Abstract
Fractional-order diffusion equations are viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this paper, in order to solve the fractional advection-diffusion equation, the fractional characteristic finite difference method is presented, which is based on the method of characteristics (MOC) and fractional finite difference (FD) procedures. The stability, consistency, convergence, and error estimate of the method are obtained. An example is also given to illustrate the applicability of theoretical results.
Highlights
The history of fractional calculus is almost as long as integerorder calculus
The standard finite difference methods (SFDM) for solving (1) were discussed in [16, 19, 20], and their stability was given in those papers, too
Let Uim be the numerical solution computed by the characteristic finite difference method (CFDM) (12)
Summary
The history of fractional calculus is almost as long as integerorder calculus. because of lack of application background, fractional calculus was developed very slowly. Wyss [10] considered the time fractional diffusion equation and the solution is given in closed form in terms of the Fox functions. Different numerical methods have been developed to solve the fractional diffusion equations [11, 12]. The standard finite difference methods (SFDM) for solving (1) were discussed in [16, 19, 20], and their stability was given in those papers, too. These methods often generate numerical solutions with severe nonphysical oscillations. The quantity ri is called the Courant (or CFL) number
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