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Journal of Scientific Computing

, Volume 70, Issue 1, pp 386–406 | Cite as

A Galerkin Finite Element Method for a Class of Time–Space Fractional Differential Equation with Nonsmooth Data

  • Zhengang ZhaoEmail author
  • Yunying Zheng
  • Peng Guo
Article

Abstract

In this article, a Galerkin finite element approximation for a class of time–space fractional differential equation is studied, under the assumption that \(u_{tt}, u_{ttt}, u_{2\alpha ,tt}\) are continuous for \(\varOmega \times (0,T]\), but discontinuous at time \(t=0\). In spatial direction, the Galerkin finite element method is presented. And in time direction, a Crank–Nicolson time-stepping is used to approximate the fractional differential term, and the product trapezoidal method is employed to treat the temporal fractional integral term. By using the properties of the fractional Ritz projection and the fractional Ritz–Volterra projection, the convergence analyses of semi-discretization scheme and full discretization scheme are derived separately. Due to the lack of smoothness of the exact solution, the numerical accuracy does not achieve second order convergence in time, which is \(O(k^{3-\beta }+k^{3}t_{n+1}^{-\beta }+k^{3}t_{n+1}^{-\beta -1})\), \(n=0,1,\ldots ,N-1\). But the convergence order in time is shown to be greater than one. Numerical examples are also included to demonstrate the effectiveness of the proposed method.

Keywords

Time–space fractional differential equation Riesz fractional derivative Crank–Nicolson scheme Product trapezoidal method Fractional Ritz–Volterra projection Galerkin finite element method 

Notes

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Number 11301333; Innovation Program of Shanghai Municipal Education Commission under Grant Number 14YZ165; Funding Scheme for Training Young Teachers in Shanghai Colleges under Grant Number zzhg12001; Natural Science Foundation of Anhui provence under Grant Number 1408085MA1; and Funding Scheme for Training Young Teachers in Shanghai Colleges under Grant Number 14AZ17.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Fundamental CoursesShanghai Customs CollegeShanghaiChina
  2. 2.School of Mathematical SciencesHuaibei Normal UniversityHuaibeiChina
  3. 3.Department of Mathematics and PhysicsShanghai DianJi UniversityShanghaiChina

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