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Numerical solutions for space fractional dispersion equations with nonlinear source terms. (English) Zbl 1208.65125
The aim of the article is to provide a finite difference scheme for a fractional differential dispersion equation with a nonlinear forcing term. Thanks to a right shifted Grünwald formula (note that an unshifted Grünwald formula for the fractional derivative of Riemann-Liouville type is unstable regardless the finite difference scheme is either explicit or implicit), the authors present an implicit Euler approximation for the problem under consideration. Existence of the finite difference approximate solution is proved thanks to the use of the Leray-Schauder fixed point theorem. To prove the convergence of the approximate solution, the authors first justify the stability of the finite difference scheme and then they combine this stability together with a consistency for the finite difference scheme. As usual, the stated stability result can be used to prove the uniqueness of the approximate finite difference solution. It is proved that the convergence order is $$h+k$$ in a discrete $$L^2$$-norm, where $$h$$ (resp. $$k$$) is the mesh size of the space (resp. time) discretization. Some numerical experiments are presented.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35K55 Nonlinear parabolic equations 35R11 Fractional partial differential equations
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