An alternating direction Galerkin method for a time-fractional partial differential equation with damping in two space dimensions

An Chen,Changpin Li

doi:10.1186/s13662-017-1414-9

Abstract

In this paper, we propose an efficient alternating direction implicit (ADI) Galerkin method for solving the time-fractional partial differential equation with damping, where the fractional derivative is in the sense of Caputo with order in (1,2). The presented numerical scheme is based on the L2-1_{sigma} method in time and the Galerkin finite element method in space. The unconditional stability and convergence of the numerical scheme are both carefully proved. Numerical results are displayed for supporting the theoretical analysis.

Highlights

In this paper, we study the following time-fractional partial differential equation with damping in two space dimensions: ⎧ ⎪⎨ C Dα,tu + λ ∂u ∂t = u + f (x, y, t), (x, y, t) ∈ ×
Several approaches are available for solving time-fractional partial differential equations in one or more than one space dimension, see [ – ] for finite difference methods, [ – ] for finite element methods, and [, ] for spectral methods
Time-fractional partial differential equations involving the Caputo derivative operator describe the history-dependent behavior well, they cause large computational cost due to the fact that all the previous numerical data need to be stored in order to obtain the current numerical solution

Summary

Introduction

We study the following time-fractional partial differential equation with damping in two space dimensions:. Several approaches are available for solving time-fractional partial differential equations in one or more than one space dimension, see [ – ] for finite difference methods, [ – ] for finite element methods, and [ , ] for spectral methods. Time-fractional partial differential equations involving the Caputo derivative operator describe the history-dependent behavior well, they cause large computational cost due to the fact that all the previous numerical data need to be stored in order to obtain the current numerical solution. In Section , an ADI Galerkin scheme for solving time-fractional hyperbolic equation with damping is derived.

Preliminaries

Conclusion

Methods