Solving singular integral equations with orthogonal rational functions

Overview & Synopsis

We suggest a new numerical method of solving the signed equilibrium with external field in logarithmic potential theory on a union of distinct real intervals. A reformulation of our problem leads us to a system of integral equations with a weakly singular Cauchy kernel. We then recall a polynomial spectral method and its error analysis, and suggest a new spectral method using orthogonal rational functions in order to solve our problem. Choosing appropriate and explicitly given poles allows to speed up computation. Such techniques have been exploited also in the famous adaptative multipole algorithm for particle simulations [8] or more recently for solving the Laplace equation [28].