Thesis (M.Sc.) - Cairo University - Faculty of Science - Department of Mathematics

Many problems in science and engineering are given in unbounded domains. In order to solve them numerically, the domain is replaced by a finite one to facilitate the numerical computations. Appropriate boundary conditions on the "artificial" boundaries is considered, and then a spectral method is employed to approximate the reduced problem.The main disadvantage of this approach is that it can be used only for problems with rapidly decaying solutions or when accurate non-reflecting or exact boundary conditions are available at the artificial boundaries, but it is difficult to obtain the exact boundary conditions. Accordingly, some additional numerical errors occur. Our first aim in this thesis is to remedy this deficiency in spectral methods by taking classical Jacobi polynomials as bases/test functions with an exponential mapping. Specifically, we develop spectral Galerkin and collocation methods based on the exponential Jacobi functions for solving linear and nonlinear partial differential equations in unbounded domains, respectively. We also establish some basic results on the exponential Jacobi orthogonal approximations in the nonuniformly weighted Sobolev space. Moreover, we derive the exponential Lagrange interpolation formula and its related error estimates.These results serve as the mathematical foundation of spectral methods for various partial differential equations in unbounded domains

Issued also as CD