performed).

MSC:
65Z05 Applications to the sciences
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35Q72 Other PDE from mechanics (MSC2000)
80A25 Combustion
35J65 Nonlinear boundary value problems for linear elliptic equations
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