Efficient Spectral Collocation Algorithm for Solving Parabolic Inverse Problems

• Published in: International journal of computational methods Vol. 13; no. 6; p. 1650036
• Main Authors: Bhrawy, A. H., Abdelkawy, M. A.
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Company 01.12.2016; World Scientific Publishing Co. Pte., Ltd
• Subjects: Boundary conditions; Boundary value problems; Collocation; Differential equations; Inverse problems; Ordinary differential equations; Runge-Kutta method; pseudo-spectral scheme; Inverse problems; nonlinear parabolic partial differential equations; Gauss–Lobatto quadrature; systems of ordinary differential equations
• ISSN: 0219-8762; 1793-6969
• DOI: 10.1142/S0219876216500365

This paper reports a new Legendre–Gauss–Lobatto collocation (SL-GL-C) method to solve numerically two partial parabolic inverse problems subject to initial-boundary conditions. The problem is reformulated by eliminating the unknown functions using some special assumptions based on Legendre–Gauss—Lobatto quadrature rule. The SL-GL-C is utilized to solve nonclassical parabolic initial-boundary value problems. Accordingly, the inverse problem is reduced into a system of ordinary differential equations (ODEs) and afterwards, such system can be solved numerically using implicit Runge–Kutta (IRK) method of order four. Four examples are introduced to demonstrate the applicability, validity, effectiveness and stable approximations of the present method. Numerical results show the exponential convergence property and error characteristics of presented method.