A simplified procedure for singular integral equations with logarithmic kernels

An exact construction of solutions to a class of singular integral equations given by Morland (1970) is approximated to reduce computer storage and time. Applications are to kernels which are the sum of a continuous bounded function and a logarithm multiplied by a second continuous bounded function, with the possible addition of a strong singularity. The exact procedure applies when both these functions are polynomials, but the use of economized Chebyshev series allows a variety of kernels to be represented accurately in this form. Here the further use of a truncated Chebyshev series during the subsequent construction leads to simple explicit algebraic expressions for the matrix of coefficients in the final system of linear equations. Numerical solutions are compared with known analytical solutions to a simple class of these equations, at different values of the parameters involved, and striking accuracy is obtained.