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Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. (English) Zbl 1171.65101
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.

##### MSC:
 65R20 Numerical methods for integral equations 45K05 Integro-partial differential equations 45G10 Other nonlinear integral equations 35K57 Reaction-diffusion equations 26A33 Fractional derivatives and integrals 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
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