A semi-analytical numerical method for solving evolution and elliptic partial differential equations

A new method for analyzing initial–boundary value problems for linear and integrable nonlinear partial differential equations (PDEs) has been introduced by one of the authors.For linear PDEs this method yields analytical solutions for certain problems that apparently cannot be solved by classical methods such as Green’s function representations, classical transforms and the method of images. Even for problems that can be solved by classical methods, the new method has advantages such as avoiding the solution of ordinary differential equations that result from the classical transforms, as well as constructing integral solutions in the complex plane which converge exponentially fast and which are uniformly convergent at the boundaries.Here we review the numerical implementation of the new method to evolution and elliptic PDEs.