Applied Mathematics and Computation 322 (2018) 55–65 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc A collocation method for fractional diffusion equation in a long time with Chebyshev functions A. Baseri, S. Abbasbandy ∗ , E. Babolian Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran a r t i c l e i n f o Keywords: Caputo derivative Rational Chebyshev functions Shifted Chebyshev polynomials Fractional diffusion equation(FDE) Error analysis a b s t r a c t In this paper, our aim is to find a new numerical method for diffusion equation with frac- tional derivative on time and space. The employed fractional derivative is in the Caputo sense. Also, by employing a class of shifted Chebyshev polynomials for the space area and a collection of rational Chebyshev functions for the time domain and then using colloca- tion method, we obtain an algebraic system of equations. The convergence estimate of the new scheme have been concluded. Finally, we evaluate results of this method with other numerical methods. © 2017 Elsevier Inc. All rights reserved. 1. Introduction Many events in engineering, fluid mechanics, physics, viscoelasticity, chemistry and finance can be explained by models using fractional calculus, i.e. containing the theory of derivatives and integrals of non-integer order [1,2]. Fractional differen- tial equations are generalized form of classical integer-order ones, which are obtained by replacing integer order derivatives by fractional ones. Using fractional derivatives have difficulties that one of them is multiple non-equivalent definitions [3]. Moreover, other problem of these definitions is that, derivatives are difficult to calculate and their geometric interpretation is obscure because of their non-local nature [4]. Nonetheless, their advantages comparing with integer-order derivatives are the capability of simulating natural physical processes and dynamic systems more accurately [5,6]. We can find the fractional calculus in many publications such [7–13]. The fractional partial differential equations based on experimental data are suggested to modeling the seepage flow in porous media [14]. Also, Mainardi [1] discussed about the applications of fractional calculus in statistical mechanics and continuum. Two principal kind of fractional partial differential equations are space-fractional differential equation and time- fractional one [15]. Because of proper properties orthogonal polynomials, they are enforceable for different problems. The problem can be reduce to an algebraic systems of equation by doing some operational calculus [16]. The fractional diffusion equation with variable coefficients considered to be: ∂ β u(x, t ) ∂ t β = d(x, t ) ∂ α u(x, t ) ∂ x α + b(x, t ) ∂ γ u(x, t ) ∂ x γ + s(x, t ), 0 < x < b, t ≥ 0, (1) the fractional order of derivatives are α, β and γ where 1 < α ≤ 2 and 0 < β , γ ≤ 1. There are three kind of functions in this equation as: s(x, t) is the source term, u(x, t) is the unknown function and d(x, t) and b(x, t) are known functions. Initial and ∗ Corresponding author. E-mail address: abbasbandy@yahoo.com (S. Abbasbandy). https://doi.org/10.1016/j.amc.2017.11.048 0096-3003/© 2017 Elsevier Inc. All rights reserved.