Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation (Q1026320)

The authors construct an explicit finite difference scheme for the approximate solution of the nonlinear diffusion equation of fractional order \[ \frac{\partial {u(x,t)}}{\partial{t}}=B(x,t)_{x}{R^{\alpha(x,t)}u(x,t)+f(u,x,t),\quad {{X_{a}}<X<{X_{b}}},0<t<T} \] with the initial and boundary conditions of usual form. The derivative of fractional order is considered in the generalized sense of Riesz. The approximate scheme can be written in matrix form \[ U^{j+1}=P^{j}U^{j}+B^{j}+F^{j}. \] The convergence and stability of this scheme are proved and some numerical examples are presented.