Numerical solutions of fractional advection–diffusion equations with a kind of new generalized fractional derivative

Abstract

In the current paper, the numerical solutions for a class of fractional advection–diffusion equations with a kind of new generalized time-fractional derivative proposed last year are discussed in a bounded domain. The fractional derivative is defined in the Caputo type. The numerical solutions are obtained by using the finite difference method. The stability of numerical scheme is also investigated. Numerical examples are solved with different fractional orders and step sizes, which illustrate that the numerical scheme is stable, simple and effective for solving the generalized advection–diffusion equations. The order of convergence of the numerical scheme is evaluated numerically, and the first-order convergence rate has been observed.

Keywords:

• fractional calculus
• generalized time-fractional derivative
• finite difference method
• advection–diffusion equations
• numerical solutions

2010 AMS Subject Classifications:

• 26A33
• 34A08
• 35R11

Acknowledgements

We appreciate the reviewers for their careful reading and providing valuable suggestions which significantly improved our manuscript. Moreover, this work was supported by the Scientific Research Innovation Project for Graduate Students in Hunan Province of China (No. CX2012B109) and the China Scholarship Council. The first author thanks the Department of Mechanical Engineering and Energy Processes at SIUC for hosting him and providing him research facilities.