Spectral regularization method for a Cauchy problem of the time fractional advection–dispersion equation

• Published in: Journal of computational and applied mathematics Vol. 233; no. 10; pp. 2631 - 2640
• Main Authors: Zheng, G.H., Wei, T.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 15-03-2010; Elsevier
• Subjects: Caputo fractional derivative; Cauchy problem; Convergence estimate; Exact sciences and technology; Fourier transform; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Real functions; Sciences and techniques of general use; Spectral regularization method; Time fractional advection–dispersion equation; Fourier transform; Convergence estimate; Time fractional advection–dispersion equation; Spectral regularization method; Caputo fractional derivative; Cauchy problem; Spectral method; Fourier transformation; equation; Fourier analysis; Numerical method; Stochastic method; Time fractional advection-dispersion; Bounded solution; Convergence; Exact solution; Regularization method; Numerical analysis; Applied mathematics
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2009.11.009

In this paper, 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
α
(
0
<
α
≤
1
)
. 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.