2018
DOI: 10.3390/e20050321
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Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative

Abstract: 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 differ… Show more

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Cited by 30 publications
(23 citation statements)
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“…Now, we will write Equation (21) in a suitable form which, after inversion, satisfies the initial and boundary conditions (10) and (11). To do this, we consider the following auxiliary functions:…”
Section: Analytical Solution Of the Problemmentioning
confidence: 99%
See 3 more Smart Citations
“…Now, we will write Equation (21) in a suitable form which, after inversion, satisfies the initial and boundary conditions (10) and (11). To do this, we consider the following auxiliary functions:…”
Section: Analytical Solution Of the Problemmentioning
confidence: 99%
“…Using the property (30), it is easy to see that function (37) satisfies boundary condition (11). The solution for the classical advection equation is obtained by making α = 1 into Equations (20) and (35)-(37).…”
Section: Analytical Solution Of the Problemmentioning
confidence: 99%
See 2 more Smart Citations
“…Up to now, there exist many kinds of fractional integrals and derivatives like Riemann-Liouville, Caputo, Riesz, Grünwald-Letnikov, and Hadamard integrals and derivatives. However, it has been noticed that most of the work is devoted to the issues related to Riemann-Liouville, Caputo, and Riesz derivatives [13,14]. Actually, the Hadamard derivative is also very worthy of in-depth study.…”
Section: Introductionmentioning
confidence: 99%