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On unsymmetric collocation by radial basis functions. (English) Zbl 1026.65107
Summary: Solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by E. J. Kansa [Comput. Math. Appl. 19, No. 8/9, 127-145 (1990; Zbl 0692.76003); ibid. 19, No. 8/9, 147-161 (1990; Zbl 0850.76048)]. It rewrites the problem as a generalized interpolation problem, and the solution is obtained by solving a (possibly large) linear system. The method has been used successfully in a variety of applications, but a proof of nonsingularity of the linear system was still missing. This paper shows that a general proof of this fact is impossible. However, numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort.

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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