Fractional Partial Differential Equation's Numerical Solution And Fundamental Solution

Differential equations of fractional order appear more and more frequently in various research areas and engineering applications. An effective and easy-to-use method for solving such equations is needed. In this paper, at first, the numerical solution of a space fractional differential equation with mixed problem is considered. This equation is obtained from the parabolic equation containing advection, diffusion and reaction terms by replacing the second order derivative in space by a fractional derivative in space of order. We use the modified Gru&&n wald formula to approximate the fractional derivative and the backward difference to approximate the first-order derivative. Therefore an explicit finite difference schemes and an implicit finite difference schemes for this equation can be presented. The stability and convergence of the finite difference schemes are proved. Some numerical examples are given. Secondly, we consider numerical solution of a space fractional advection-dispersion equation with mixed problem, which is obtained from the advection-dispersion equation by replacing the second order derivative in space by a fractional derivative in space of order. Using integration method or finite-volume method, we can give an explicit finite difference schemes and an implicit finite difference schemes for this equation, where the fractional derivative is approximated by the Gru&&n wald formula. The stability and convergence of the finite difference schemes are proved. Some numerical results are given. Finally, the analytic solution of a space and time fractional advection-dispersion equation is considered. This equation is obtained from an advection-dispersion equation by replacing the second order derivative in space by α (1 < α≤ 2) order derivate in space of order, the first order derivative in space by γ (0 < γ≤ 1) order derivative in space of order and the first order derivative at time by β (0 < β≤ 1) order derivative at time of order. Used the Fourier transform, the Laplace transform and their inverse transforms, the analytic solution of this equation can be arrived at. The fundamental solution of this equation is discussed.