Space-time Fractional Diffusion Equation Research Articles

Identification Source Term of the Distributed Order Time-Space Fractional Diffusion Equation

Abstract The inverse source problem for time-distributed order spatial fractional diffusion equations is examined in this study. The forward problem is solved by combining finite difference methods with matrix transformation techniques. The Tikhonov regularization approach is then applied to convert the inverse problem into a variational problem. The optimal perturbation approach is used to find the minimizer of the variational problem, leading to an approximation solution that depends solely on the time-dependent source term. The algorithm’s effectiveness and stability are demonstrated using numerical examples in one-dimensional domains.

Combining approach of collocation and finite difference methods for fractional parabolic PDEs

This research aims to estimate the solutions of fractional-order partial differential equations of spacial fractional and both time-space fractional order. For this, we use finite differences for time derivatives and the well-known collocation method for space derivatives with lower-order Bernstein polynomials as basis functions. We explain the mathematical formulations in detail. Convergence and stability analysis of the space–time fractional diffusion equation with the source term is reported subsequently. Three numerical examples are considered for demonstrating the accuracy and reliability of the proposed method.

A New Perspective for Scientific Modelling: Sparse Reconstruction Based Approach for Learning Time-Space Fractional Differential Equations

Abstract This paper studies a sparse reconstruction based approach to learn time-space fractional differential equations, i.e. to identify parameter values and particularly the order of the fractional derivatives. The approach uses a generalized Taylor series expansion to generate, in every iteration, a feature matrix which is used to learn the fractional orders of both, temporal and spatial derivatives by minimizing the LASSO operator using Differential Evolution algorithm. To verify the robustness of the method, numerical results for time-space fractional diffusion equation, wave equation, and Burgers? equation at different noise levels in the data are presented. Finally, the methodology is applied to a realistic example where underlying fractional differential equation associated to published experimental data obtained from an in-vitro cell culture assay is learned.

Sine Transform Based Preconditioning for an Inverse Source Problem of Time-Space Fractional Diffusion Equations

Sine Transform Based Preconditioning for an Inverse Source Problem of Time-Space Fractional Diffusion Equations

Numerical Analysis for Stochastic Time-Space Fractional Diffusion Equation Driven by Fractional Gaussian Noise

Numerical Analysis for Stochastic Time-Space Fractional Diffusion Equation Driven by Fractional Gaussian Noise

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.

Solution of Inverse Problem for Diffusion Equation with Fractional Derivatives Using Metaheuristic Optimization Algorithm

The article focuses on the presentation and comparison of selected heuristic algorithms for solving the inverse problem for the anomalous diffusion model. Considered mathematical model consists of time-space fractional diffusion equation with initial boundary conditions. Those kind of models are used in modelling the phenomena of heat flow in porous materials. In the model, Caputo’s and Riemann-Liouville’s fractional derivatives were used. The inverse problem was based on identifying orders of the derivatives and recreating fractional boundary condition. Taking into consideration the fact that inverse problems of this kind are ill-conditioned, the problem should be considered as hard to solve. Therefore,to solve it, metaheuristic optimization algorithms popular in scientific literature were used and their performance were compared: Group Teaching Optimization Algorithm (GTOA), Equilibrium Optimizer (EO), Grey Wolf Optimizer (GWO), War Strategy Optimizer (WSO), Tuna Swarm Optimization (TSO), Ant Colony Optimization (ACO), Jellyfish Search (JS) and Artificial Bee Colony (ABC). This paper presents computational examples showing effectiveness of considered metaheuristic optimization algorithms in solving inverse problem for anomalous diffusion model.

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 fast finite difference scheme for the time-space fractional diffusion equation

This paper proposes a fast numerical scheme for time-space fractional diffusion equation in two spatial dimensions, where the temporal fractional partial derivative is in the Caputo sense of order α ϵ (0; 1) and the spatial fractional derivative is given by fractional Laplacian (−Δ)β/2 in two space dimensions with β ϵ (1, 2). The temporal Caputo derivative is discretized by a fast evaluation based on the L2-1σ formula. The spatial discretization adopts fractional centered difference formula in two dimensions. The proposed fully discrete scheme is proved to be stable and convergent with 2nd order accuracy both in time and space. The feasibility, stability, and convergence of the proposed scheme are demonstrated by the numerical example.

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