Journal of Computational Mathematics Vol.xx, No.x, 200x, 1–24. http://www.global-sci.org/jcm doi:10.4208/jcm.1906-m2018-0131 TWO-VARIABLE JACOBI POLYNOMIALS FOR SOLVING SOME FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS * Jafar Biazar 1) and Khadijeh Sadri Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box 41335-1914, Guilan, Rasht, Iran, E-mail: biazar@guilan.ac.ir, jafar.biazar@gmail.com; kh.sadri@uma.ac.ir Abstract Two-variable Jacobi polynomials, as a two-dimensional basis, are applied to solve a class of temporal fractional partial differential equations. The fractional derivative operators are in the Caputo sense. The operational matrices of the integration of integer and fractional orders are presented. Using these matrices together with the Tau Jacobi method converts the main problem into the corresponding system of algebraic equations. An error bound is obtained in a two-dimensional Jacobi-weighted Sobolev space. Finally, the efficiency of the proposed method is demonstrated by implementing the algorithm to several illustrative examples. Results will be compared with those obtained from some existing methods. Mathematics subject classification: 35R11, 65M15, 65M70. Key words: Fractional partial differential equation, Two-variable Jacobi polynomials, Ca- puto derivative, Error bound. 1. Introduction Fractional partial differential equations (FPDEs) are used as modeling tools of various phe- nomena in different branches of science. For example, diffusive processes associated with sub- diffusion (fractional in time), super-diffusion (fractional in space), or both, advection-diffusion, and convection-diffusion processes can be modeled by FPDEs [1–5]. The advantage of these equations in compared to integer-order partial differential equations is the ability of natural simulation of physical processes and dynamical systems more accurately [6]. For instance, some phenomena in fluid and continuum mechanics [7], viscoplastic and viscoelastic flows [8], biology, and acoustics [9], describing chemical and pollute transport in heterogeneous aquifers [10–12], pricing mechanisms and heavy stochastic processes in finance [13], and describing convection process of liquid in medium [14]. Therefore, it helps mathematicians and engineers in the better understanding of the nature and behavior of physical phenomena. For this reason, FPDEs are increasingly studied, but their analytic solving is difficult. Hence, mathematicians have been attracted to solve this class of equations numerically. For example, in [14], the normalized and rational Bernstein polynomials are applied to solve a kind of time-space fractional diffusive equa- tion. The finite difference method is used to solve the fractional reaction-subdiffusion equation in [15]. Authors in [16] propose a wavelet method to solve a class of fractional convection- diffusion equation with variable coefficients. Chen and et al. use generalized fractional-order Legendre functions to obtain numerical solutions of FPDEs with variable coefficients [17]. Ding * Received June 25, 2018 / Revised version received February 21, 2019 / Accepted June 20, 2019 / Published online xxxxxx xx, 20xx / 1) Corresponding author