Proceedings of the 13th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2013 24–27 June, 2013. A meshfree numerical method for the time-fractional diffusion equation N. F. Martins 1 , M. L. Morgado 2 and M. Rebelo 3 2 CM-UTAD, Department of Mathematics, University of Tr´as-os-Montes e Alto Douro, UTAD, Quinta de Prados 5001-801, Vila Real, Portugal 1,3 CEMAT-IST and Department of Mathematics, Faculdade de Ciˆ encias e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal emails: nfm@fct.unl.pt, luisam@utad.pt, msjr@fct.unl.pt Abstract In this work we provide an application of the method of fundamental solutions to the one-dimensional time-fractional diffusion equation. The proposed scheme is a meshfree method based on fundamental solutions basis functions for the one-dimensional time-fractional diffusion equation. Some numerical examples are presented in order to illustrate the feasibility and accuracy of the method. Key words: Fractional differential equations, Caputo derivative, sub-diffusion equa- tion, Method of fundamental solutions 1 Introduction Nowadays, the time-fractional diffusion equation (TFDE) is recognized as an important model for processes where anomalous diffusion occurs ([1], [21], [27], [28], [30], [32]). It is obtained from the classical (integer order) linear diffusion equation by replacing the first- order time derivative with a derivative of arbitrary positive real order α (see for example [22], [23] and [24] for the physical interpretation of the noninteger derivative). Here, we are concerned with the numerical solution of the one-dimensional TFDE: ∂ α u(x, t) ∂t α = D α ∂ 2 u(x, t) ∂x 2 , 0 <t ≤ T, 0 < x < L, (1) c CMMSE ISBN: 978-84-616-2723-3