Jacobi–Gauss–Lobatto collocation method for solving nonlinear reaction–diffusion equations subject to Dirichlet boundary conditions

• Published in: Applied mathematical modelling Vol. 40; no. 3; pp. 1703 - 1716
• Main Authors: Bhrawy, A.H., Doha, E.H., Abdelkawy, M.A., Van Gorder, R.A.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.02.2016
• Subjects: Collocation method; Jacobi–Gauss–Lobatto quadrature; Nonlinear reaction–diffusion equations; Nonlinear reaction–diffusion equations; Collocation method; Jacobi–Gauss–Lobatto quadrature
• ISSN: 0307-904X
• DOI: 10.1016/j.apm.2015.09.009

This paper extends the application of the spectral Jacobi–Gauss–Lobatto collocation (J-GL-C) method based on Gauss–Lobatto nodes to obtain semi-analytical solutions of nonlinear time-dependent reaction–diffusion equations (RDEs) subject to Dirichlet boundary conditions. This approach has the advantage of allowing us to obtain the solution in terms of the Jacobi parameters α and β, which therefore means that the method holds a number of collocation methods as a special case. In addition, the problem is reduced to the solution of system of ordinary differential equations (SODEs) in the time variable, which may then be solved by any standard numerical technique. We consider five applications of the general method to concrete examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving nonlinear time-dependent RDEs.