In this paper we construct a new difference analog of the Caputo fractional derivative (called the L2-1 σ formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid L 2 -norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by the numerical calculations carried out for some test problems.
We consider boundary value problems of the first and third kind for the diffusionwave equation. By using the method of energy inequalities, we find a priori estimates for the solutions of these boundary value problems.Fractional calculus is used for the description of a large class of physical and chemical processes that occur in media with fractal geometry as well as in the mathematical modeling of economic and social-biological phenomena [1, Chap. 5]. It was proved in [1] that fractional differentiation is a positive operator; this result permits one to obtain a priori estimates for solutions of a wide class of boundary value problems for equations with fractional derivatives. The paper [2] deals with the generalization of the differentiation and integration operations from integer to fractional, real, and complex orders and with applications of fractional integration and differentiation to integral and differential equations and in function theory. In the paper [3], an a priori estimate in terms of a fractional Riemann-Liouville integral of the solution was obtained for the solution of the first initial-boundary value problem for the fractional diffusion equation. The general fractional diffusion equation (0 < α ≤ 1) with regularized fractional derivative was considered in [4]. A more detailed bibliography on fractional partial differential equations, including the diffusion-wave equation, can be found, for example, in [5].In the present paper, we use the method of energy inequalities to obtain a priori estimates for solutions of boundary value problems for the diffusion-wave equation with Caputo fractional derivative [6].
In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of time-space fractional convection-diffusion equations (TSFCDEs). To start with, an implicit difference method based on two-sided weighted shifted Grünwald formulae is proposed with a discussion of the stability and convergence. We construct an implicit difference scheme (IDS) and show that it converges with second order accuracy in both time and space. Then, we develop fast solution methods for handling the resulting system of linear equation with the Toeplitz matrix. The fast Krylov subspace solvers with suitable circulant preconditioners are designed to deal with the resulting Toeplitz linear systems. * Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from O(N 2 ) to O(N ) and the computational complexity from O(N 3 ) to O(N log N ) in each iterative step, where N is the number of grid nodes. Extensive numerical example runs show the utility of these methods over the traditional direct solvers of the implicit difference methods, in terms of computational cost and memory requirements.
Solutions of boundary value problems for a diffusion equation of fractional and variable order in differential and difference settings are studied. It is shown that the method of the energy inequalities is applicable to obtaining a priori estimates for these problems exactly as in the classical case. The credibility of the obtained results is verified by performing numerical calculations for a test problem.
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