Abstract
The performance of the parameter-free conical radial basis functions accompanied with the Chebyshev node generation is investigated for the solution of boundary value problems. In contrast to the traditional conical radial basis function method, where the collocation points are placed uniformly or quasi-uniformly in the physical domain of the boundary value problems in question, we consider three different Chebyshev-type schemes to generate the collocation points. This simple scheme improves accuracy of the method with no additional computational cost. Several numerical experiments are given to show the validity of the newly proposed method.
Highlights
E source points and the collocation points, usually uniformly located in the physical domain, are taken to be the same in the traditional RBF approach
E main differences among these three newly proposed schemes lie in the position of the collocation points. e figures of the three schemes will be shown in the following numerical section to verify the performance of the proposed schemes
We can see that the relative average errors of the CGR-RBF, CG-RBF, and CGL-RBF are superior than the tradition Conical Radial Basis Functions (CRBF) case with two decimals for m 5 and m 7
Summary
We consider the boundary value problems (BVPs) for elliptic partial differential equation (PDE) of second order: z2u z2u zu zu a(P) zx2 + b(P) zy2 + c(P) zx + d(P) zy [1]. E source points and the collocation points, usually uniformly located in the physical domain, are taken to be the same in the traditional RBF approach. We consider the source points generation in a new way. 3. The Chebyshev-Type Schemes e main idea is to use the Chebyshev-type schemes, which is generated in the interval (−1, 1), instead of the traditional uniformly distributed source points. It should be noted that the Chebyshev collocation method is usually used to find the approximate solutions of differential equations using a truncated Chebyshev series [19, 20]. E novelty of the idea in this paper lies in that the points generated by the Chebyshev-type schemes are combined with the CRBF, while the computational cost remains the same as the traditional way and there is no need to consider the fictitious points. Is transformation can be extended to the following two schemes
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