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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 $$ \frac{\partial u(u,t)}{\partial t}= \kappa(x,t)R_{\alpha(x,t)}u(x,t)-\nu(x,t)\frac{\partial u}{\partial x}+ f(u,x,t), \quad (x,t)\in \Omega =[a,b]\times[0,T],\tag1$$ and initial boundary conditions $$ u(x,0)=\phi(x), \qquad u(a,t)=0, \qquad u(b,t)=0,$$ where $1\le \underline{\alpha}\le \alpha(x,t)\le \overline{\alpha}\le 2$; $nu(x,t)$ $(0\le \nu(x,t)\le \overline{\nu}$ represents the average fluid velocity, $f(u,x,t) $ is the source term which satisfies the Lipschitz condition $$|f(u_1 ,x,t)-f(u_2, x,t)|\le L|u_1 -u_2|, \quad \text{ for }u_1, u_2\in \Omega$$ and $0\le \kappa(x,t)\le \overline{\kappa}$ 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 $0<c_+(x,t)\le c_1$, $0< c_-(x,t)\le c_2$. 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.

26A33Fractional derivatives and integrals (real functions)
45K05Integro-partial differential equations
35K57Reaction-diffusion equations
65M12Stability and convergence of numerical methods (IVP of PDE)
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