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

In this paper, we consider the variable-order nonlinear fractional diffusion equation∂u(x,t)∂t=B(x,t)xRα(x,t)u(x,t)+f(u,x,t),where xRα(x,t)xRα(x,t) is a generalized Riesz fractional derivative of variable order α(x,t)(1<α(x,t)⩽2) and the nonlinear reaction term f(u,x,t)f(u,x,t) satisfies the Lipschitz condition |f(u1,x,t)-f(u2,x,t)|⩽L|u1-u2||f(u1,x,t)-f(u2,x,t)|⩽L|u1-u2|. A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.