Abstract
In this paper, a fractional diffusion equation by using the explicit numerical method in a finite domain with second-order accuracy which includes the Riesz fractional derivative approximation is studied. For the Riesz fractional derivative approximation, ''fractional centered derivative'' approach is used. The error of the Riesz fractional derivative to the fractional centered difference is calculated. We used the implicit numerical method to solve the fractional diffusion equation and also investigated the stability of explicit and implicit methods. The maximum error of the implicit method for fractional diffusion equation with using fractional centered difference approach is shown by using the numerical results.
Highlights
Fractional differential equations are used frequently in science and engineering, such as: fractional diffusion and wave equations [1, 2], electrical systems [3], viscoelastic-We consider the following equation in a finite domain associated with initial and Dirichlet boundary conditions ∂u (x, t ) ∂t = D ∂αu (x, t ) ∂ xα +f (x,t), a < x < b, 0
4882 Explicit and Implicit Methods for Fractional Diffusion Equations with the Riesz Fractional Derivative derivative approximation to a linear diffusion equation, which has an independent fractional derivative, represented that the used method is unconditionally stable for given problems
Shen and et al [13] applied implicit and explicit finite difference methods with Grunwald–Letnikov derivative approximation to a linear Riesz fractional diffusion equation, and showed that the explicit method is conditionally and the implicit models is unconditionally stable for given problems
Summary
Fractional differential equations are used frequently in science and engineering, such as: fractional diffusion and wave equations [1, 2], electrical systems [3], viscoelastic-. 4882 Explicit and Implicit Methods for Fractional Diffusion Equations with the Riesz Fractional Derivative derivative approximation to a linear diffusion equation, which has an independent fractional derivative, represented that the used method is unconditionally stable for given problems. Shen and et al [13] applied implicit and explicit finite difference methods with Grunwald–Letnikov derivative approximation to a linear Riesz fractional diffusion equation, and showed that the explicit method is conditionally and the implicit models is unconditionally stable for given problems. Zhang and Liu [15] applied the implicit finite difference method with Grunwald–Letnikov derivative approximation to a nonlinear Riesz fractional diffusion equation and showed that the used method is stable for small time. ∂α f (x) When h → 0 and ∂ x α is the Riesz fractional derivative for 1
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