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IMA Journal of Applied Mathematics 1979 23(2):139-166; doi:10.1093/imamat/23.2.139
© 1979 by Institute of Mathematics and its Applications
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A Fast Galerkin Algorithm for Singular Integral Equations

L. M. DELVES, L. F. ABD-ELAL and J. A. HENDRY

Department of Computational and Statistical Science, University of Liverpool
Department of Mathematics, University of Cairo
Computer Centre, University of Birmingham

A recent paper (Delves, 1977) described a variant of the Galerkin method for linear Fredholm integral equations of the second kind with smooth kernels, for which the total solution time using N expansion functions is {theta}(N2 ln N) compared with the standard Galerkin count of {theta}(N3). We describe here a modification of this method which retains this operations count and which is applicable to weakly singular Fredholm equations of the form


where K0(x, y) is a smooth kernel and Q contains a known singularity. Particular cases treated in detail include Fredholm equations with Green's function kernels, or with kernels having logarithmic singularities; and linear Volterra equations with either regular kernels or of Abel type. The case when g(x) and/or f(x) contains a known singularity is also treated. The method described yields both a priori and a posteriori error estimates which are cheap to compute; for smooth kernels (Q = 1) it yields a modified form of the algorithm described in Delves (1977) with the advantage that the iterative scheme required to solve the equations in {theta}(N2) operations is rather simpler than that given there.


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