Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations

The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L2-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.

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.

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.

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.

A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.

High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Diffusion Equation

In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.The unconditional stability of the LDG scheme is proved,and an a priori error estimate with O(h^(k+1)+At^(2))is derived,where k≥0 denotes the index of the basis function.Extensive numerical results with Q^(k)(k=0,1,2,3)elements are provided to confirm our theoretical results,which also show that the second-order convergence rate in time is not impacted by the changed parameter θ.

Finite Element Method for a Kind of Two-Dimensional Space-Fractional Diffusion Equation with Its Implementation

In this article, we consider a two-dimensional symmetric space-fractional diffusion equation in which the space fractional derivatives are defined in Riesz potential sense. The well-posed feature is guaranteed by energy inequality. To solve the diffusion equation, a fully discrete form is established by employing Crank-Nicolson technique in time and Galerkin finite element method in space. The stability and convergence are proved and the stiffness matrix is given analytically. Three numerical examples are given to confirm our theoretical analysis in which we find that even with the same initial condition, the classical and fractional diffusion equations perform differently but tend to be uniform diffusion at last.

Lagrange’s Spectral Collocation Method for Numerical Approximations of Two-Dimensional Space Fractional Diffusion Equation

Due to the ability to model various complex phenomena where classical calculus failed, fractional calculus is getting enormous attention recently. There are several approaches available for numerical approximations of various types of fractional differential equations. For fractional diffusion equations spectral collocation is one of the efficient and most popular ap-proximation techniques. In this research, we introduce spectral collocation method based on Lagrange’s basis polynomials for numerical approximations of two-dimensional (2D) space fractional diffusion equations where spatial fractional derivative is described in Riemann-Liouville sense. We consider four different types of nodes to generate Lagrange’s basis polynomials and as collocation points in the proposed spectral collocation technique. Spectral collocation method converts the diffusion equation into a system of ordinary differential equations (ODE) for time variable and we use 4th order Runge-Kutta method to solve the resulting system of ODE. Two examples are considered to verify the efficiency of different types of nodes in the proposed method. We compare approximated solution with exact solution and find that Lagrange’s spectral collocation method gives very high accuracy approximation. Among the four types of nodes, nodes from Jacobi polynomial give highest accuracy and nodes from Chebyshev polynomials of 1st kind give lowest accuracy in the proposed method.

A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Diffusion Equations

For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable.Numerical results indicate the effectiveness and accuracy of the method and con-firm the analysis.

Exact Solutions of a Generalized Multi-Fractional Nonlinear Diffusion Equation in Radical Symmetry

This paper is devoted to investigating exact solutions of a generalized fractional nonlinear anomalousdiffusion equation in radical symmetry.The presence of external force and absorption is also considered.We firstinvestigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones.Inboth situations,we obtain the corresponding exact solutions,and the solutions found here can have a compact behavioror a long tailed behavior.

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.

A Numerical Algorithm Based on Quadratic Finite Element for Two-Dimensional Nonlinear Time Fractional Thermal Diffusion Model

In this article,a high-order scheme,which is formulated by combining the quadratic finite element method in space with a second-order time discrete scheme,is developed for looking for the numerical solution of a two-dimensional nonlinear time fractional thermal diffusion model.The time Caputo fractional derivative is approximated by using the L2-1formula,the first-order derivative and nonlinear term are discretized by some second-order approximation formulas,and the quadratic finite element is used to approximate the spatial direction.The error accuracy O(h3+
t2)is obtained,which is verified by the numerical results.

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.

Comparison of Numerical Approximations of One-Dimensional Space Fractional Diffusion Equation Using Different Types of Collocation Points in Spectral Method Based on Lagrange’s Basis Polynomials

Recently many research works have been conducted and published regarding fractional order differential equations. There are several approaches available for numerical approximations of the solution of fractional order diffusion equations. Spectral collocation method based on Lagrange’s basis polynomials to approximate numerical solutions of one-dimensional (1D) space fractional diffusion equations are introduced in this research paper. The proposed form of approximate solution satisfies non-zero Dirichlet’s boundary conditions on both boundaries. Collocation scheme produce a system of first order Ordinary Differential Equations (ODE) from the fractional diffusion equation. We applied this method with four different sets of collocation points to compare their performance.

Enriched reproducing kernel particle method for fractional advection–diffusion equation

The reproducing kernel particle method （RKPM） has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advectiondiffusion equation （FADE）, which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.

IDENTIFYING AN UNKNOWN SOURCE IN SPACE-FRACTIONAL DIFFUSION EQUATION

In this paper, we identify a space-dependent source for a fractional diffusion equation. This problem is ill-posed, i.e., the solution （if it exists） does not depend continuously on the data. The generalized Tikhonov regularization method is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained, Numerical examples are presented to illustrate the validity and effectiveness of this method.

A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation

In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.

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.

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.