Next Article in Journal
A New Two-Dimensional Map with Hidden Attractors
Next Article in Special Issue
Time-Fractional Diffusion with Mass Absorption in a Half-Line Domain due to Boundary Value of Concentration Varying Harmonically in Time
Previous Article in Journal
The Relationship between Postural Stability and Lower-Limb Muscle Activity Using an Entropy-Based Similarity Index
Previous Article in Special Issue
The Power Law Characteristics of Stock Price Jump Intervals: An Empirical and Computational Experimental Study
Article Menu
Issue 5 (May) cover image

Export Article

Open AccessArticle
Entropy 2018, 20(5), 321; https://doi.org/10.3390/e20050321

Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative

1
The State Key Laboratory of Scientific and Engineering Computing (LSEC), The Institute of Computational Mathematics and Scientific/Engineering Computing (ICMSEC), Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
2
COMSATS Institute of Information Technology, Lahore 54500, Pakistan
3
Department of Mathematics, Cankaya University, Ankara 06530, Turkey
4
Institute of Space Sciences, Magurele-Bucharest 077125, Romania
5
College of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China
6
Department of Mathematics, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia
7
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
*
Authors to whom correspondence should be addressed.
Received: 1 March 2018 / Revised: 20 April 2018 / Accepted: 20 April 2018 / Published: 26 April 2018
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)
View Full-Text   |   Download PDF [1710 KB, uploaded 3 May 2018]   |  

Abstract

In this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection–diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated by the second-order fractional weighted and shifted Grünwald–Letnikov formula. Based on the equivalence between the fractional differential equation and the integral equation, we have transformed the fractional differential equation into an equivalent integral equation. Then, the integral is approximated by the trapezoidal formula. Further, the stability and convergence analysis are discussed rigorously. The resulting scheme is formally proved with the second order accuracy both in space and time. Numerical experiments are also presented to verify the theoretical analysis. View Full-Text
Keywords: fractional advection dispersion equation; riesz derivative; caputo derivative; trapezoidal formula fractional advection dispersion equation; riesz derivative; caputo derivative; trapezoidal formula
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Arshad, S.; Baleanu, D.; Huang, J.; Al Qurashi, M.M.; Tang, Y.; Zhao, Y. Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative. Entropy 2018, 20, 321.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top