Two-dimensional Fractional Diffusion Equation Research Articles

Two fast finite difference methods for a class of variable-coefficient fractional diffusion equations with time delay

This paper introduces the fast Crank–Nicolson (CN) and compact difference schemes for solving the one- and two-dimensional fractional diffusion equations with time delay. The CN method is employed for temporal discretization, while the fractional centered difference (FCD) formula discretizes the Riesz space derivative. Additionally, a novel fourth-order scheme is developed using compact difference operators to improve accuracy. The convergence and stability of these schemes are rigorously proven. The discretized systems combining Toeplitz-like structures can be effectively solved by Krylov subspace solvers with suitable preconditioners. Each time level of these methods requires a computational complexity of O(plogp) per iteration and a memory of O(p), where p represents the total number of grid points in space. Numerical examples are given to illustrate both the theoretical results and the computational efficiency of the fast algorithm.

On numerical solution of two-dimensional variable-order fractional diffusion equation arising in transport phenomena

<abstract><p>In recent years, the application of variable-order (VO) fractional differential equations for describing complex physical phenomena ranging from biology, hydrology, mechanics and viscoelasticity to fluid dynamics has become one of the most hot topics in the context of scientific modeling. An interesting aspect of VO operators is their capability to address the behavior of scientific and engineering systems with time and spatially varying properties. The VO fractional diffusion equation is a fundamental model that allows transitions among sub-diffusive, diffusive and super-diffusive behaviors without altering the underlying governing equations. In this paper, we considered the two-dimensional fractional diffusion equation with the Caputo time VO derivative, which is essential for describing anomalous diffusion in real-world complex systems. A new Crank-Nicolson (C-N) difference scheme and an efficient explicit decoupled group (EDG) method were proposed to solve the problem under consideration. The proposed EDG method is based on a skewed difference scheme in conjunction with a grouping procedure of the solution grid points. Special attention was devoted to investigating the stability and convergence of the proposed methods. Three numerical examples with known exact analytical solutions were provided to illustrate our considerations. The proposed methods were shown to be stable and convergent theoretically as well as numerically. In addition, a comparative study was done between the EDG method and the C-N difference scheme. It was found that the proposed methods are accurate in simulating the considered problem, while the EDG method is superior to the C-N difference method in terms of Central Processing Unit (CPU) timing, verifying the efficiency of the former method in solving the VO problem.</p></abstract>

LAPLACE DECOMPOSITION METHOD FOR SOLVING THE TWO-DIMENSIONAL DIFFUSION PROBLEM IN FRACTAL HEAT TRANSFER

In this paper, the Local Fractional Laplace Decomposition Method (LFLDM) is used for solving a type of Two-Dimensional Fractional Diffusion Equation (TDFDE). In this method, first we apply the Laplace transform and its inverse to the main equation, and then the Adomian decomposition is used to obtain approximate/analytical solution. The accuracy and applicability of the LFLDM is shown through two examples. The LFLDM results are in good agreement with the exact solution of the problems.

On the analysis and application of a spectral collocation scheme for the nonlinear two-dimensional fractional diffusion equation

In this paper, we propose and analyze a novel spectral scheme for the numerical solution of a two-dimensional time-fractional diffusion equation. The proposed scheme approximates the unknown function and its derivatives in space by finite Lucas and Fibonacci polynomial expansion, and time derivative/variable by L1 formula with θ-weighted difference scheme. Error estimates are derived and convergence is proved. Six test problems are then solved to verify the theory numerically. The obtained results confirm the low computational cost and better accuracy than known results in the literature.

Solution of two-dimensional fractional diffusion equation by a novel hybrid D(TQ) method

Abstract This work is an experiment to solve the fractional diffusion equation in two dimensions with a novel hybrid method. The method involves an amalgamation of the well-known differential transform method and the differential quadrature method. This work is not about the superiority of one method over the other, instead this is an idea that can be worked upon for possible greatness. Numerical examples are discussed with tables and figures.

Inexact block SSOR-like preconditioners for non-Hermitian positive definite linear systems of strong Hermitian parts

In this work, a class of inexact block SSOR-like preconditioners are proposed for the non-Hermitian positive definite linear system whose coefficient matrix is of a blockwise form. The novel preconditioners are based on the SSOR-like iteration method proposed by Bai (Numer. Linear Algebra Appl. 23, 37-60, 2016), but each of the diagonal blocks in the SSOR-like method is replaced by an approximation matrix which is easier to deal with. The estimated bounds for eigenvalues of the new preconditioned matrix, as well as the convergence property about the corresponding iteration method, are discussed. Finally, numerical experiments arise from the discretization of two-dimensional fractional diffusion equation and two-dimensional linear integro-differential equation are presented to confirm the theoretical analyses and illustrate the efficiency of the new preconditioners.

A numerical method based on three-dimensional Legendre wavelet method for two-dimensional time-fractional diffusion equation

In this paper, an efficient numerical method is proposed to handle two-dimensional fractional diffusion equations on a finite domain. The proposed method combines the product of Legendre wavelet bases for two spatial dimensions and a time direction. The operational matrix of the proposed method is obtained. Tikhonov regularization is employed to stabilize the system in cases where the final linear system of equations is large. The convergence analysis of the method is studied and some numerical examples are presented to investigate the efficiency and accuracy of the method.

A preconditioned fast quadratic spline collocation method for two-sided space-fractional partial differential equations

Abstract Quadratic spline collocation methods for the one and two-dimensional fractional diffusion equations are proposed. By carefully exploring the mathematical structure of the coefficient matrix, we propose a matrix-free fast Krylov subspace iterative solver for the corresponding quadratic spline collocation scheme. And then, preconditioning technique is applied to further accelerate the convergence of the fast Krylov subspace iterative method. It shows that the fast quadratic spline collocation scheme has greatly reduced the computational cost from O ( N 2 ) to O ( N log N ) per iteration and memory requirement from O ( N 2 ) to O ( N ) , while still well approximates the fractional diffusion equations without any accuracy lost. Numerical experiments are given to demonstrate the efficiency and effectiveness of the fast method.

Fast Solvers for Two-Dimensional Fractional Diffusion Equations Using Rank Structured Matrices

We consider the discretization of time-space diffusion equations with fractional derivatives in space and either one-dimensional (1D) or 2D spatial domains. The use of an implicit Euler scheme in time and finite differences or finite elements in space leads to a sequence of dense large scale linear systems describing the behavior of the solution over a time interval. We prove that the coefficient matrices arising in the 1D context are rank structured and can be efficiently represented using hierarchical formats ($\mathcal H$-matrices, HODLR). Quantitative estimates for the rank of the off-diagonal blocks of these matrices are presented. We analyze the use of HODLR arithmetic for solving the 1D case and we compare this strategy with existing methods that exploit the Toeplitz-like structure to precondition the GMRES iteration. The numerical tests demonstrate the convenience of the HODLR format when at least a reasonably low number of time steps is needed. Finally, we explain how these properties can be leveraged to design fast solvers for problems with 2D spatial domains that can be reformulated as matrix equations. The experiments show that the approach based on the use of rank-structured arithmetic is particularly effective and outperforms current state of the art techniques.

The numerical solution for the time-fractional inverse problem of diffusion equation

In this study, the spectral meshless radial point interpolation (SMRPI) technique is applied to the Cauchy problem of two-dimensional fractional diffusion equation. We obtain the unknown data on the inner boundary when overspecified boundary data is imposed on the outer boundary. The SMRPI is based on a combination of meshfree methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. Here, similar to other meshless methods, localization in SMRPI can reduce the ill-posedness of the Cauchy problem. However, it does not require to use regularization algorithms and therefore reduces computational time. Two numerical examples, are tested to show that the SMRPI can overcome the ill-posedness of the Cauchy problem and has acceptable accuracy. Also, by adding some large perturbations, the proposed method is still stable.

High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations

As the generalization of the integer order partial differential equations (PDE), the fractional order PDEs are drawing more and more attention for their applications in fluid flow, finance and other areas. This paper presents high-order accurate Runge-Kutta local discontinuous Galerkin (DG) methods for one- and two-dimensional fractional diffusion equations containing derivatives of fractional order in space. The Caputo derivative is chosen as the representation of spatial derivative, because it may represent the fractional derivative by an integral operator. Some numerical examples show that the convergence orders of the proposed local Pk-DG methods are O(hk+1) both in one and two dimensions, where Pk denotes the space of the real-valued polynomials with degree at most k.

English

English

Numerical method in reproducing kernel space for an inverse source problem for the fractional diffusion equation

We propose a new numerical method for reproducing kernel Hilbert space to solve an inverse source problem for a two-dimensional fractional diffusion equation, where we are required to determine an x-dependent function in a source term by data at the final time. The exact solution is represented in the form of a series and the approximation solution is obtained by truncating the series. Furthermore, a technique is proposed to improve some of the existing methods. We prove that the numerical method is convergent under an a priori assumption of the regularity of solutions. The method is simple to implement. Our numerical result shows that our method is effective and that it is robust against noise in L2-space in reconstructing a source function.

Parameter identification in fractional differential equations

This article investigates the fractional derivative order identification, the coefficient identification, and the source identification in the fractional diffusion problems. If 1 < α < 2, we prove the unique determination of the fractional derivative order and the diffusion coefficient p(x) by ∫0tu(0,s)ds, 0 < t < T for one-dimensional fractional diffusion-wave equations. Besides, if 0 < α < 1, we show the unique determination of the source term f(x,y) by U(0,0,t), 0 <t <T for two-dimensional fractional diffusion equations. Here, α denotes the fractional derivative order over t.

High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations

High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations

An O( N log 2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices. Consequently, the numerical solution of these methods often require computational work of O( N 3) per time step and memory of O( N 2) for where N is the number of grid points. In this paper we develop a fast alternating-direction implicit finite difference method for space-fractional diffusion equations in two space dimensions. The method only requires computational work of O( N log 2 N) per time step and memory of O( N), while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination. Our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 2 months and 1 week consumed by a traditional finite difference method to 1.5 h, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method.

Application of Adomian’s Decomposition Method for the Analytical Solution of Space Fractional Diffusion Equation

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow, finance and others areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by Adomian’s decomposition method (ADM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present techniques. The present method performs extremely well in terms of efficiency and simplicity.

Application of He’s Variational Iteration Method for the Analytical Solution of Space Fractional Diffusion Equation

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow, finance and others areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by variational iteration method (VIM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present techniques. The present method performs extremely well in terms of efficiency and simplicity.

Analytical solution for the space fractional diffusion equation by two-step Adomian Decomposition Method

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical superdiffusive problems in fluid flow, finance and other areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by two-step Adomian Decomposition Method (TSADM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form and then their solutions have been represented graphically. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present technique. The solutions obtained by the standard decomposition method have been numerically evaluated and presented in the form of tables and then compared with those obtained by TSADM. The present TSADM performs extremely well in terms of efficiency and simplicity.

Application of modified decomposition method for the analytical solution of space fractional diffusion equation

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow, finance and others areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by Modified decomposition method (MDM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form. The decomposition series analytic solution of the problem is quickly obtained by observing the existence of the self-cancelling “noise” phenomenon. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present technique. The present method performs extremely well in terms of efficiency and simplicity.