SINGULARITY EXPANSIONS FOR THE SOLUTIONS OF SECOND KIND FREDHOLM INTEGRAL EQUATIONS WITH WEAKLY SINGULAR CONVOLUTION KERNELS

Considered are Fredholm integral equations of the second kind, y equals f plus lambda Ky, defined on some interval left bracket a, b right bracket , where f is a given function, lambda is a given scalar, K is an integral operator with weakly singular convolution kernel k, and y is the solution. By examining the theoretical properties of K in certain Banach spaces, it is shown that, provided f is an element of L//1 left bracket a, b right bracket , y may be expressed as a linear combination (of arbitrary length) of singular terms plus an unknown smoother function. Such an expansion is called ″singularity expansion″ . The singular terms are integrals, which in most practical cases may be evaluated explicitly in terms of algebraic functions. Four examples are given in which the method is applied to equations arising in practice.