Numerical study of boundary problems for partial differential equations

Abstract

The aim of this work is to study various problems of mathematical equations using spectral methods. It develops four numerical techniques suitable for every studied problem and shows efficiency throughout different numerical illustrations. This study proposes a Legendre Galerkin method coupled with finite differences technique for the advection-diffusion equation with perturbed Robin boundary conditions. The obtained results provide two ways to confirm the efficiency: firstly, by calculating the error of approximation, and simultaneously by comparing the obtained approximate solution to the exact solution of the problem with Dirichlet boundary conditions. For the same equation, a second scheme is proposed using the spectral Galerkin method for both temporal and spatial discretizations. On the same axis, a transition to integral/integro-differential equations is introduced. A novel way of writing the basis functions as compact combinations of orthogonal polynomials using the set of initial conditions is elaborated in a Galerkin method for the integral and integro-differential equations, depending on the order of derivation. Additionally, some numerical techniques, such as using Gauss types quadrature, are also investigated for more accuracy. The last set of results pertains to the study of integro-differential equations of fractional order. Some interestingestimations are formulated to approximate the solution using orthogonal polynomials in a collocation method. All the presented techniques are supported by numerical examples that cover a vast range of cases, to demonstrate the efficiency of the proposed algorithms