Method For Time-fractional Diffusion Equation Research Articles

A direct sampling method for time-fractional diffusion equation

This paper introduces a direct sampling method tailored for identifying the location of the source term within a time-fractional diffusion equation (TFDE). The key aspect of our approach involves the utilization of a versatile family of index functions, which can be chosen according to the specific characteristics of the source term. Recognizing the key role of the TFDE’s fundamental solution within the index function, we further enhance our method by deriving its asymptotic expansions. This advancement not only enhances the accuracy, but also significantly improves the computational efficiency of our method. To validate the effectiveness and robustness of the proposed sampling method, we conduct a series of comprehensive numerical experiments.

Error analysis of a fully discrete method for time-fractional diffusion equations with a tempered fractional Gaussian noise

We develop and analyze a fully discrete method, in order to solve numerically time-fractional diffusion equations driven by additive tempered fractional Gaussian noise. The stochastic model involves a Caputo fractional derivative of order α∈(0,1). We discuss the regularity results of the solution by using the inversion of Laplace transform, the Wright function and covariance function of stochastic process. A spectral Galerkin method is applied to approximate the stochastic model in space. Time discretization is achieved by using firstly a Riemann–Liouville derivative to reformulate the spatial semi-discrete form, and then applying the Grünwald–Letnikov formula. We derive the error estimates of the discrete methods, based on regularity results of the solution. Finally, extensive numerical experiments are presented to confirm our theoretical analysis.

A priori and a posteriori error estimates of a space–time Petrov–Galerkin spectral method for time-fractional diffusion equation

Time-fractional diffusion equation is an important transport dynamical model for simulating fractal time random walk. This article is devoted to investigating the a priori and a posteriori error estimates for this model equation. A Petrov–Galerkin spectral method is revisited in this paper to address our problem, which the generalized Jacobi functions and Fourier-like basis functions are utilized as basis for constructing efficient and accurate space–time spectral approximations. Rigorous proofs are given for the stability of our spectral scheme. And then the convergence of the proposed method is proved by establishing an a priori error estimate. Specifically, an efficient and reliable a posteriori error estimator is introduced, and we derive that the residual-based error indicator provides an upper bound and a lower bound for the numerical error. Finally, several numerical experiments are provided to examine our theoretical claims.

A Local Discontinuous Galerkin Method for Time-Fractional Diffusion Equations

A Local Discontinuous Galerkin Method for Time-Fractional Diffusion Equations

A high-order and extra-dof-free generalized finite element method for time-fractional diffusion equation

A high-order and extra-dof-free generalized finite element method for time-fractional diffusion equation

Non-polynomial Spectral-Galerkin Method for Time-Fractional Diffusion Equation on Unbounded Domain

Non-polynomial Spectral-Galerkin Method for Time-Fractional Diffusion Equation on Unbounded Domain

Green's function method for time-fractional diffusion equation on the star graph with equal bonds

Green's function method for time-fractional diffusion equation on the star graph with equal bonds

A local discontinuous Galerkin method for time-fractional diffusion equation with discontinuous coefficient

In this paper, a fully discrete Local discontinuous Galerkin (LDG) method is discussed for a time-fractional reaction diffusion initial-boundary value problem with discontinuous diffusive coefficient. In this method the well-known L2-1σ scheme on graded meshes is presented to deal with the weak singularity at initial time t=0, while in the spatial direction a LDG method on a uniform mesh is presented to handle the discontinuous coefficient. By the discrete fractional Gronwall inequality, the L2-norm stability and consistency estimate results are derived for the proposed fully discrete LDG method. Numerical experiments are presented to verify the sharpness of the error analysis.

A direct discontinuous Galerkin method for time-fractional diffusion equation with discontinuous diffusive coefficient

In this work, a time-fractional reaction diffusion initial-boundary value problem with discontinuous diffusive coefficient is considered. A fully discrete direct discontinuous Galerkin (DDG) method is presented to solve it. In this method, the well-known L1 scheme with graded mesh is presented to deal with the weak singularity at initial time t = 0, while in spatial a DDG method with uniform mesh is presented to handle the discontinuous diffusive coefficient. Then norm stability and consistency estimate results are derived. Numerical experiments are presented to confirm the sharpness of the error analysis.

Error Analysis of a High Order Method for Time-Fractional Diffusion Equations

In this paper, we consider a numerical method for the time-fractional diffusion equation. The method uses a high order finite difference method to approximate the fractional derivative in time, resulting in a time stepping scheme for the underlying equation. Then the resulting equation is discretized in space by using a spectral method based on the Legendre polynomials. The main body of this paper is devoted to carry out a rigorous analysis for the stability and convergence of the time stepping scheme. As a by-product and direct extension of our previous work, an error estimate for the spatial discretization is also provided. The key contribution of the paper is the proof of the ($3-\alpha$)-order convergence of the time scheme, where $\alpha$ is the order of the time-fractional derivative. Then the theoretical result is validated by a number of numerical tests. To the best of our knowledge, this is the first proof for the stability of the ($3-\alpha$)-order scheme for the time-fractional diffusion equation.

A Computational Study of an Implicit Local Discontinuous Galerkin Method for Time-Fractional Diffusion Equations

We propose, analyze, and test a fully discrete local discontinuous Galerkin (LDG) finite element method for a time-fractional diffusion equation. The proposed method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. By choosing the numerical fluxes carefully, we prove that our scheme is unconditionally stable and convergent. Finally, numerical examples are performed to illustrate the effectiveness and the accuracy of the method.