Solving Cauchy integral equations of the first kind by the Adomian decomposition method

Abstract

A new approach is presented to resolve Cauchy integral equations of the first kind in the general case by first considering a regularized integral equation and then transforming it into a canonical form suitable for applying the Adomian decomposition method (ADM). We obtain a decomposition solution φϵ of the regularized integral equation and prove the convergence of our new combined method. As the regularization parameter ϵ→1, the obtained solution is shown to be a sufficiently good approximate solution for the particular Cauchy integral equation. The proposed method has been tested for a variety of Cauchy integral equations, which are particularly important in engineering applications, e.g. airfoil design, etc.