Matrix Transfer Technique Research Articles

Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian

This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional diffusion equation, which uses linear finite element or fourth-order compact difference method combining with matrix transfer technique to approximate spectral fractional Laplacian. Then we introduce a fast time-stepping L1 scheme for time discretization. The proposed scheme can exactly evaluate fractional power of matrix and perform matrix-vector multiplication at per time level by using discrete sine transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory requirement. Further, we address stability and convergence analyses of full discrete scheme based on fast time-stepping L1 scheme on graded time mesh. Our error analysis shows that the choice of graded mesh factor ω=(2−α)/α shall give an optimal temporal convergence O(N−(2−α)) with N denoting the number of time mesh. Finally, ample numerical examples are delivered to illustrate our theoretical analysis and the efficiency of the suggested scheme.

Time–space fractional Euler–Poisson–Darboux equation with Bessel fractional derivative in infinite and finite domains

In this paper, we study the time-fractional Euler–Poisson–Darboux equation with the Bessel fractional derivative. The Laplacian operator of this equation is considered in the ordinary and fractional derivatives and also in different coordinates. For the multi-dimensional Euler–Poisson–Darboux equation in the infinite domain (the whole space), we use the joint modified Meijer–Fourier transforms and establish a complex inversion formula for deriving the fundamental solution. The fractional moment of this solution is also presented in different dimensions. For studying the time-fractional Euler–Poisson–Darboux equation by the numerical methods in finite domain, we sketch the semi- and fully-discrete methods along with the matrix transfer technique to analyze the equation with fractional Laplacian operators in the cartesian, polar and spherical coordinates. The associated error and convergence theorems are also discussed. The illustrative examples are finally presented to verify our results in different coordinates.

A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion Problems

Invoking the matrix transfer technique, we propose a novel numerical scheme to solve the time-fractional advection–dispersion equation (ADE) with distributed-order Riesz-space fractional derivatives (FDs). The method adopts the midpoint rule to reformulate the distributed-order Riesz-space FDs by means of a second-order linear combination of Riesz-space FDs. Then, a central difference approximation is used side by side with the matrix transform technique for approximating the Riesz-space FDs. Based on this, the distributed-order time-fractional ADE is transformed into a time-fractional ordinary differential equation in the Caputo sense, which has an equivalent Volterra integral form. The Simpson method is used to discretize the weakly singular kernel of the resulting Volterra integral equation. Stability, convergence, and error analysis are presented. Finally, simulations are performed to substantiate the theoretical findings.

A fast implicit difference scheme for solving high-dimensional time-space fractional nonlinear Schrödinger equation

In this work, an efficient implicit difference scheme is developed for solving the high-dimensional time-space fractional nonlinear Schrödinger equation. The derived scheme is constructed by utilizing a fast evaluation of Caputo fractional derivative based on the - formula; meanwhile, the compact finite difference with matrix transfer technique is adopted for the spatial discretization. Moreover, a linearized iteration method based on the fast discrete sine transform technique is considered to solve the nonlinear system effectively. Because the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. The stability, truncation error and convergence analysis of the discrete scheme are discussed in detail. Furthermore, a fast iterative algorithm is provided. Finally, several numerical examples are given to verify the efficiency and accuracy of the derived scheme, and a comparison with similar work is presented.

An Inverse Source Problem for A One-dimensional Time-Space Fractional Diffusion Equation

For the fractional diffusion equation, it has received extensive study from various angles on the inverse problem. This paper is dedicated to determining a source term with time-dependence of the time-space fractional diffusion equation with additional observation data. First, the implicit difference scheme and the matrix transfer technique are used to solve an initial boundary value direct problem, in which the time-space fractional diffusion equation for the homogeneous Dirichlet boundary condition is considered. For the conclusive solution of the given inverse source problem, a numerical method is proposed based on the optimal perturbation algorithm with optimized Tikhonov regularization. The numerical examples illustrate that our proposed numerical method is highly effective and relatively stable.

Numerical Solution of Backward Problem of a Multi-term Time-space Fractional Diffusion Equation

There have been many studies from a variety of perspectives about the fractional diffusion equation for the backward problem. This work looks at the backward problem in a multi-term time-space fractional diffusion equation (TSFDE), which uses the observational final data. This paper proposes a numerical solution based on a Tikhonov regularization technique combined with an optimal perturbation algorithm (OPA) to take care of this poorly posed problem. An implicit finite difference technology is introduced by the matrix transfer technique as a numerical approach to solving the problem of the initial boundary value. It is both highly effective and relatively stable, as our suggested numerical method’s numerical results illustrate.

A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes

A weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order α,), where time accuracy is of the order (2−α) or (1+α). To deal with this problem, we present a second-order numerical scheme for nonlinear time–space fractional reaction–diffusion equations. For spatial resolution, we employ a matrix transfer technique. Using graded meshes in time, we improve the convergence rate of the algorithm. Furthermore, some sharp error estimates that give an optimal second-order rate of convergence are presented and proven. We discuss the stability properties of the numerical scheme and elaborate on several empirical examples that corroborate our theoretical observations.

On spectral polar fractional Laplacian

In this paper, we introduce the fractional Sturm–Liouville operators and present the polar fractional Laplacian as a particular case of these operators. We also consider the distributed order space–time fractional diffusion equation involving the polar fractional Laplacian and propose three approaches for studying this problem on the circle and ring domains. First, we apply the integral transforms as the analytical methods to get the solution in terms of the Laplace-type integrals using the Titchmarsh theorem. Second, we use the finite difference method for discretization of the space variable and employ the matrix transfer technique to obtain a system of the distributed order fractional equations. For this system, we modify the Putzer’s algorithm and get the solution in terms of the eigenvalues of discretization matrix. Third, we employ the backward Euler numerical method for discretization of the time variable and find the corresponding error bound. Moreover, we compare and verify the approximate solutions with the exact solutions in analytical form.

A vertex-centred finite volume method for the 3D multi-term time and space fractional Bloch–Torrey equation with fractional Laplacian

We investigate the suitability of a fractional Laplacian in the construction of a three-dimensional (3D) multi-term time–space fractional Bloch–Torrey equation. Instead of using the Riesz fractional derivative as in most previous research, we consider the fractional Laplacian in space and multi-term Caputo fractional derivative in time. A vertex-centred finite volume method is firstly derived to discretise this 3D fractional dynamic system in combination with the matrix transfer technique and the weighted and shifted Grünwald–Letnikov formula. Since computing the fractional power of the matrix representation m(−Δ) of the Laplacian operator results in a highly dense matrix, we use matrix function approximations developed in a Krylov subspace framework and exploit the sparsity of m(−Δ) to reduce the computational overhead. Then the Lanczos method is employed to approximate vector products of matrix functions. A preconditioner is applied to accelerate the convergence process, which shows excellent performance and reduces the memory considerably. In particular, we introduce block matrix functions to deal with the coupled system. Furthermore, the analytical solution of the two-term time–space fractional diffusion equation is derived and expressed by the multinomial Mittag-Leffler function. The feasibility and effectiveness of the proposed numerical techniques are verified by some numerical examples. Compared with two-dimensional fractional models or single-term time fractional dynamic systems, this multi-term time–space fractional model provides more flexibility for the characterisation of anomalous diffusion in biological tissues.

Numerical simulation of high-dimensional two-component reaction–diffusion systems with fractional derivatives

ABSTRACT Two-component reaction–diffusion models in high dimensions involving space-fractional derivatives are used as a powerful modelling approach for understanding several aspects of spatial heterogeneity and nonlocality. In this paper, we propose an accurate, unconditionally stable, and maximum principle preserving fourth-order method both in space and time variables to study the complex dynamical processes of two-component nonlinear space-fractional reaction–diffusion systems posed in high dimensions. To achieve a fast fourth-order accurate method, we adapt the matrix transfer technique with a fast Fourier transform-based implementation in space and an exponential integrator in time. The main advantage of the method is that it avoids storing the large dense matrix resulting from discretizing the fractional operator with the matrix transform approach and significantly reduces the computational costs. Some numerical experiments are carried out in two and three space dimensions to demonstrate the accuracy and computational efficiency of the method.

Matrix transfer technique for anomalous diffusion equation involving fractional Laplacian

The fractional Laplacian, (−△)s, s∈(0,1), appears in a wide range of physical systems, including Lévy flights, some stochastic interfaces, and theoretical physics in connection to the problem of stability of the matter. In this paper, a matrix transfer technique (MTT) is employed combining with spectral/element method to solve fractional diffusion equations involving the fractional Laplacian. The convergence of the MTT method is analyzed by the abstract operator theory. Our method can be applied to solve various fractional equation involving fractional Laplacian on some complex domains. Numerical results indicate exponential convergence in the spatial discretization which is in good agreement with the theoretical analysis.

Fast numerical approximation for the space-fractional semilinear parabolic equations on surfaces

In this work, a fast numerical method is considered to solve the space-fractional semilinear parabolic equations on closed surfaces. It is a challenge in that how to define the space fractional operator and corresponding semilinear parabolic equation with the energy functional defined on surface. To overcome it, using the local tangential space, we construct the spectral approximation for the space fractional operator on surfaces and apply the matrix transfer technique to avoid the difficulty of fractional nonlocality. The main advantage of the matrix transfer method is the completed diagonal representation of fractional operators from eigenvalue decomposition. Moreover, the time-discrete error estimates are presented as well as the energy stability. Various numerical examples are carried out to verify the theoretical results.

Numerical approximation of 2D multi-term time and space fractional Bloch–Torrey equations involving the fractional Laplacian

In this paper, we propose a novel numerical technique to 2D multi-term time and space fractional Bloch–Torrey equations defined on an irregular convex domain. First, we consider the problem with space integral Laplacian operator. We present the semi-discrete and fully-discrete schemes by using the L1 formula on a temporal graded mesh and an unstructured-mesh Galerkin finite element method (FEM) based on the matrix transfer technique (MTT). Moreover, the unconditional stability and convergence of the schemes are discussed and theoretically proved. Then we extend the technique and develop a numerical scheme for the case of space fractional Laplacian operator. In addition, the recent L1+ formula is utilized to improving the temporal accuracy. Finally, three numerical examples in rectangular, elliptical and human brain-like irregular domains are given to demonstrate the accuracy and efficiency of the proposed numerical scheme.

Fast high-order method for multi-dimensional space-fractional reaction–diffusion equations with general boundary conditions

To achieve the efficient and accurate long-time integration, we propose a fast and stable high-order numerical method for solving fractional-in-space reaction–diffusion equations. The proposed method is explicit in nature and utilizes the fourth-order compact finite difference scheme and matrix transfer technique (MTT) in space with FFT-based implementation. Time integration is done through the modified fourth-order exponential time differencing Runge–Kutta scheme. The linear stability analysis and various numerical experiments including two-dimensional (2D) Fitzhugh–Nagumo, Allen–Cahn, Gierer–Meinhardt, Gray–Scott and three-dimensional (3D) Schnakenberg models are presented to demonstrate the accuracy, efficiency, and stability of the proposed method.

Fourth-order time-stepping compact finite difference method for multi-dimensional space-fractional coupled nonlinear Schrödinger equations

In this work, an efficient fourth-order time-stepping compact finite difference scheme is devised for the numerical solution of multi-dimensional space-fractional coupled nonlinear Schrodinger equations. Some existing numerical schemes for these equations lead to full and dense matrices due to the non-locality of the fractional operator. To overcome this challenge, the spatial discretization in our method is carried out by using the compact finite difference scheme and matrix transfer technique in which FFT-based computations can be utilized. This avoids storing the large matrix from discretizing the fractional operator and also significantly reduces the computational costs. The amplification symbol of this scheme is investigated by plotting its stability regions, which indicates the stability of the scheme. Numerical experiments show that this scheme preserves the conservation laws of mass and energy, and achieves the fourth-order accuracy in both space and time.

Predictor-corrector schemes for nonlinear space-fractional parabolic PDEs with time-dependent boundary conditions

We propose novel second-order accurate methods for nonlinear space-fractional PDEs with time-dependent boundary conditions. The matrix transfer technique is used for spatial discretization of the parabolic PDEs to obtain systems of ordinary differential equations. The treatment of the boundary conditions gave rise to a source term which depends on the temporal variable only. Methods based on rational approximations to the exponential function and which uses Gaussian quadrature points are developed for integrating the systems of differential equations. We develop new schemes based on the (1,1)-, (0,2)-Padé approximations and the real distinct pole approximation to the exponential function with Gaussian quadrature points. Error estimates are provided for the convergence analysis. Stability and reliability of the resulting schemes are discussed. A partial fraction decomposition technique is used for the efficient implementation of the schemes. Finally, numerical results in one and two spatial dimensions are presented to corroborate our theoretical observations.

Fractional Laplace operator in two dimensions, approximating matrices, and related spectral analysis

In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by considering the matrix-sequences involved, we analyze the extreme eigenvalues, we give estimates on conditioning, and we study the spectral distribution in the Weyl sense using the tools of the theory of Generalized Locally Toeplitz matrix-sequences. Furthermore, we give a concise description of the spectral properties when non-constant coefficients come into play. Several numerical experiments are reported and critically discussed.

An efficient split-step method for distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions

A split-step predictor-corrector method, based on Adams-Moulton formula, is presented for the solution of nonlinear distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions. The distributed-order space-fractional equation is first discretized to a multi-term space-fractional equation by applying a quadrature rule. Multi-term space-fractional equation is spatially discretized by matrix transfer technique and a linearly implicit split-step predictor-corrector method is applied for time-stepping to avoid solving nonlinear systems at each time step. The method is shown to be unconditionally stable and second order convergent. Numerical experiments are performed to confirm the stability and second order convergence of the method. The split-step predictor-corrector method is also compared with a second order IMEX predictor-corrector method. The IMEX predictor-corrector method is found to incur oscillatory behavior when implemented on problems with nonsmooth initial conditions or with mismatch between the initial and boundary conditions. Our method, which is an L-stable method, produces reliable and oscillation free results for such problems.

Fourth-order time stepping methods with matrix transfer technique for space-fractional reaction-diffusion equations

In this paper, we develop two high-order methods for space-fractional reaction-diffusion equations. The schemes are based on fourth-order Matrix Transfer Technique (MTT) for spatial discretization and fourth-order Exponential Time Differencing Runge–Kutta (ETDRK) for temporal discretization. The ETDRK schemes are based on diagonal and sub-diagonal Padé approximations to the matrix exponential functions. It is observed that the A-stable scheme incurs unwanted oscillations due to high-frequency components present in the solution. These oscillations diminish as the order of the space-fractional derivative decreases. We propose a reliability constraint which is dependent on the order of the space-fractional derivative to avoid these oscillations. However, the L-stable scheme is oscillation-free for any time step. Partial fraction splitting technique is used to compose computationally efficient versions of the schemes. The amplification factor of the schemes is investigated by plotting their stability regions. Convergence analysis is performed on numerical experiments to demonstrate the fourth-order accuracy of the developed schemes in space and time. Numerical experiments made on multidimensional space-fractional reaction-diffusion equations show the reliability, stability, and efficiency of the schemes.

A Split-Step Predictor–Corrector Method for Space-Fractional Reaction–Diffusion Equations with Nonhomogeneous Boundary Conditions

A split-step second-order predictor–corrector method for space-fractional reaction–diffusion equations with nonhomogeneous boundary conditions is presented and analyzed for the stability and convergence. The matrix transfer technique is used for spatial discretization of the problem. The method is shown to be unconditionally stable and second-order convergent. Numerical experiments are performed to confirm the stability and second-order convergence of the method. The split-step predictor–corrector method is also compared with an IMEX predictor–corrector method which is found to incur oscillatory behavior for some time steps. Our method is seen to produce reliable and oscillation-free results for any time step when implemented on numerical examples with nonsmooth initial data. We also present a priori reliability constraint for the IMEX predictor–corrector method to avoid unwanted oscillations and show its validity numerically.