The Numerical Solutions of a Two-Dimensional Space-Time Riesz-Caputo Fractional Diffusion Equation

Abstract

This paper is concerned with the numerical solutions of a two-dimensional space-time fractional differential equation used to model the dynamic properties of complex systems governed by anomalous diffusion. The space-time fractional anomalous diffusion equation is defined by replacing the second order space and the first order time derivatives with Riesz and Caputo operators, respectively. Using the Laplace and Fourier transforms, a general representation of analytical solution is obtained in terms of the Mittag-Leffler function. Grünwald-Letnikov (GL) approximation is also used to find numerical solution of the problem. Finally, simulation results for two examples illustrate the comparison of the analytical and numerical solutions and also validity of the GL approach to this problem.

Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation

“…Many powerful numerical and analytical methods have been presented in literature on finance. Among them, homotopy perturbation method with Sumudu transform and Laplace transform [7][8][9], homotopy analysis method [10], fractional variational iteration method [11], variational iteration method with Sumudu transform, finite difference method [12] and fractional diffusion models [13,14] are relatively new approaches providing an analytical and numerical approximation to Black-Scholes option pricing equation. Methods described in [15,16] are the other numerical methods, used in order to solve dynamic problems in elastic media and generalized semi-infinite programming.…”

The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions

“…This generalization, in association with the convolution theorem, allows for the derivation of some new results on the inverse Laplace transform of irrational functions that appear in problems with fractional relaxation and fractional diffusion [4][5][6][7][8][9][10][11]. This work complements recent progress on the numerical and approximate Laplace-transform solutions of fractional diffusion equations [12][13][14][15]. Most materials are viscoelastic; they both dissipate and store energy in a way that depends on the frequency of loading.…”

On the (k,s)-Hilfer-Prabhakar Fractional Derivative With Applications to Mathematical Physics

“…This paper is motivated by the rich applications of fractional differential equations (FDEs) in physics, economics, engineering, and many other branches of science [8,10,13,17]. Since no general method exists that can be used to analytically solve every FDE, one of the most pressing and challenging tasks is to develop suitable methods for finding analytical solutions to certain classes of FDEs [29][30][31]. Researchers have become interested in fractional interpretations of the classical integral transforms, i.e., Laplace and Fourier transforms [32][33][34], in the past few years.…”