Analytical approximants of time-dependent partial differential equations with tau methods

Tau spectral methods and Adomian’s decomposition can be fruitfully combined to quickly approximate the analytical solution of any time-dependent partial differential equation with boundary conditions defined on the four sides of a rectangle. In this work, combinations of Legendre polynomials have been used to generate orthogonal two-dimensional polynomials on a rectangular domain. The time evolution of the solution is condensed in a set of nonlinear differential equations for the polynomial coefficients. This system can be integrated by using Adomian’s decomposition method with analytic extension or, alternatively, successive approximations, which generate a time series that can be truncated at the required precision order. The result is an analytic approximation to the final solution which can be easily obtained by using any commercial symbolic processor.