A direct numerical method for approximate solution of inverse reaction diffusion equation via two-dimensional Legendre hybrid functions

• Published in: Numerical algorithms Vol. 83; no. 2; pp. 511 - 528
• Main Authors: Gholampoor, I., Kajani, M. Tavassoli
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2020; Springer Nature B.V
• Subjects: Algebra; Algorithms; Boundary conditions; Computer Science; Functions (mathematics); Inverse problems; Numeric Computing; Numerical Analysis; Numerical methods; Original Paper; Partial differential equations; Polynomials; Reaction-diffusion equations; Regularization; Regularization methods; Spectral methods; Theory of Computation; Ill-posed problems; Inverse problems; Reaction diffusion equation; Hybrid of block-pulse functions and Legendre polynomials; Regularization; Spectral methods
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-019-00691-0

In this paper, we propose an efficient numerical method based on two-dimensional hybrid of block-pulse functions and Legendre polynomials for numerically solving an inverse reaction diffusion equation. The main idea of the present method is based upon some of the important benefits of the hybrid functions such as high accuracy, wide applicability, and adjustability of the orders of the block-pulse functions and Legendre polynomials to achieve highly accurate numerical solutions. By using the spectral method, inverse reaction diffusion equation with initial and boundary conditions would reduce to a system of nonlinear algebraic equations. Due to the ill-posed system of nonlinear algebraic equations, a regularization scheme is employed to obtain a numerical stable solution. Finally, some numerical examples are presented to show the accuracy and effectiveness of this method.