Space-time Fractional Diffusion Equation Research Articles

A fast algorithm for multi-term time-space fractional diffusion equation with fractional boundary condition

A fast algorithm for multi-term time-space fractional diffusion equation with fractional boundary condition

Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian

This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional diffusion equation, which uses linear finite element or fourth-order compact difference method combining with matrix transfer technique to approximate spectral fractional Laplacian. Then we introduce a fast time-stepping L1 scheme for time discretization. The proposed scheme can exactly evaluate fractional power of matrix and perform matrix-vector multiplication at per time level by using discrete sine transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory requirement. Further, we address stability and convergence analyses of full discrete scheme based on fast time-stepping L1 scheme on graded time mesh. Our error analysis shows that the choice of graded mesh factor ω=(2−α)/α shall give an optimal temporal convergence O(N−(2−α)) with N denoting the number of time mesh. Finally, ample numerical examples are delivered to illustrate our theoretical analysis and the efficiency of the suggested scheme.

An $$\alpha $$-robust analysis of finite element method for space-time fractional diffusion equation

An $$\alpha $$-robust analysis of finite element method for space-time fractional diffusion equation

Existence of solutions to a nonlinear fractional diffusion equation with exponential growth

In this paper, we study a Cauchy problem for a space–time fractional diffusion equation with exponential nonlinearity. Based on the standard Lp-Lq estimates of strongly continuous semigroup generated by fractional Laplace operator, we investigate the existence of global solutions for initial data with small norm in Orlicz space exp Lp(Rd) and a time weighted Lr(Rd) space. In the framework of the Hölder interpolation inequality, we also discuss the existence of local solutions without the Orlicz space.

The quasi-reversibility regularization method for backward problem of the multi-term time-space fractional diffusion equation

In this paper, we investigate a backward problem of 1D multi-term time–space fractional diffusion equation, which is ill-posed. With the help of quasi-reversibility regularization method, we give the regularization solution, based on some properties of Fourier transform and Mittag-Leffler functions. Besides, we establish a-priori and a-posteriori convergence estimates, respectively. Finally, we illustrate two numerical examples to verify the effectiveness and feasibility of our proposed method.

An efficient numerical method based on Chelyshkov operation matrix for solving a type of time-space fractional reaction diffusion equation

An efficient numerical method based on Chelyshkov operation matrix for solving a type of time-space fractional reaction diffusion equation

Solution of the fractional diffusion equation by using Caputo-Fabrizio derivative: application to intrinsic arsenic diffusion in germanium

In this work, we focused on solving the space-time fractional diffusion equation with an application on the intrinsic arsenic diffusion in germanium. At first we have treated the differential equation in a semi-infinite medium by using Caputo-Fabrizio fractional derivative. We have introduced the Laplace transform to solve this type of equations. Secondly, Based on the obtained solution, we have simulated an profile of arsenic diffusion in germanium under intrinsic conditions. Accurate simulations have been achieved showing that the fractional derivative orders affect on the estimation of the diffusion coefficient, where increasing the time fractional derivative order α reduces the value of the diffusion coefficient, while increasing the space fractional derivative order β increases the value of the diffusion coefficient.

A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation

<abstract><p>Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method.</p></abstract>

A Preconditioner for Galerkin–Legendre Spectral All-at-Once System from Time-Space Fractional Diffusion Equation

As a model that possesses both the potentialities of Caputo time fractional diffusion equation (Caputo-TFDE) and symmetric two-sided space fractional diffusion equation (Riesz-SFDE), time-space fractional diffusion equation (TSFDE) is widely applied in scientific and engineering fields to model anomalous diffusion phenomena including subdiffusion and superdiffusion. Due to the fact that fractional operators act on both temporal and spatial derivative terms in TSFDE, efficient solving for TSFDE is important, where the key is solving the corresponding discrete system efficiently. In this paper, we derive a Galerkin–Legendre spectral all-at-once system from the TSFDE, and then we develop a preconditioner to solve this system. Symmetry property of the coefficient matrix in this all-at-once system is destroyed so that the deduced all-at-once system is more convenient for parallel computing than the traditional timing-step scheme, and the proposed preconditioner can efficiently solve the corresponding all-at-once system from TSFDE with nonsmooth solution. Moreover, some relevant theoretical analyses are provided, and several numerical results are presented to show competitiveness of the proposed method.

Accurate numerical simulations for fractional diffusion equations using spectral deferred correction methods

This paper mainly studies the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for discretization in space and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. Therefore, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of the numerical scheme.

Based on Gradient Algorithm for the Inverse Source Problem of a Class of Time-space Fractional Diffusion Equations

Since entering the 21st century, the establishment of fractional-order diffusion equations in various fields has been of great value and has garnered widespread attention. This study focuses on inverse source term problem for time-space fractional diffusion equation (TSFDE) using given boundary data. First, the identification source problem is transformed into a functional minimization problem utilize the Tikhonov-type regularization method. Then, the sensitivity and the adjoint problem are derived, and the gradient of functional is obtained. The conjugate gradient algorithm is used to solve the minimization problem. Finally, three xamplel with different types of source terms are used to stated the effectiveness and stability, the impact of various parameters on the numerical results is analyzed.

A nonstationary iterated quasi-boundary value method for reconstructing the source term in a time–space fractional diffusion equation

An inverse source problem of a time–space fractional diffusion equation is considered in this paper. Due to the ill-posedness of the inverse problem, we propose a novel nonstationary iterated quasi-boundary value regularization method for reconstructing the source function, and show that the regularization problem is well-posed. The convergence rates are established under a priori and a posterior choice rules of regularization parameters, respectively. A numerical scheme for solving the regularization problem in one-dimensional case is derived from a finite difference method. Moreover, various of numerical examples are performed to test the efficiency of our method.

The solution of the time-space fractional diffusion equation based on the Chebyshev collocation method

The solution of the time-space fractional diffusion equation based on the Chebyshev collocation method

Finite Difference Approximation for the Space-Time Fractional Linear Diffusion Equation Involving the Caputo-Hadamard Fractional Derivative

Finite Difference Approximation for the Space-Time Fractional Linear Diffusion Equation Involving the Caputo-Hadamard Fractional Derivative

Solution of space–time fractional diffusion equation involving fractional Laplacian with a local radial basis function approximation

Solution of space–time fractional diffusion equation involving fractional Laplacian with a local radial basis function approximation

On the Solution of Mathematical Model Including Space-Time Fractional Diffusion Equation in Conformable Derivative, Via Weighted Inner Product

This research aims to accomplish an analytic solution to mathematical models involving space-time fractional differential equations in the conformable sense in series form through the weighted inner product and separation of variables method. The main advantage of this method is that various linear problems of any kind of differential equations can be solved by using this method. First, the corresponding eigenfunctions are established by solving the Sturm-Liouville eigenvalue problem. Secondly, the coefficients of the eigenfunctions are determined by employing weighted inner product and initial condition. Thirdly, the analytic solution to the problem is constructed in the series form. Finally, an illustrative example is presented to show how this method is implemented for fractional problems and exhibit its effectiveness and accuracy.

Numerical solution of the space-time fractional diffusion equation based on fractional reproducing kernel collocation method

Numerical solution of the space-time fractional diffusion equation based on fractional reproducing kernel collocation method

Galerkin Method for a Backward Problem of Time-Space Fractional Symmetric Diffusion Equation

We investigate a backward problem of the time-space fractional symmetric diffusion equation with a source term, wherein the negative Laplace operator −Δ contained in the main equation belongs to the category of uniformly symmetric elliptic operators. The problem is ill-posed because the solution does not depend continuously on the measured data. In this paper, the existence and uniqueness of the solution and the conditional stability for the inverse problem are given and proven. Based on the least squares technique, we construct a Galerkin regularization method to overcome the ill-posedness of the considered problem. Under a priori and a posteriori selection rules for the regularization parameter, the Hölder-type convergence results of optimal order for the proposed method are derived. Meanwhile, we verify the regularized effect of our method by carrying out some numerical experiments where the initial value function is a smooth function or a non-smooth one. Numerical results show that this method works well in dealing with the backward problem of the time-space fractional parabolic equation.

The Quasi-Boundary Regularization Method for Recovering the Initial Value in a Nonlinear Time–Space Fractional Diffusion Equation

In this paper, we consider the inverse problem for identifying the initial value problem of the time–space fractional nonlinear diffusion equation. The uniqueness of the solution is proved by taking the fixed point theorem of Banach compression, and the ill-posedness of the problem is analyzed through the exact solution. The quasi-boundary regularization method is chosen to solve the ill-posed problem, and the error estimate between the regularization solution and the exact solution is given. Moreover, several numerical examples are chosen to prove the effectiveness of the quasi-boundary regularization method. Finally, our method can be used to solve high dimensional time–space fractional nonlinear diffusion equation, especially in cylindrical and spherical symmetric regions.

The non-uniform L1-type scheme coupling the finite volume method for the time–space fractional diffusion equation with variable coefficients

In this paper, we consider the time–space fractional diffusion equation on a bounded interval with the Caputo time fractional derivative and the two-sided Riemann–Liouville spatial fractional derivative. |The diffusion coefficients are variable with respect to x and t. The temporal non-uniform L1-type discretization and the finite volume method based on the nodal basis functions are adopted to discrete this fractional partial derivative equation. We strictly prove that our scheme is conditionally stable with convergence rate Oh2+N−min{2−α,rα}, where h is the spatial step and N,α,r are the temporal nodal numbers, the fractional parameter and the graded parameter respectively. Finally, some numerical results are presented to validate the theoretical analysis.