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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.

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