THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 16, Number 4/2015, pp. 490–498 A CHEBYSHEV-LAGUERRE-GAUSS-RADAU COLLOCATION SCHEME FOR SOLVING A TIME FRACTIONAL SUB-DIFFUSION EQUATION ON A SEMI-INFINITE DOMAIN A.H. BHRAWY 1,2 , M.A. ABDELKAWY 2 , A.A. ALZAHRANI 3 , D. BALEANU 4,5 , E.O. ALZAHRANI 1 1 King Abdulaziz University, Faculty of Science, Department of Mathematics, Jeddah, Saudi Arabia 2 Beni-Suef University, Faculty of Science, Department of Mathematics, Beni-Suef, Egypt 3 King Abdulaziz University, Department of Chemical and Materials Engineering, Faculty of Engineering, Jeddah, Saudi Arabia 4 Cankaya University, Faculty of Arts and Sciences, Department of Mathematics and Computer Sciences, Ankara, Turkey 5 Institute of Space Sciences, Magurele-Bucharest, Romania Corresponding author: D. Baleanu, E-mail: dumitru@cankaya.edu.tr We propose a new efficient spectral collocation method for solving a time fractional sub-diffusion equation on a semi-infinite domain. The shifted Chebyshev-Gauss-Radau interpolation method is adapted for time discretization along with the Laguerre-Gauss-Radau collocation scheme that is used for space discretization on a semi-infinite domain. The main advantage of the proposed approach is that a spectral method is implemented for both time and space discretizations, which allows us to present a new efficient algorithm for solving time fractional sub-diffusion equations. Key words: time fractional sub-diffusion equation, semi-infinite domain, Chebyshev-Gauss-Radau collocation scheme, Laguerre-Gauss-Radau collocation scheme, Caputo derivatives. 1. INTRODUCTION Several computational problems in diverse research areas are considered on semi-infinite domains. The earthquake engineering field and underwater acoustic problems can be modeled as partial differential equations on semi-infinite domains. Spectral methods provide a computational approach that become popular during the last decade [1]. They have gained new popularity in automatic computations for a wide class of physical problems in fluid and heat flows. Recently, spectral methods were used to numerically solve problems on semi-infinite domains [2–7]; in such a case, the choice of the basis functions for a truncated series expansion of the solution depends on orthogonal systems of infinitely differentiable global functions defined on the half line. In recent years there has been a high level of interest of employing spectral methods for numerically solving many types of integral and differential equations, due to their ease of applying them for both finite and infinite domains [1, 8, 9, 10]. Spectral methods not only have exponential rates of convergence but also have a high level of accuracy. There are three main types of spectral methods namely collocation [11, 12, 13], tau [14, 15], and Galerkin [16, 17, 18] methods. Fractional differential equations (FDEs) model many phenomena in several fields such as fluid mechanics, chemistry, biology, viscoelasticity, engineering, finance and physics [19–29]. Most FDEs do not have exact analytic solutions, so approximation techniques must be used. Finite element methods were presented in [30–33] to obtain the numerical solutions of FDEs. Meanwhile, the numerical treatment based on finite difference methods for FDEs was proposed in [34–36]. Moreover, several spectral algorithms were also designed for FDEs, see for example [37]. Time fractional sub-diffusion equation on a semi-infinite domain is studied in this paper using a new collocation method. The collocation method has a wide range of applications, due to its ease of use and adaptability in various problems [38–40]. The aim of this article is to extend the application of shifted Chebyshev-Gauss-Radau interpolation method in combination with generalized Laguerre-Gauss-Radau collocation scheme for the numerical treatment of the time fractional sub-diffusion equations on semi-infinite domains. The shifted Chebyshev- Gauss-Radau interpolation method is adapted for time discretization and the Laguerre-Gauss-Radau