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Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. (English) Zbl 1204.26013
The authors study the variable-order fractional advection-diffusion equation with a nonlinear source term
$\frac{\partial u(u,t)}{\partial t}= \kappa(x,t)R_{\alpha(x,t)}u(x,t)-\nu(x,t)\frac{\partial u}{\partial x}+ f(u,x,t), \quad (x,t)\in \Omega =[a,b]\times[0,T],\tag{1}$ and initial boundary conditions
$u(x,0)=\phi(x), \qquad u(a,t)=0, \qquad u(b,t)=0,$
where $$1\leq \underline{\alpha}\leq \alpha(x,t)\leq \overline{\alpha}\leq 2$$; $$nu(x,t)$$ $$(0\leq \nu(x,t)\leq \overline{\nu}$$ represents the average fluid velocity, $$f(u,x,t)$$ is the source term which satisfies the Lipschitz condition
$|f(u_1 ,x,t)-f(u_2, x,t)|\leq L|u_1 -u_2|, \quad \text{ for }u_1, u_2\in \Omega$ and $$0\leq \kappa(x,t)\leq \overline{\kappa}$$ and $$R_{\alpha(x,t)}u(x,t)$$ is a variable-order fractional derivative defined by
$R_{\alpha(x,t)}u(x,t)=c_+(x,t)^a D^{\alpha(x,t)}_x u(x,t)+c_-(x,t)_x D^{\alpha(x,t)}_b u(x,t),$ where $$0<c_+(x,t)\leq c_1$$, $$0< c_-(x,t)\leq c_2$$. The explicit and implicit Euler methods of approximations for (1) are used and the stability and convergence of the methods are well discussed. Furthermore, other numerical methods such as fractional methods of lines, matrix transfer technique and extrapolation method are also presented and well discussed. Finally, numerical examples were given to demonstrate the effectiveness of the theoretical analysis used.

##### MSC:
 26A33 Fractional derivatives and integrals 45K05 Integro-partial differential equations 35K57 Reaction-diffusion equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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