Applied and Computational Mathematics 2014; 3(6): 330-336 Published online December 31, 2014 (http://www.sciencepublishinggroup.com/j/acm) doi: 10.11648/j.acm.20140306.17 ISSN: 2328-5605 (Print); ISSN: 2328-5613 (Online) The line method combined with spectral chebyshev for space-time fractional diffusion equation I. K. Youssef 1, * , A. M. Shukur 2 1 Department of Mathematics, Ain Shams University, Cairo, Egypt 2 Department of Applied Mathematics, University of Technology, Baghdad, Iraq Email address: kaoud22@hotmail.com (I. K. Youssef), ahmhshald@yahoo.com (A. M. Shukur) To cite this article: I. K. Youssef, A. M. Shukur. The Line Method Combined with Spectral Chebyshev for Space-Time Fractional Diffusion Equation. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 330-336. doi: 10.11648/j.acm.20140306.17 Abstract: The Method of Lines Combined with Chebyshev Spectral Method respect to weighted residual (Collocation Points) for Space-Time fractional diffusion equation is considered, the direct way will be used for approximating Time fractional and the expiation of shifted first kind of Chebyshev polynomial will be used to approximate unknown functions, the structure of the systems and the matrices will be fund, the algorithm steps is illustrated, The tables and figures of the results of the implementation by using this method at different values of fractional order will be shown, with the helping of programs of matlab. Keywords: Space-Time Fractional Diffusion Equation, Chebyshev-Spectral Method, Finite Difference Method 1. Introduction Diffusion equation provide an important tool for modeling a numerous problems in engineering, physics and others science, the generalization of the integer order diffusion equation to space fractional order, time fractional order or more general is the space-time fractional order diffusion equation (S-TFPDE), is more important to study it and to find the easier ways to solve it. When time dependent PDEs are solved numerically by spectral methods[1, 2], the pattern is usually the same: employ spectral treatment to space dependency and use finite difference in time or leave the time dependency to obtain a system of ordinary differential equations (ODE) in time. In this paper, S-TFPDE has been considered   ,    =   ,    + , ; 0 <  ≤ 1 <  ≤ 2, 0 ≤  ≤ , 0 ≤  ≤ , (1) With initial and boundary conditions 0,  = ,  = 0 ;  0 ≤  ≤ , , 0 = ;  0 ≤  ≤ . (2) The idea of the treatment in this paper depends on the method of discretization in time (the method of lines) [3]. In this method the time interval 0,  is divided into  subintervals   ,  ,⋯, " (  # = $ #% , # &, ’ = 1, 2, ⋯ ,  of length (= ) " , and the problem is transformed to solve fractional boundary value problem along each time level. For the fractional boundary value problem a spectral collocation method is employed. Spectral method [1, 2] involve seeking the solution to differential equations in terms of a series of known, smooth functions. Spectral methods may be viewed as an extreme development of the class of discretization schemes for differential equations known generally as the method of weighted residuals (WRM). The key elements of the WRM are the trial functions and the test functions. The trial functions are used as the basis functions for a truncated series expansion of the solution. The test functions are used to ensure that the differential equation is satisfied as closely as possible by the truncated series expansion. The choice of test function distinguishes which used spectral schemes, is collocation. In the collocation approach the test functions are the translated Dirac delta functions centered at the collocation points. The treatment considered in this work depends on the use of the collocation approach. Collocation The core of the collocation method is definition of the residue function and the collocation points. Consider the boundary value problem, [4, 5].