Numerical solutions and analysis of diffusion for new generalized fractional advection-diffusion equations

Abstract

Abstract:In this paper we study a class of new Generalized Fractional Advection-Diffusion Equations (GFADEs) with a new Generalized Fractional Derivative (GFD) proposed last year. The new GFD is defined in the Caputo sense using a weight function and a scale function. The GFADE is discussed in a bounded domain, and numerical solutions for two examples consisting of a linear and a nonlinear GFADE are obtained using an implicit finite difference approach. The stability of the numerical scheme is investigated, and the order of convergence is estimated numerically. Numerical results illustrate that the finite difference scheme is simple and effective for solving the GFADEs. We investigate the influence of weight and scale functions on the diffusion of GFADEs. Linear and nonlinear stretching and contracting functions are considered. It is found that an increasing weight function increases the rate of diffusion, and a scale function can stretch or contract the diffusion on the time domain. PACS

Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation

“…Substituting (24) to the semi-discretization (22), we have (27) where f k (x) = f (x, t k ) and R s is the error in the space direction caused by replacing u (x, t) with u N (x, t). The classical first-and third-order partial derivatives in (27) are derived as…”

“…Taking a scale function and a weight function into consideration, fractional integrals and derivatives were essentially generalized. Partial differential equations with such generalized fractional derivatives are not studied much [25][26][27][28]. Therefore, in this paper, we choose the KdV equation to further study this topic.…”

Linear Fractional Diffusion-Wave Equation for Scientists and Engineers

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“…Equations with the Riemann-Liouville fractional derivative have also been studied in [69, 113, 134-139, 203, 225, 226, 229] as well as more complicated models, in particular fractional diffusion equations of distributed order (see [7, 8, 27-29, 55, 84, 108, 109, 125, 213] and references therein) and the generalized diffusion equation containing the fractional time-derivative with a weight and a scale [232]. The following integrals [196] appear in the case of the wave equation …”

Numerical scheme with convergence for a generalized time‐fractional Telegraph‐type equation

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