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Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation. (English) Zbl 1186.65128
Authors’ abstract: A Cauchy problem for the time fractional advection-dispersion equation (TFADE) is investigated. Such a problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order $\alpha \left(0<\alpha \le 1\right)$. We show that the Cauchy problem of TFADE is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. The convergence estimate is obtained under a priori bound assumptions for the exact solution. Numerical examples are given to show the effectiveness of the proposed numerical method.
MSC:
 65M30 Improperly posed problems (IVP of PDE, numerical methods) 35R11 Fractional partial differential equations 35R25 Improperly posed problems for PDE 65M12 Stability and convergence of numerical methods (IVP of PDE) 65M70 Spectral, collocation and related methods (IVP of PDE)