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.
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Xu, Y., Agrawal, O.P. Numerical solutions and analysis of diffusion for new generalized fractional advection-diffusion equations. centr.eur.j.phys. 11, 1178–1193 (2013). https://doi.org/10.2478/s11534-013-0295-0
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DOI: https://doi.org/10.2478/s11534-013-0295-0