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Numerical Solution of Space-Time-Fractional Reaction-Diffusion Equations via the Caputo and Riesz Derivatives

  • Kolade M. OwolabiEmail author
  • Hemen Dutta
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 200)

Abstract

The present chapter considers the numerical solution of space-time-fractional reaction-diffusion problems used to model complex phenomena that are governed by dynamic of anomalous diffusion. The time- and space-fractional reaction-diffusion equation is modelled by replacing the first order derivative in time and the second-order derivative in space respectively with the Caputo and Riesz operators. We propose some numerical approximation schemes such as the matrix method, average central difference operator and L2 method. To give a general two-dimensional representation of the analytical solution in terms of the Mittag-Leffler function, we apply the Laplace transform technique in time and the Fourier transform method in space. The effectiveness and applicability of the proposed methods are tested on a range of practical problems that are current and recurring interests in one, two and three dimensions are chosen to cover pitfalls that may arise.

Keywords

Caputo fractional derivative Fractional reaction-diffusion equations Numerical simulation Left- and right- Riemann-Liouville fractional derivatives Riesz fractional derivative 

2010 Mathematics Subject Classification

26A33 65L05 65M06 93C10 

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Copyright information

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Authors and Affiliations

  1. 1.Faculty of Natural and Agricultural Sciences, Institute for Groundwater StudiesUniversity of the Free StateBloemfonteinSouth Africa
  2. 2.Department of Mathematical SciencesFederal University of TechnologyAkureNigeria
  3. 3.Department of MathematicsGauhati UniversityGuwahatiIndia

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