Numerical solution of singular integral equations via initial value methods

摘要

Quadrature methods for solving singular integral equations, especially with logarithmic kernels, have been known to break down completely; even if they succeed, the results are often inaccurate and the accuracy difficult to estimate. Product integration methods (to eliminate the singularity in the kernel function) can tackle the problem effectively if the required solution is a sufficiently smooth function, but at a prohibitive cost of computational time. The author proposes a different type of method for solving singular integral equations by reducing them into equivalent Fredholm integral equations (whenever possible) and then imbedding the latter in a class of equivalent initial value problems, which tend to be stable and can be solved in a routine manner. The validity of the method is also discussed.