A new reliable algorithm based on the sinc function for the time fractional diffusion equation

This paper deals with the numerical solution of time fractional diffusion equation. In this work, we consider the fractional derivative in the sense of Riemann-Liouville. At first, the time fractional derivative is discretized by integrating both sides of the equation with respect to the time variable and we arrive at a semi–discrete scheme. The stability and convergence of time discretized scheme are proven by using the energy method. Also we show that the convergence order of this scheme is O(τ2?α). Then we use the sinc collocation method to approximate the solution of semi–discrete scheme and show that the problem is reduced to a Sylvester matrix equation. Besides by performing some theorems, the exponential convergence rate of sinc method is illustrated. The numerical experiments are presented to show the excellent behavior and high accuracy of the proposed hybrid method in comparison with some other well known methods.     

Quenching phenomenon of a time‐fractional diffusion equation with singular source term

In this paper, a time‐fractional diffusion equation with singular source term is considered. The Caputo fractional derivative with order 0<α ?1 is applied to the temporal variable. Under specific initial and boundary conditions, we find that the time‐fractional diffusion equation presents quenching solution that is not globally well‐defined as time goes to infinity. The quenching time is estimated by using the eigenfunction of linear fractional diffusion equation. Moreover, by implementing a finite difference scheme, we give some numerical simulations to demonstrate the theoretical analysis. Copyright © 2017 John Wiley & Sons, Ltd.     

Finite Difference/Collocation Method for Two-Dimensional Sub-Diffusion Equation with Generalized Time Fractional Derivative

In this paper, we propose a finite difference/collocation method for two-dimensional time fractional diffusion equation with generalized fractional operator. The main purpose of this paper is to design a high order numerical scheme for the new generalized time fractional diffusion equation. First, a finite difference approximation formula is derived for the generalized time fractional derivative, which is verified with order $2-\alpha$ $(0<\alpha<1)$. Then, collocation method is introduced for the two-dimensional space approximation. Unconditional stability of the scheme is proved. To make the method more efficient, the alternating direction implicit method is introduced to reduce the computational cost. At last, numerical experiments are carried out to verify the effectiveness of the scheme.     

带非线性源项的Riesz回火分数阶扩散方程的预估校正方法

本文针对带非线性源项的Riesz回火分数阶扩散方程,利用预估校正方法离散时间偏导数,并用修正的二阶Lubich回火差分算子逼近Riesz空间回火的分数阶偏导数,构造出一类新的数值格式.给出了数值格式在一定条件下的稳定性与收敛性分析,且该格式的时间与空间收敛阶均为二阶.数值试验表明数值方法是有效的.     

带非线性源项的双侧空间分数阶扩散方程的隐式中点方法

本文用隐式中点方法离散一阶时间偏导数,并用拟紧差分算子逼近Riemann-Liouville空间分数阶偏导数,构造了求解带非线性源项的空间分数阶扩散方程的数值格式.给出了数值方法的稳定性和收敛性分析.数值试验表明数值方法是有效的.     

Maximum principles for a time–space fractional diffusion equation

In this paper, we focus on maximum principles of a time–space fractional diffusion equation. Maximum principles for classical solution and weak solution are all obtained by using properties of the time fractional derivative operator and the fractional Laplace operator. We deduce maximum principles for a full fractional diffusion equation, other than time-fractional and spatial-integer order diffusion equations.     

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α? (0,1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.     

The dependence on fractional orders of mild solutions to the fractional diffusion equation with memory

In this paper we investigate the Cauchy problem for a fractional diffusion equation and the time-fractional derivative is taken in the Caputo type sense. We give a representation of solutions under Fourier series and analyze initial value problems for the semi-linear fractional diffusion equation with a memory term. We also discuss the stability of the fractional derivative order for the time under some assumptions on the input data. Our key idea is to use Mittag-Leffler functions, the Banach fixed point theorem, and some Sobolev embeddings.     

Inverse problem for a space‐time fractional diffusion equation: Application of fractional Sturm‐Liouville operator

An inverse problem of determining a time‐dependent source term from the total energy measurement of the system (the over‐specified condition) for a space‐time fractional diffusion equation is considered. The space‐time fractional diffusion equation is obtained from classical diffusion equation by replacing time derivative with fractional‐order time derivative and Sturm‐Liouville operator by fractional‐order Sturm‐Liouville operator. The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases are discussed, and particular examples are provided.     

Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order 0 < α < 1 ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent withO(Τ +h ²), where Τ andh are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.     

A numerical technique based on B-spline for a class of time-fractional diffusion equation

This paper presents an efficient numerical technique for solving a class of time-fractional diffusion equation. The time-fractional derivative is described in the Caputo form. The L1 scheme is used for discretization of Caputo fractional derivative and a collocation approach based on sextic B-spline basis function is employed for discretization of space variable. The unconditional stability of the fully-discrete scheme is analyzed. Two numerical examples are considered to demonstrate the accuracy and applicability of our scheme. The proposed scheme is shown to be sixth order accuracy with respect to space variable and (2 − α)-th order accuracy with respect to time variable, where α is the order of temporal fractional derivative. The numerical results obtained are compared with other existing numerical methods to justify the advantage of present method. The CPU time for the proposed scheme is provided.     

Analysis of the time fractional nonlinear diffusion equation from diffusion process

Under investigation in this paper is a time fractional nonlinear diffusion equation which can be utilized to express various diffusion processes. The symmetry of this considered equation has been obtained via fractional Lie group approach with the sense of Riemann-Liouville (R-L) fractional derivative. Based on the symmetry, this equation can be changed into an ordinary differential equation of fractional order. Moreover, some new invariant solutions of this considered equation are found. Lastly, utilising the Noether theorem and the general form of Noether type theorem, the conservation laws are yielded to the time fractional nonlinear diffusion equation, respectively. Our discovery that there are no conservation laws under the general form of Noether type theorem case. This result tells us the symmetry of this equation is not variational symmetry of the considered functional. These rich results can give us more information to interpret this equation.     

The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equations

In this paper, we apply the dual reciprocity boundary elements method for the numerical solution of two‐dimensional linear and nonlinear time‐fractional modified anomalous subdiffusion equations and time‐fractional convection–diffusion equation. The fractional derivative of problems is described in the Riemann–Liouville and Caputo senses. We employ the linear radial basis function for interpolation of the nonlinear, inhomogeneous and time derivative terms. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity which appears in the nonlinear problems under consideration. The accuracy and efficiency of the proposed schemes are checked by five test problems. The proposed method is employed for solving some examples in two dimensions on unit square and also in complex regions to demonstrate the efficiency of the new technique. Copyright © 2016 John Wiley & Sons, Ltd.     

Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation

In this article, an efficient algorithm for the evaluation of the Caputo fractional derivative and the superconvergence property of fully discrete finite element approximation for the time fractional subdiffusion equation are considered. First, the space semidiscrete finite element approximation scheme for the constant coefficient problem is derived and supercloseness result is proved. The time discretization is based on the L1‐type formula, whereas the space discretization is done using, the fully discrete scheme is developed. Under some regularity assumptions, the superconvergence estimate is proposed and analyzed. Then, extension to the case of variable coefficients is also discussed. To reduce the computational cost, the fast evaluation scheme of the Caputo fractional derivative to solve the fractional diffusion equations is designed. Finally, numerical experiments are presented to support the theoretical results.     

Quenching of combustion explosion model with balanced space-fractional derivative

Balanced space-fractional derivative is usually applied in modelling the state-dependence, isotropy, and anisotropy in diffusion phenomena. In this paper, we introduce a class of space-fractional reaction-diffusion model with singular source term arising in combustion process. The fractional derivative employed in this model is defined in the sum of left-sided and right-sided Riemann-Liouville fractional derivatives. With assistance of Kaplan's first eigenvalue method, we prove that the classic solution of this model may not be globally well-defined, and the heat conduction governed by this model depends on the order of fractional derivative, the parameters in the equation, and the length of spatial interval. Finite difference method is implemented to solve this model, and an adaptive strategy is applied to improve the computational efficiency. The positivity, monotonicity, and stability of the numerical scheme are discussed. Numerical simulation and observation of the quenching and stationary solutions coincide the theoretical studies.     

The space-time fractional diffusion equation with Caputo derivatives

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order β∈(0, 2] and the first-order time derivative with Caputo derivative of order α∈(0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.     

Solving a final value fractional diffusion problem by boundary condition regularization

The evolution process of fractional order describes some phenomenon of anomalous diffusion and transport dynamics in complex system. The equation containing fractional derivatives provides a suitable mathematical model for describing such a process. The initial boundary value problem is hard to solve due to the nonlocal property of the fractional order derivative. We consider a final value problem in a bounded domain for fractional evolution process with respect to time, which means to recover the initial state for some slow diffusion process from its present status. For this ill-posed problem, we construct a regularizing solution using quasi-reversible method. The well-posedness of the regularizing solution as well as the convergence property is rigorously analyzed. The advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented. Numerical examples are presented to show the validity of the proposed scheme.     

On a terminal value problem for a generalization of the fractional diffusion equation with hyper-Bessel operator

In this paper, we consider an inverse problem of recovering the initial value for a generalization of time-fractional diffusion equation, where the time derivative is replaced by a regularized hyper-Bessel operator. First, we investigate the existence and regularity of our terminal value problem. Then we show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Tikhonov regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.     

Modeling non-Darcian flow and solute transport in porous media with the Caputo–Fabrizio derivative

In this study, the non-Darcian flow and solute transport in porous media are modeled with a revised Caputo derivative called the Caputo–Fabrizio fractional derivative. The fractional Swartzendruber model is proposed for the non-Darcian flow in porous media. Furthermore, the normal diffusion equation is converted into a fractional diffusion equation in order to describe the diffusive transport in porous media. The proposed Caputo–Fabrizio fractional derivative models are addressed analytically by applying the Laplace transform method. Sensitivity analyses were performed for the proposed models according to the fractional derivative order. The fractional Swartzendruber model was validated based on experimental data for water flows in soil–rock mixtures. In addition , the fractional diffusion model was illustrated by fitting experimental data obtained for fluid flows and chloride transport in porous media. Both of the proposed fractional derivative models were highly consistent with the experimental results.     

NUMERICAL SOLUTION OF THE SPACE FRACTIONAL DIFFERENTIAL EQUATION

In this paper, a space fractional differential equation is considered. The equation is obtained from the parabolic equation containing advection, diffusion and reaction terms by replacing the second order derivative in space by a fractional derivative in space of order. An implicit finite difference approximation for this equation is presented. The stability and convergence of the finite difference approximation are proved. A fractional-order method of lines is also presented. Finally, some numerical results are given.