Meshless simulations of the two-dimensional fractional-time convection–diffusion–reaction equations Ahmad Shirzadi a,n , Leevan Ling b , Saeid Abbasbandy c a Department of Mathematics, Persian Gulf University, Bushehr, Iran b Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong c Department of Mathematics, Imam Khomeini International University, Ghazvin 34149, Iran article info Article history: Received 24 January 2012 Received in revised form 10 May 2012 Accepted 11 May 2012 Available online 29 May 2012 Keywords: Fractional differential equations Meshless local Petrov–Galerkin Moving least-squares Geometric time grids Memory effect abstract The aim of this work is to propose a numerical approach based on the local weak formulations and finite difference scheme to solve the two-dimensional fractional-time convection–diffusion–reaction equations. The numerical studies on sensitivity analysis to parameter and convergence analysis show that our approach is stable. Moreover, numerical demonstrations are given to show that the weak-form approach is applicable to a wide range of problems; in particular, a forced-subdiffusion–convection equation previously solved by a strong-form approach with weak convection is considered. It is shown that our approach can obtain comparable simulations not only in weak convection but also in convection dominant cases. The simulations to a subdiffusion–convection–reaction equation are also presented. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction The fractional diffusion equation has recently been widely used to model for anomalous diffusion. In particular, a 20-month natural gradient tracer study from [1] first showed that water flow in aquifer is better modeled by anomalous diffusion instead of normal diffusion governed by Fick’s second law. Fractional derivatives appear in mathematical modeling of the anomalous diffusion [2,3] and are now used to model diffusion processes of contaminants in porous media, see [4–7] and the references therein. The growing number of applications of fractional deriva- tives in various fields of science and engineering indicates that there is a significant demand for better mathematical algorithms for models with real objects and processes. Several numerical methods to solve fractional differential equations have been presented before, Refs. [8–12] are among them. Due to the memory effect in fractional derivatives, only recently, the focus of numerical methods jumps from one dimensional to higher dimensions. We consider a two-dimensional fractional differen- tial equation in the form of D a t uðx, tÞþ n uðx, tÞ¼ Duðx, tÞþ o  ruðx, tÞþ f ðx, tÞ ð1:1Þ subject to a compatible initial condition uðx, 0Þ¼ gðxÞ for x A O and boundary conditions uðx, tÞ¼ hðx, tÞ for x A @O, where O  R 2 is a regular bounded domain, n is a real constant, o is a vector in R 2 , f is the source function with sufficient smoothness, and D a t denotes the Caputo fractional derivative of order a with respect to t defined by D a t uðx, tÞ¼ 1 Gð1aÞ Z t 0 ðtsÞ a @uðx, sÞ @s ds, 0 oa o1: ð1:2Þ Unlike the integer-order time-derivative of the standard diffusion equation, the integro-differential definition in (1.2) is influenced by the history of the solution. The non-Markov process greatly increases the memory requirement for any numerical algorithms designed to solve (1.1). In our previous work [13], the subdiffu- sion problems (1.1) with a weak convection coefficient was solved by an algorithm that couples an adaptive time stepping and adaptive spatial basis selection approach [14,15]. In this paper, we are interested in knowing how well the local weak formulation is in solving (1.1) in comparison to the strong form collocation. A new finite differences scheme is proposed to deal with the time variable and its derivatives in (1.1). Then the local weak forms of the time-free equations are constructed in local subdomains. The moving least-squares basis and Heaviside test functions are deployed for our numerical computations; a brief review is given in Section 2. In Section 3, we begin our numerical study by a sensitivity and convergence analysis on problems with known solutions. With the basic knowledge acquired, solutions Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements 0955-7997/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.enganabound.2012.05.005 n Corresponding author. E-mail addresses: shirzadi.a@gmail.com (A. Shirzadi), lling@hkbu.edu.hk (L. Ling), abbasbandy@yahoo.com (S. Abbasbandy). Engineering Analysis with Boundary Elements 36 (2012) 1522–1527