A fast implicit difference scheme for solving the generalized time-space fractional diffusion equations with variable coefficients

In this paper, we first propose an unconditionally stable implicit difference scheme for solving generalized time-space fractional diffusion equations (GTSFDEs) with variable coefficients. The numerical scheme utilizes the $L1$-type formula for the generalized Caputo fractional derivative in time discretization and the second-order weighted and shifted Gr\"{u}nwald difference (WSGD) formula in spatial discretization, respectively. Theoretical results and numerical tests are conducted to verify the $(2 - \gamma)$-order and 2-order of temporal and spatial convergence with $\gamma\in(0,1)$ the order of Caputo fractional derivative, respectively. The fast sum-of-exponential approximation of the generalized Caputo fractional derivative and Toeplitz-like coefficient matrices are also developed to accelerate the proposed implicit difference scheme. Numerical experiments show the effectiveness of the proposed numerical scheme and its good potential for large-scale simulation of GTSFDEs.Comment: 23 pages, 10 tables, 1 figure. Make several corrections again and have been submitted to a journal at Sept. 20, 2019. Version 2: Make some necessary corrections and symbols, 13 Jan. 2020. Revised manuscript has been resubmitted to journa

On the bilateral preconditioning for an L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations

Time-space fractional Bloch-Torrey equations (TSFBTEs) are developed by some researchers to investigate the relationship between diffusion and fractional-order dynamics. In this paper, we first propose a second-order implicit difference scheme for TSFBTEs by employing the recently proposed L2-type formula [A. A. Alikhanov, C. Huang, Appl. Math. Comput. (2021) 126545]. Then, we prove the stability and the convergence of the proposed scheme. Based on such a numerical scheme, an L2-type all-at-once system is derived. In order to solve this system in a parallel-in-time pattern, a bilateral preconditioning technique is designed to accelerate the convergence of Krylov subspace solvers according to the special structure of the coefficient matrix of the system. We theoretically show that the condition number of the preconditioned matrix is uniformly bounded by a constant for the time fractional order $\alpha \in (0,0.3624)$. Numerical results are reported to show the efficiency of our method.Comment: 24 pages, 6 tables, 4 figure

Numerical solution methods for fractional partial differential equations

Fractional partial differential equations have been developed in many different fields such as physics, finance, fluid mechanics, viscoelasticity, engineering and biology. These models are used to describe anomalous diffusion. The main feature of these equations is their nonlocal property, due to the fractional derivative, which makes their solution challenging. However, analytic solutions of the fractional partial differential equations either do not exist or involve special functions, such as the Fox (H-function) function (Mathai & Saxena 1978) and the Mittag-Leffler function (Podlubny 1998) which are diffcult to evaluate. Consequently, numerical techniques are required to find the solution of fractional partial differential equations. This thesis can be considered as two parts, the first part considers the approximation of the Riemann-Liouville fractional derivative and the second part develops numerical techniques for the solution of linear and nonlinear fractional partial differential equations where the fractional derivative is defied as a Riemann-Liouville derivative. In the first part we modify the L1 scheme, developed initially by Oldham & Spanier (1974), to develop the three schemes which will be defined as the C1, C2 and C3 schemes. The accuracy of each method is considered. Then the memory effect of the fractional derivative due to nonlocal property is discussed. Methods of reduction of the computation L1 scheme are proposed using regression approximations. In the second part of this study, we consider numerical solution schemes for linear fractional partial differential equations. Here the numerical approximation schemes are developed using an approximation of the fractional derivative and a spatial discretization scheme. In this thesis the L1, C1, C2, C3 fractional derivative approximation schemes, developed in the first part of the thesis, are used in conjunction with either the Centred-finite difference scheme, the Dufort-Frankel scheme or the Keller Box scheme. The stability of these numerical schemes are investigated via the technique of the Fourier analysis (Von Neumann stability analysis). The convergence of each the numerical schemes is also discussed. Numerical tests were used to conform the accuracy and stability of each proposed method. In the last part of the thesis numerical schemes are developed to handle nonlinear partial differential equations and systems of nonlinear fractional partial differential equations. We considered two models of a reversible reaction in the presence of anomalous subdiffusion. The Centred-finite difference scheme and the Keller Box methods are used to spatially discretise the spatial domain in these schemes. Here the L1 scheme and a modification of the L1 scheme are used to approximate the fractional derivative. The accuracy of the methods are discussed and the convergence of the scheme are demonstrated by numerical experiments. We also give numerical examples to illustrate the e�ciency of the proposed scheme

New Trends in Differential and Difference Equations and Applications

This is a reprint of articles from the Special Issue published online in the open-access journal Axioms (ISSN 2075-1680) from 2018 to 2019 (available at https://www.mdpi.com/journal/axioms/special issues/differential difference equations)