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# Second Order Accuracy Finite Difference Methods for Fractional Diffusion Equations

[+] Author Affiliations
Yuki Takeuchi, Reiji Suda

The University of Tokyo, Tokyo, Japan

Paper No. DETC2013-12059, pp. V004T08A015; 10 pages
doi:10.1115/DETC2013-12059
From:
• ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
• Volume 4: 18th Design for Manufacturing and the Life Cycle Conference; 2013 ASME/IEEE International Conference on Mechatronic and Embedded Systems and Applications
• Portland, Oregon, USA, August 4–7, 2013
• Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
• ISBN: 978-0-7918-5591-1

## abstract

Finite difference methods for fractional differential equation are ever proposed. However, precise error orders have not been analyzed for the methods higher than first order accuracy. This paper proposes a few finite difference methods for fractional diffusion equations and shows our methods have second order accuracy under the conditions that the solution functions have higher order than second order at boundaries. In addition, we show that the accuracy may decrease in the case that the solution functions have lower order than second order at boundaries when we use second order accuracy scheme. In this paper, we treat schemes based on Grunwald-Letnikov definition and apply them to three kinds of fractional diffusion equations using Riemann-Liouville derivative operator including time-fractional diffusion equation, space-fractional diffusion equation and time-space-fractional diffusion equation. Finally, we show the simulation results which indicate that our methods are stable and have successfully second order accuracy under the assumed conditions.

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