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Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation. (English) Zbl 1214.65046
The authors consider, in one space dimension, the advection-diffusion equation, but with Canuto-type fractional time derivative and Riesz-type fractional space derivatives. They develop difference approximations for the nonlocal operator corresponding to the equation and its Cauchy resp. initial-boundary value problem. They prove conditional stability in the explicit case and unconditional stability for the implicit approximation and get first-order convergence (and slightly better results for the implicit case) for sufficiently smooth (in the sense of C-spaces) solutions. Their numerical results show that Richardson extrapolation is very helpful in improving the accuracy whereas a technique to reduce the computational effort, the short-memory principle, is less effective.

MSC:
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
35K15 Initial value problems for second-order parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35R11 Fractional partial differential equations
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