A SEMI-ANALYTIC SPECTRAL METHOD FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

In this article we present a semi-analytic method for solving elliptic partial differential equations. The technique is based on using a spectral method approximation for the second-order derivative in one of the spatial directions followed by solving the resulting system of second-order differential equations by an analytic method. That is, the system of second-order two-point boundary-value problems are solved analytically by casting them in first-order form and solving the resulting set of first-order equations by using the matrix exponential. An important aspect of our technique is that the solution obtained is semi-analytic, e.i., using analytic method in y and spectral method in x. The new method can be used for both linear and non-linear boundary conditions as well as for nonlinear elliptic partial differential equations.