In this paper, a new approach for solving space fractional order diffusion equations is proposed. The fractional derivative in this problem is in the Caputo sense. This approach is based on shifted Chebyshev polynomials of the fourth kind with the collocation method. | Turk J Math (2016) 40: 1283 – 1297 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Numerical approach for solving space fractional order diffusion equations using shifted Chebyshev polynomials of the fourth kind Nasser Hassan SWEILAM1 , Abdelhameed Mohamed NAGY2 , Adel Abd Elaziz El-SAYED3,∗ 1 Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt 2 Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt 3 Department of Mathematics, Faculty of Science, Fayoum University, Fayoum, Egypt Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, a new approach for solving space fractional order diffusion equations is proposed. The fractional derivative in this problem is in the Caputo sense. This approach is based on shifted Chebyshev polynomials of the fourth kind with the collocation method. The finite difference method is used to reduce the equations obtained by our approach for a system of algebraic equations that can be efficiently solved. Numerical results obtained with our approach are presented and compared with the results obtained by other numerical methods. The numerical results show the efficiency of the proposed approach. Key words: Space fractional order diffusion equation, Caputo derivative, Chebyshev collocation method, finite difference method, Chebyshev polynomials of the fourth kind; Euler approximation 1. Introduction Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of many materials and processes [12, 14]. This is the main advantage of the fractional order derivatives in comparison with the classical integer order models, in which such effects are in fact neglected [19, 26]. The advantages of the fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the
