Vol.:(0123456789) 1 3 Engineering with Computers https://doi.org/10.1007/s00366-018-0665-8 ORIGINAL ARTICLE The local discontinuous Galerkin method for 2D nonlinear time‑fractional advection–diffusion equations Jafar Eshaghi 1  · Saeed Kazem 1  · Hojjatollah Adibi 1 Received: 2 June 2018 / Accepted: 20 November 2018 © Springer-Verlag London Ltd., part of Springer Nature 2018 Abstract This paper presents a numerical solution of time-fractional nonlinear advection–diffusion equations (TFADEs) based on the local discontinuous Galerkin method. The trapezoidal quadrature scheme (TQS) for the fractional order part of TFADEs is investigated. In TQS, the fractional derivative is replaced by the Volterra integral equation which is computed by the trap- ezoidal quadrature formula. Then the local discontinuous Galerkin method has been applied for space-discretization in this scheme. Additionally, the stability and convergence analysis of the proposed method has been discussed. Finally some test problems have been investigated to confirm the validity and convergence of the proposed method. Keywords Time-fractional advection–diffusion equations · Discontinuous Galerkin method · Local discontinuous Galerkin method · Caputo derivative · Stability and convergence analysis Mathematics Subject Classification 45D05 · 45G05 · 41A30 1 Introduction During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been suc- cessfully applied to problems in system biology, physics, chemistry, engineering, etc. [1–8]. The historical background development of fractional cal- culus begins with step by step introduction of the fractional derivative and fractional integral of functions. The fractional calculus is a branch of mathematics with a long history in earlier work, the main application of fractional calculus is considered as a technique for solving integral equations. The fractional calculus started by some speculations of Leibniz [9] and Euler [10], and has been developed, pro- gressively up till now. A list of mathematicians, who have provided important contributions, includes: Laplace [11] who presented expressions for certain frac- tional derivatives. Liouville in [12] suggested a better way of writing Fouriers formula, and gave the definition of a fractional derivative as an infinite series: d u y dx u = ∑ A m e mx m u . Riemann presented a generalization of Taylor series expan- sion and derived the following definition for fractional integration: Also, O’shaughnessy [13] solved the fractional equation Some of the fundamental works on the subject of fractional calculus, which have investigated the theory of derivatives and integrals of fractional order are the books of McBride and Roach [14], Katsuyuki Nishimoto [15], Kiryakova [16], d − dx − u(x)= 1 Γ()  x a (x − t) −1 u(t)dt. d 1 2 dx 1 2 y = y x . * Saeed Kazem saeedkazem@gmail.com Jafar Eshaghi j_eshaghi@aut.ac.ir Hojjatollah Adibi adibih@aut.ac.ir 1 Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., 15914 Tehran, Iran