Numerical Solution of Space-Time-Fractional Reaction-Diffusion Equations via the Caputo and Riesz Derivatives
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 mo
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Abstract The present chapter considers the numerical solution of space-timefractional 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 twodimensional 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
K. M. Owolabi (B) Faculty of Natural and Agricultural Sciences, Institute for Groundwater Studies, University of the Free State, Bloemfontein 9300, South Africa e-mail: [email protected] K. M. Owolabi Department of Mathematical Sciences, Federal University of Technology, PMB 704, Akure, Ondo State, Nigeria H. Dutta Department of Mathematics, Gauhati University, Guwahati 781014, India e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_5
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1 Introduction Fractional calculus is characterized as as extension of ordinary differentiation to arbitrarily noninteger order case. Its study has gained an appreciable significance and popularity in the last few years due to its robust applications in areas of engineering and science. Fractional calculus has been used to model physical phenomena through fractional differential equations. Over the years, the study of nonlinear problems of traveling wave solutions played an effective role in studying nonlinear real-life processes. Nowadays, many other applicable areas of fractional calculus are encountered in various application fields such as biology, chemistry, electricity, mechanics, economics, geology and medicine, notably the signal-image processing and control theory. The primary topics include the continuous time random walk (CTRW), anomalous diffusion, vibration and control, fractional Brownian motion, power law, fractional filters, biomedical engineering, Riesz potential, fractals, fractional neutron point kinetic model, nonlocal phenomena, memory-dependent scenario, computational fractional derivative equations, porous media, fractio
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