Abstract
The Galerkin–Chebyshev matrix is the coefficient matrix for the Galerkin method (or the degenerate kernel approximation method) using Chebyshev polynomials. Each entry of the matrix is defined by a double integral. For convolution kernels K(x-y) on finite intervals, this paper obtains a general recursion relation connecting the matrix entries. This relation provides a fast generation of the Galerkin–Chebyshev matrix by reducing the construction of a matrix of order N from N 2+O(N) double integral evaluations to 3N+O(1) evaluations. For the special cases (a) K(x-y)=|x-y|α-1(-ln|x-y|)p and (b) K(x-y)=K ν(σ|x-y|) (modified Bessel functions), the number of double integral evaluations to generate a Galerkin–Chebyshev matrix of arbitrary order can be further reduced to 2p+2 double integral evaluations in case (a) and to 8 double integral evaluations in case (b).
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Cope, D.K. Recursive generation of the Galerkin–Chebyshev matrix for convolution kernels. Advances in Computational Mathematics 9, 21–35 (1998). https://doi.org/10.1023/A:1018933305810
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DOI: https://doi.org/10.1023/A:1018933305810