Bulletin of the
Korean Mathematical Society BKMS

Numerical solutions for the fractional differential dispersion equations with nonlinear forcing terms are considered. The backward Euler finite difference scheme is applied in order to obtain numerical solutions for the equation. Existence and stability of the approximate solutions are carried out by using the right shifted Gr\"unwald formula for the fractional derivative term in the spatial direction. Error estimate of order $O(\Delta x + \Delta t)$ is obtained in the discrete $L_2$ norm. The method is applied to a linear fractional dispersion equations in order to see the theoretical order of convergence. Numerical results for a nonlinear problem show that the numerical solution approach the solution of classical diffusion equation as fractional order approaches $2$.

Keywords: fractional differential equation, Riemann--Liouville fractional derivative, Caputo fractional derivative, finite difference scheme, stability, convergence, error estimate

MSC numbers: 65N06, 65N12, 65N15, 65R20