A new regularization method for a Cauchy problem of the time fractional diffusion equation

In this paper, we consider a Cauchy problem of the time fractional diffusion equation (TFDE). Such problem is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α (0 < α ≤ 1). We show that the Cauchy problem of TFDE is severely ill-posed and further apply a new regularization method to solve it based on the solution given by the Fourier method. Convergence estimates in the interior and on the boundary of solution domain are obtained respectively under different a-priori bound assumptions for the exact solution and suitable choices of regularization parameters. Finally, numerical examples are given to show that the proposed numerical method is effective.     

A local radial basis function collocation method to solve the variable‐order time fractional diffusion equation in a two‐dimensional irregular domain

The local radial basis function (RBF) method is a promising solver for variable‐order time fractional diffusion equation (TFDE), as it overcomes the computational burden of the traditional global method. Application of the local RBF method is limited to Fickian diffusion, while real‐world diffusion is usually non‐Fickian in multiple dimensions. This article is the first to extend the application of the local RBF method to two‐dimensional, variable‐order, time fractional diffusion equation in complex shaped domains. One of the main advantages of the local RBF method is that only the nodes located in the subdomain, surrounding the local point, need to be considered when calculating the numerical solution at this point. This approach can perform well with large scale problems and can also mitigate otherwise ill‐conditioned problems. The proposed numerical approach is checked against two examples with curved boundaries and known analytical solutions. Shape parameter and subdomain node number are investigated for their influence on the accuracy of the local RBF solution. Furthermore, quantitative analysis, based on root‐mean‐square error, maximum absolute error, and maximum error of the partial derivative indicates that the local RBF method is accurate and effective in approximating the variable‐order TFDE in two‐dimensional irregular domains.     

Analysis of an Implicit Finite Difference Scheme for Time Fractional Diffusion Equation

Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper.     

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.     

Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations

We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg–deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained.     

On the numerical solutions for the fractional diffusion equation

Fractional differential equations have recently been applied in various area of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional diffusion equation (FDE) is considered. The fractional derivative is described in the Caputo sense. The method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FDE to a system of ordinary differential equations, which solved by the finite difference method. Numerical simulation of FDE is presented and the results are compared with the exact solution and other methods.     

The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. Numerical methods and analysis of stability and convergence of numerical scheme for the variable fractional order partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the space-time variable fractional order diffusion equation on a finite domain. It is worth mentioning that here we use the Coimbra-definition variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation is proposed and then the stability and convergence of the numerical scheme are investigated. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.     

Numerical techniques for simulating a fractional mathematical model of epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider the numerical simulation of a fractional mathematical model of epidermal wound healing (FMM-EWH), which is based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in the advection and diffusion terms belong to the intervals (0,1) or (1,2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of Riemann-Liouville and Grünwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.     

Finite element method for space-time fractional diffusion equation

In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order β (1 < β ≤ 2), and the first order time derivative by a Caputo fractional derivative of order γ (0 < γ ≤ 1). For the 0 < γ < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(τ^2?γ + h²) and O(τ² + h²), respectively, in which τ is the time step size and h is the space step size. And for the case γ = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(τ² + h²) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.     

RBFs approximation method for time fractional partial differential equations

In this paper, radial basis functions (RBFs) approximation method is implemented for time fractional advection–diffusion equation on a bounded domain. In this method the first order time derivative is replaced by the Caputo fractional derivative of order α ∈ (0, 1], and spatial derivatives are approximated by the derivative of interpolation in the Kansa method. Stability and convergence of the method is discussed. Several numerical examples are include to demonstrate effectiveness and accuracy of the method.     

Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β ₁∈(0,1) and β ₂∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.     

Analytical approximate solutions of Riesz fractional diffusion equation and Riesz fractional advection–dispersion equation involving nonlocal space fractional derivatives

In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second‐order space derivative with the Riesz fractional derivative of order α∈(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first‐order and second‐order space derivatives with the Riesz fractional derivatives of order β∈(0,1] and of order α∈(1,2] respectively. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

A new fractional finite volume method for solving the fractional diffusion equation

The inherent heterogeneities of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement. One approach to model solute transport through such media is by fractional diffusion equations with a space–time dependent variable coefficient. In this paper, a two-sided space fractional diffusion model with a space–time dependent variable coefficient and a nonlinear source term subject to zero Dirichlet boundary conditions is considered.Some finite volume methods to solve a fractional differential equation with a constant dispersion coefficient have been proposed. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann–Liouville fractional derivatives at control volume faces in terms of function values at the nodes. However, these finite volume methods have not been extended to two-dimensional and three-dimensional problems in a natural manner. In this paper, a new weighted fractional finite volume method with a nonlocal operator (using nodal basis functions) for solving this two-sided space fractional diffusion equation is proposed. Some numerical results for the Crank–Nicholson fractional finite volume method are given to show the stability, consistency and convergence of our computational approach. This novel simulation technique provides excellent tools for practical problems even when a complex transition zone is involved. This technique can be extend to two-dimensional and three-dimensional problems with complex regions.     

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order α(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order β(0,1) and of order α(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.     

Collocation method with Lagrange polynomials for variable-order time-fractional advection–diffusion problems

In this study, we proposed a numerical technique to solve a class of variable-order time-fractional advection–diffusion equations (VOTFADEs) by applying an operational matrix of differentiation based on fractional-order Lagrange polynomials (FOLPs). The variable-order fractional derivative is assumed to be Caputo's derivative. Using the operational matrix and collocation method, the advection–diffusion equation can be reduced to an algebraic system of equations that can be solved using Newton's iterative method. Error analysis also has been carried out for the proposed method. The current approach is simple to use and computer oriented and provides highly accurate approximate solutions. The effectiveness and accuracy of the proposed method are demonstrated using a few numerical examples.     

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.     

On evolutionary equations with material laws containing fractional integrals

A well‐posedness result for a time‐shift invariant class of evolutionary operator equations involving material laws with fractional time‐integrals of order α ? ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time‐)derivative established as a normal operator in a suitable L² type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann‐Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker‐Planck equation, equations describing super‐diffusion and sub‐diffusion processes, and a Kelvin‐Voigt type model in fractional visco‐elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright © 2014 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.     

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.