A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS

In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finite difference method with an accuracy of order3-α,and the space discretization is based on the LDG method.For the finite difference method,we summarize and supplement some previous work by others,and apply it to the analysis of the convergence and stability of the proposed scheme.The optimal error estimate is obtained in the L2norm,indicating that the scheme has temporal(3-α)th-order accuracy and spatial(k+1)th-order accuracy,where k denotes the highest degree of a piecewise polynomial in discontinuous finite element space.The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.

Finite Difference Schemes for Time-Space Fractional Diffusion Equations in One-and Two-Dimensions

In this paper,finite difference schemes for solving time-space fractional diffusion equations in one dimension and two dimensions are proposed.The temporal derivative is in the Caputo-Hadamard sense for both cases.The spatial derivative for the one-dimensional equation is of Riesz definition and the two-dimensional spatial derivative is given by the fractional Laplacian.The schemes are proved to be unconditionally stable and convergent.The numerical results are in line with the theoretical analysis.

THE QUASI-BOUNDARY VALUE METHOD FOR IDENTIFYING THE INITIAL VALUE OF THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION

In this article,we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation.This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem.We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule.Some numerical results in one-dimensional case and two-dimensional case show that our method is efficient and stable.

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.

Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Diffusion Equation

In this paper,we develop a novel fi nite-diff erence scheme for the time-Caputo and space-Riesz fractional diff usion equation with convergence order O(τ^2−α+h^2).The stability and convergence of the scheme are analyzed by mathematical induction.Moreover,some numerical results are provided to verify the eff ectiveness of the developed diff erence scheme.

Analysis of anomalous transport based on radial fractional diffusion equation

Anomalous transport in magnetically confined plasmas is investigated by radial fractional transport equations.It is shown that for fractional transport models,hollow density profiles are formed and uphill transports can be observed regardless of whether the fractional diffusion coefficients(FDCs)are radially dependent or not.When a radially dependent FDC<D_(α)(r)1 is imposed,compared with the case under=D_(α)(r)1.0,it is observed that the position of the peak of the density profile is closer to the core.Further,it is found that when FDCs at the positions of source injections increase,the peak values of density profiles decrease.The non-local effect becomes significant as the order of fractional derivative a 1 and causes the uphill transport.However,as a 2,the fractional diffusion model returns to the standard model governed by Fick’s law.

Local Discontinuous Galerkin Methods with Novel Basis for Fractional Diffusion Equations with Non-smooth Solutions

In this paper,we develop novel local discontinuous Galerkin(LDG)methods for fractional diffusion equations with non-smooth solutions.We consider such problems,for which the solutions are not smooth at boundary,and therefore the traditional LDG methods with piecewise polynomial solutions suffer accuracy degeneracy.The novel LDG methods utilize a solution information enriched basis,simulate the problem on a paired special mesh,and achieve optimal order of accuracy.We analyze the L2 stability and optimal error estimate in L2-norm.Finally,numerical examples are presented for validating the theoretical conclusions.

A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations

After discretization by the finite volume method,the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure.The theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized matrix.Moreover,we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations,which is unconditionally convergent for any positive constant.Meanwhile,the iteration methods introduce a new preconditioner for Krylov subspace methods.Numerical experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner,compared with the existing approaches.

An Indirect Finite Element Method for Variable-Coefficient Space-Fractional Diffusion Equations and Its Optimal-Order Error Estimates

We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension.By the representation formula of the solutions u(x)to the proposed variable coefficient models in terms of v(x),the solutions to the constant coefficient analogues,we apply finite element methods for the constant coefficient fractional diffusion equations to solve for the approximations vh(x)to v(x)and then obtain the approximations uh(x)of u(x)by plugging vh(x)into the representation of u(x).Optimal-order convergence estimates of u(x)−uh(x)are proved in both L2 and Hα∕2 norms.Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.

A Note on Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Diffusion Equation

Recently,Zhang and Ding developed a novel finite difference scheme for the time-Caputo and space-Riesz fractional diffusion equation with the convergence order 0(ι^(2-a)+h^(2))in Zhang and Ding(Commun.Appl.Math.Comput.2(1):57-72,2020).Unfortunately,they only gave the stability and convergence results for a∈(0,1)andβ∈[7/8+^(3)√621+48√87+19/8^(3)√621+48√87,2]In this paper,using a new analysis method,we find that the original difference scheme is unconditionally stable and convergent with orderΟ(ι^(2-a)+h^(2))for all a∈(0,1)andβ∈(1,2].Finally,some numerical examples are given to verify the correctness of the results.

Green Function of Generalized Time Fractional Diffusion Equation Using Addition Formula of Mittag-Leffler Function

In this paper, we use the Mittag-Leffler addition formula to solve the Green function of generalized time fractional diffusion equation in the whole plane and prove the convergence of the Green function.

A DIRECT DISCONTINUOUS GALERKIN METHOD FOR TIME FRACTIONAL DIFFUSION EQUATIONS WITH FRACTIONAL DYNAMIC BOUNDARY CONDITIONS

This paper deals with the numerical approximation for the time fractional diffusion problem with fractional dynamic boundary conditions.The well-posedness for the weak solutions is studied.A direct discontinuous Galerkin approach is used in spatial direction under the uniform meshes,together with a second-order Alikhanov scheme is utilized in temporal direction on the graded mesh,and then the fully discrete scheme is constructed.Furthermore,the stability and the error estimate for the full scheme are analyzed in detail.Numerical experiments are also given to illustrate the effectiveness of the proposed method.

EXPONENTIAL TIKHONOV REGULARIZATION METHOD FOR SOLVING AN INVERSE SOURCE PROBLEM OF TIME FRACTIONAL DIFFUSION EQUATION

In this paper,we mainly study an inverse source problem of time fractional diffusion equation in a bounded domain with an over-specified terminal condition at a fixed time.A novel regularization method,which we call the exponential Tikhonov regularization method with a parameter γ,is proposed to solve the inverse source problem,and the corresponding convergence analysis is given under a-priori and a-posteriori regularization parameter choice rules.Whenγis less than or equal to zero,the optimal convergence rate can be achieved and it is independent of the value of γ.However,when γ is greater than zero,the optimal convergence rate depends on the value of γ which is related to the regularity of the unknown source.Finally,numerical experiments are conducted for showing the effectiveness of the proposed exponential regularization method.

A Posteriori Error Estimates of the Galerkin Spectral Methods for Space-Time Fractional Diffusion Equations

In this paper,an initial boundary value problem of the space-time fractional diffusion equation is studied.Both temporal and spatial directions for this equation are discreted by the Galerkin spectral methods.And then based on the discretization scheme,reliable a posteriori error estimates for the spectral approximation are derived.Some numerical examples are presented to verify the validity and applicability of the derived a posteriori error estimator.

An analytical solution of multi-dimensional space fractional diffusion equations with variable coefficients

In this paper,we have considered the multi-dimensional space fractional diffusion equations with variable coefficients.The fractional operators(derivative/integral)are used based on the Caputo definition.This study provides an analytical approach to determine the analytical solution of the considered problems with the help of the two-step Adomian decomposition method(TSADM).Moreover,new results have been obtained for the existence and uniqueness of a solution by using the Banach contraction principle and a fixed point theorem.We have extended the dimension of the space fractional diffusion equations with variable coefficients into multi-dimensions.Finally,the generalized problems with two different types of the forcing term have been included demonstrating the applicability and high efficiency of the TSADM in comparison to other existing numerical methods.The diffusion coefficients do not require to satisfy any certain conditions/restrictions for using the TSADM.There are no restrictions imposed on the problems for diffusion coefficients,and a similar procedures of the TSADM has followed to the obtained analytical solution for the multi-dimensional space fractional diffusion equations with variable coefficients.

Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation

In this paper,we investigate initial boundary value problems of the spacetime fractional diffusion equation and its numerical solutions.Two definitions,i.e.,Riemann-Liouville definition and Caputo one,of the fractional derivative are considered in parallel.In both cases,we establish the well-posedness of the weak solution.Moveover,based on the proposed weak formulation,we construct an efficient spectral method for numerical approximations of the weak solution.The main contribution of this work are threefold:First,a theoretical framework for the variational solutions of the space-time fractional diffusion equation is developed.We find suitable functional spaces and norms in which the space-time fractional diffusion problem can be formulated into an elliptic weak problem,and the existence and uniqueness of the weak solution are then proved by using existing theory for elliptic problems.Secondly,we show that in the case of Riemann-Liouville definition,the well-posedness of the space-time fractional diffusion equation does not require any initial conditions.This contrasts with the case of Caputo definition,in which the initial condition has to be integrated into the weak formulation in order to establish the well-posedness.Finally,thanks to the weak formulation,we are able to construct an efficient numerical method for solving the space-time fractional diffusion problem.

Numerical Methods for Semilinear Fractional Diffusion Equations with Time Delay

In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical significance.We develop a novel implicit alternating direction method that can effectively and efficiently tackle the RSFDEs in both two and three dimensions.The numerical method is proved to be uniquely solvable,stable and convergent with second order accuracy in both space and time.Numerical results are presented to verify the accuracy and efficiency of the proposed numerical scheme.

Finite element methods for fractional diffusion equations

Due to the successful applications in engineering,physics,biology,finance,etc.,there has been substantial interest in fractional diffusion equations over the past few decades,and literatures on developing and analyzing efficient and accurate numerical methods for reliably simulating such equations are vast and fast growing.This paper gives a concise overview on finite element methods for these equations,which are divided into time fractional,space fractional and time-space fractional diffusion equations.Besides,we also involve some relevant topics on the regularity theory,the well-posedness,and the fast algorithm.

Application of low-dimensional finite element method to fractional diffusion equation

Classical finite element method(FEM)has been applied to solve some fractional differential equations,but its scheme has too many degrees of freedom.In this paper,a low-dimensional FEM,whose number of basis functions is reduced by the theory of proper orthogonal decomposition(POD)technique,is proposed for the time fractional diffusion equation in two-dimensional space.The presented method has the properties of low dimensions and high accuracy so that the amount of computation is decreased and the calculation time is saved.Moreover,error estimation of the method is obtained.Numerical example is given to illustrate the feasibility and validity of the low-dimensional FEM in comparison with traditional FEM for the time fractional differential equations.

Non-probabilistic solutions of imprecisely defined fractional-order diffusion equations

The fractional diffusion equation is one of the most important partial differential equations（PDEs） to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0 〈 α≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method（ADM） symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases.