Approximate Solutions of Systems of Singular Integral Equations

Using the properties of the related orthogonal polynomials, approximate solutions of systems of simultaneous singular integral equations are obtained, in which the essential features of the singularity of the unknown functions are preserved. In the system of integral equations of the first kind, the fundamental solution is the weight function of the Chebyshev polynomials of first or second kind. In the system of singular integral equations of the second kind with constant coefficients, the elements of the fundamental matrix are the weights of Jacobi polynomials. A direct method is introduced to obtain the fundamental matrix of the system. The approximate solution is then expressed as the fundamental function, representing the singular behavior of the unknown functions, multiplied by a series of proper orthogonal polynomials with unknown coefficients. The techniques of deriving the system of algebraic equations to determine these coefficients are described. In order to have an idea about the effectiveness of the method and the convergence problems arising from the truncation of the series, numerical results of an elastostatic problem are included.