Method For Fractional Diffusion Equations Research Articles

Exponential Convergence and Computational Efficiency of BURA-SD Method for Fractional Diffusion Equations in Polygons

In this paper, we develop a new Best Uniform Rational Approximation-Semi-Discrete (BURA-SD) method taking into account the singularities of the solution of fractional diffusion problems in polygonal domains. The complementary capabilities of the exponential convergence rate of BURA-SD and the hp FEM are explored with the aim of maximizing the overall performance. A challenge here is the emerging singularly perturbed diffusion–reaction equations. The main contributions of this paper include asymptotically accurate error estimates, ending with sufficient conditions to balance errors of different origins, thereby guaranteeing the high computational efficiency of the method.

RBF-Based Local Meshless Method for Fractional Diffusion Equations

The fractional diffusion equation is one of the important recent models that can efficiently characterize various complex diffusion processes, such as in inhomogeneous or heterogeneous media or in porous media. This article provides a method for the numerical simulation of time-fractional diffusion equations. The proposed scheme combines the local meshless method based on a radial basis function (RBF) with Laplace transform. This scheme first implements the Laplace transform to reduce the given problem to a time-independent inhomogeneous problem in the Laplace domain, and then the RBF-based local meshless method is utilized to obtain the solution of the reduced problem in the Laplace domain. Finally, Stehfest’s method is utilized to convert the solution from the Laplace domain into the real domain. The proposed method uses Laplace transform to handle the fractional order derivative, which avoids the computation of a convolution integral in a fractional order derivative and overcomes the effect of time-stepping on stability and accuracy. The method is tested using four numerical examples. All the results demonstrate that the proposed method is easy to implement, accurate, efficient and has low computational costs.

A New Modified Technique of Adomian Decomposition Method for Fractional Diffusion Equations with Initial-Boundary Conditions

In general, solving fractional partial differential equations either numerically or analytically is a difficult task. However, mathematicians have tried their best to make the task easy and promoted various techniques for their solutions. In this regard, a very prominent and accurate technique, which is known as the new technique of the Adomian decomposition method, is developed and presented for the solution of the initial-boundary value problem of the diffusion equation with fractional view analysis. The suggested model is an important mathematical model to study the behavior of degrees of memory in diffusing materials. Some important results for the given model at different fractional orders of the derivatives are achieved. Graphs show the obtained results to confirm the accuracy and validity of the suggested technique. These results are in good contact with the physical dynamics of the targeted problems. The obtained results for both fractional and integer orders problems are explained through graphs and tables. Tables and graphs support the physical behavior of each problem and the best of physical analysis. From the results, it is concluded that as the fractional order derivative is changed, the graphs or paths of dynamics are also changed. Therefore, we now choose the best solution or dynamic of the problem at a particular derivative order. It is analyzed that the present technique is one of the best techniques to handle the solutions of fractional partial differential equations having initial and boundary conditions (BCs), which are very rare in literature. Furthermore, a small number of calculations are done to achieve a very high rate of convergence, which is the novelty of the present research work. The proposed method provides the series solution with twice recursive formulae to increase the desired accuracy and is preferred among the best techniques to find the solution of fractional partial differential equations with mixed initials and BCs.

Preconditioning for symmetric positive definite systems in balanced fractional diffusion equations

In this paper, we study the finite volume discretization method for balanced fractional diffusion equations where the fractional differential operators are comprised of both Riemann-Liouville and Caputo fractional derivatives. The main advantage of this approach is that new symmetric positive definite Toeplitz-like linear systems can be constructed for solving balanced fractional diffusion equations when diffusion functions are non-constant. It is different from non-symmetric Toeplitz-like linear systems usually obtained by existing numerical methods for fractional diffusion equations. The preconditioned conjugate gradient method with circulant and banded preconditioners can be applied to solve the proposed symmetric positive definite Toeplitz-like linear systems. Numerical examples, for both of one- and two- dimensional cases, are given to demonstrate the good accuracy of the finite volume discretization method and the fast convergence of the preconditioned conjugate gradient method.

A GPIU method for fractional diffusion equations

The fractional diffusion equations can be discretized by applying the implicit finite difference scheme and the unconditionally stable shifted Grünwald formula. Hence, the generating linear system has a real Toeplitz structure when the two diffusion coefficients are non-negative constants. Through a similarity transformation, the Toeplitz linear system can be converted to a generalized saddle point problem. We use the generalization of a parameterized inexact Uzawa (GPIU) method to solve such a kind of saddle point problem and give a new algorithm based on the GPIU method. Numerical results show the effectiveness and accuracy for the new algorithm.

Analytical solutions and numerical schemes of certain generalized fractional diffusion models

We give the analytical solutions of the fractional diffusion equations in one-, and two-dimensional space described by the Caputo left generalized fractional derivative. We introduce the forward Euler method for fractional diffusion equations represented by the Caputo left generalized fractional derivative. The contribution of this paper is to evaluate the impact of the second parameter of the Caputo left generalized fractional derivative in the behavior of the analytical solutions of the fractional diffusion equations, and to propose a numerical method for the generalized fractional diffusion equations. We will present the difference existing between the classical diffusion equation, the fractional diffusion equation described by Caputo fractional derivative and the fractional diffusion equation expressed by the Caputo left generalized fractional derivative. The Fourier-sine-Laplace-transform method is used to determine the analytical solutions of the fractional diffusion equations described by the Caputo left generalized fractional derivative. Some particular cases of diffusion equations are discussed, and the numerical simulations of their analytical solutions are presented and analyzed.

Solving fractional diffusion equations by Sinc and radial basis functions

In this study, we approximate the solution of the fractional diffusion equations based on Gaussian radial basis function (GRBF). Our approach is based on the Caputo fractional derivative and the combination of GRBF and Sinc function, here the GRBF direct and GRBF-QR methods are developed. The Sinc quadrature rule combined with double exponential transformation (DE) has been used to approximate the fractional integral. Three examples have been considered to test the presented methods, we compared the results to verify the effectiveness of the presented methods against recent existing methods diven by [A radial basis functions method for fractional diffusion equations, J. Comput. Phys. 238 (2013) 71–81; An efficient Chebyshev-tau method for solving the space fractional diffusion equations, Appl. Math. Comput. 224 (2013) 259–267; A tau approach for solution of the space fractional diffusion equations, Comput. Math. Appl. 62 (2011) 1135–1142], and also conclude that GRBF-QR-Sinc method demonstrates better accuracy.

Residual Power Series Method for Fractional Diffusion Equations

Residual Power Series Method for Fractional Diffusion Equations

On numerical contour integral method for fractional diffusion equations with variable coefficients

In this note, we study numerical contour integral method for fractional diffusion equations with variable coefficients. We find that the method can be applied if the diffusion coefficients and order of differentiation satisfy a certain condition. This work extends a recent result.

On ADI-like iteration method for fractional diffusion equations

The fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of the identity matrix with two diagonal-times-Toeplitz matrices. In this paper, the alternating direction implicit (ADI)-like method based on the classical ADI iteration is proposed to solve the discretized linear system. Theoretical analyses show that the ADI-like iteration method is convergent. Each iteration of this method requires the solutions of two linear systems with Hessenberg coefficient matrices. These two systems can be solved effectively by Hessenberg LU factorization. Numerical examples are presented to illustrate the effectiveness of the ADI-like iteration method.

Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes

This paper, as the sequel to previous work, develops numerical schemes for fractional diffusion equations on a two-dimensional finite domain with triangular meshes. We adopt the nodal discontinuous Galerkin methods for the full spatial discretization by the use of high-order nodal basis, employing multivariate Lagrange polynomials defined on the triangles. Stability analysis and error estimates are provided, which shows that if polynomials of degree N are used, the methods are (N+1)-th order accurate for general triangulations. Finally, the performed numerical experiments confirm the optimal order of convergence.

Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations

The computation time required by standard finite difference methods with fixed timesteps for solving fractional diffusion equations is usually very large because the number of operations required to find the solution scales as the square of the number of timesteps. Besides, the solutions of these problems usually involve markedly different time scales, which leads to quite inhomogeneous numerical errors. A natural way to address these difficulties is by resorting to adaptive numerical methods where the size of the timesteps is chosen according to the behaviour of the solution. A key feature of these methods is then the efficiency of the adaptive algorithm employed to dynamically set the size of every timestep. Here we discuss two adaptive methods based on the step-doubling technique. These methods are, in many cases, immensely faster than the corresponding standard method with fixed timesteps and they allow a tolerance level to be set for the numerical errors that turns out to be a good indicator of the actual errors.

Local Discontinuous Galerkin methods for fractional diffusion equations

We consider the development and analysis of local discontinuous Galerkin methods for fractional diffusion problems in one space dimension, characterized by having fractional derivatives, parameterized by beta in [1,2]. After demonstrating that a classic approach fails to deliver optimal order of convergence, we introduce a modified local numerical flux which exhibits optimal order of convergence O(h^(k+1)) uniformly across the continuous range between pure advection (beta=1) and pure diffusion (beta=2). In the two classic limits, known schemes are recovered. We discuss stability and present an error analysis for the space semi-discretized scheme, which is supported through a few examples.

Explicit and Implicit Methods for Fractional Diffusion Equations with the Riesz Fractional Derivative

In this paper, a fractional diffusion equation by using the explicit numerical method in a finite domain with second-order accuracy which includes the Riesz fractional derivative approximation is studied. For the Riesz fractional derivative approximation, ''fractional centered derivative'' approach is used. The error of the Riesz fractional derivative to the fractional centered difference is calculated. We used the implicit numerical method to solve the fractional diffusion equation and also investigated the stability of explicit and implicit methods. The maximum error of the implicit method for fractional diffusion equation with using fractional centered difference approach is shown by using the numerical results.

A finite difference method with non-uniform timesteps for fractional diffusion equations

An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the behavior of the solution in order to keep the numerical errors small without the penalty of a huge computational cost. The method is unconditionally stable and convergent. In fact, it is shown that consistency and stability implies convergence for a rather general class of fractional finite difference methods to which the present method belongs. The huge computational advantage of adaptive methods against fixed step methods for fractional diffusion equations is illustrated by solving the problem of the dispersion of a flux of subdiffusive particles stemming from a point source.

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.

A direct O( N log 2 N) finite difference method for fractional diffusion equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods have full coefficient matrices which require storage of O( N 2) and computational cost of O( N 3) where N is the number of grid points. In this paper we develop a fast finite difference method for fractional diffusion equations, which only requires storage of O( N) and computational cost of O( N log 2 N) while retaining the same accuracy and approximation property as the regular finite difference method. Numerical experiments are presented to show the utility of the method.

Weighted average finite difference methods for fractional diffusion equations

A class of finite difference methods for solving fractional diffusion equations is considered. These methods are an extension of the weighted average methods for ordinary (non-fractional) diffusion equations. Their accuracy is of order (Δx)2 and Δt, except for the fractional version of the Crank–Nicholson method, where the accuracy with respect to the timestep is of order (Δt)2 if a second-order approximation to the fractional time-derivative is used. Their stability is analyzed by means of a recently proposed procedure akin to the standard von Neumann stability analysis. A simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, is found and checked numerically. Some examples are provided in which the new methods’ numerical solutions are obtained and compared against exact solutions.