Abstract
An efficient Legendre-Galerkin spectral method and its error analysis for a one-dimensional parabolic equation with Dirichlet-type non-local boundary conditions are presented in this paper. The spatial discretization is based on Galerkin formulation and the Legendre orthogonal polynomials, while the time derivative is discretized by using the symmetric Euler finite difference schema. The stability and convergence of the semi-discrete spectral approximation are rigorously set up by following a novel approach to overcome difficulties caused by the non-locality of the boundary condition. Several numerical tests are included to confirm the efficacy of the proposed method and to support the theoretical results.
Highlights
Initial boundary value problems for partial differential equations with non-local boundary conditions NLBCs have been extensively investigated in many papers and textbooks
An efficient Legendre-Galerkin spectral method and its error analysis for a one-dimensional parabolic equation with Dirichlet-type non-local boundary conditions are presented in this paper
This section is devoted to set up the convergence and stability results for semidiscrete Legendre-Galerkin spectral method, we aim to show that the proposed method enjoys the spectral accuracy
Summary
Initial boundary value problems for partial differential equations with non-local boundary conditions NLBCs have been extensively investigated in many papers and textbooks. This paper is devoted to develop and analyse the implementation of Legendre-Galerkin spectral method to non-local boundary value problem for a linear parabolic equation. There is no research available in the literature devoted to analysis the implementation of spectral methods to nonlocal boundary value problems for parabolic equations. Motivated by this fact, we mainly aim in this paper to provide a suitable approach to solve the one-dimensional parabolic equation (1.1) subject to non-local boundary conditions (1.2) by a spectral method with efficient implementation and exponential rate of convergence as in the spectral methods for problems with classical boundary conditions. We conclude the paper with some remarks on the main features of the method presented in previous sections and highlight some possible extensions of our method
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