Two-dimensional Time Fractional Diffusion Equation Research Articles

On the analysis and application of a spectral collocation scheme for the nonlinear two-dimensional fractional diffusion equation

In this paper, we propose and analyze a novel spectral scheme for the numerical solution of a two-dimensional time-fractional diffusion equation. The proposed scheme approximates the unknown function and its derivatives in space by finite Lucas and Fibonacci polynomial expansion, and time derivative/variable by L1 formula with θ-weighted difference scheme. Error estimates are derived and convergence is proved. Six test problems are then solved to verify the theory numerically. The obtained results confirm the low computational cost and better accuracy than known results in the literature.

Error estimate of L1-ADI scheme for two-dimensional multi-term time fractional diffusion equation

<abstract><p>A two-dimensional multi-term time fractional diffusion equation $ {D}_{t}^{\mathit{\boldsymbol{\alpha}}} u(x, y, t)- \Delta u(x, y, t) = f(x, y, t) $ is considered in this paper, where $ {D}_{t}^{\mathit{\boldsymbol{\alpha}}} $ is the multi-term time Caputo fractional derivative. To solve the equation numerically, L1 discretisation to each fractional derivative is used on a graded temporal mesh, together with a standard finite difference method for the spatial derivatives on a uniform spatial mesh. We provide a rigorous stability and convergence analysis of a fully discrete L1-ADI scheme for solving the multi-term time fractional diffusion problem. Numerical results show that the error estimate is sharp.</p></abstract>

Optimal H1-Norm Estimation of Nonconforming FEM for Time-Fractional Diffusion Equation on Anisotropic Meshes

In this paper, based on the L2-1σ scheme and nonconforming EQ1rot finite element method (FEM), a numerical approximation is presented for a class of two-dimensional time-fractional diffusion equations involving variable coefficients. A novel and detailed analysis of the equations with an initial singularity is described on anisotropic meshes. The fully discrete scheme is shown to be unconditionally stable, and optimal second-order accuracy for convergence and superconvergence can be achieved in both time and space directions. Finally, the obtained numerical results are compared with the theoretical analysis, which verifies the accuracy of the proposed method.

An alternating direction implicit legendre spectral method for simulating a 2D multi-term time-fractional Oldroyd-B fluid type diffusion equation

In this paper, an alternating direction implicit legendre spectral (ADILS) method is developed for the two-dimensional (2D) multi-term time fractional Oldroyd-B fluid type diffusion equation. This method is provided based on legendre spectral approximation in space and a new difference discretization in time, and it can convert 2D computation to two 1D computations. Meanwhile, a very important lemma is proposed to prove the unconditional stability and convergence of this ADILS scheme. Especially, the corrected scheme is utilized to increase the convergence accuracy for the non-smooth solution. With three numerical experiments, the validity of the method is confirmed. The present method also can be extended to solve the generalized non-Newtonian fluid type equations, multi-term time fractional mixed diffusion and diffusion-wave equations.

A noniterative reconstruction method for solving a time-fractional inverse source problem from partial boundary measurements

In this paper, a noniterative method for solving an inverse source problem governed by the two-dimensional time-fractional diffusion equation is proposed. The basic idea consists in reconstructing the geometrical support of the unknown source from partial boundary measurements of the associated potential. A Kohn–Vogelius type shape functional is considered together with a regularization term penalizing the relative perimeter of the unknown set of anomalies. Identifiability result is derived and uniqueness of a minimizer is ensured. The shape functional measuring the misfit between the solutions of two auxiliary problems containing information about the boundary measurements is minimized with respect to a finite number of ball-shaped trial anomalies by using the topological derivative method. In particular, the second-order topological gradient is exploited to devise an efficient and fast noniterative reconstruction algorithm. Finally, some numerical experiments are presented, showing different features of the proposed approach in reconstructing multiple anomalies of varying shapes and sizes by taking noisy data into account.

Formula omitted]-robust [formula omitted]-norm error estimate of nonuniform Alikhanov scheme for fractional sub-diffusion equation

The sharp H1-norm error estimate of a finite difference method for two-dimensional time-fractional diffusion equation is established, where the Caputo time-fractional derivative term is approximated by Alikhanov scheme on graded mesh. Under reasonable assumption that the exact solution has typical weak singularity at initial time, the temporal convergence order of the fully scheme is O(N−min{rα,2}) and the error bound does not blow up when α approximates 1−. The theoretical analysis utilizes an improved discrete fractional Grönwall inequality, and the correctness of the theoretical result is verified by numerical experiments.

Pointwise error estimate of an alternating direction implicit difference scheme for two-dimensional time-fractional diffusion equation

An alternating direction implicit (ADI) difference method is adopted to solve the two-dimensional time-fractional diffusion equation with Dirichlet boundary condition whose solution has some weak singularity at initial time. L1 scheme on uniform mesh is used to discretize the temporal Caputo fractional derivative. Pointwise-in-time error estimate is given for the fully discrete ADI scheme, where the error bound does not blowup when α (the order of fractional derivative) approaches 1−. It is shown both in theoretically and numerically that the temporal convergence order of the ADI scheme is O(τ2α+τtnα−1) at time t=tn; hence the scheme is globally O(τα) accurate in temporal direction, but it is O(τmin⁡{2α,1}) when t is away from 0.

α-Robust H1-norm convergence analysis of ADI scheme for two-dimensional time-fractional diffusion equation

A fully discrete ADI scheme is proposed for solving the two-dimensional time-fractional diffusion equation with weakly singular solutions, where L1 scheme on graded mesh is adopted to tackle the initial singularity. An improved discrete fractional Grönwall inequality is employed to give an α-robust H1-norm convergence analysis of the fully discrete ADI scheme, where the error bound does not blow up when the order of fractional derivative α→1−. Numerical results show that the theoretical analysis is sharp.

Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations

<p style='text-indent:20px;'>In this work, two fully novel finite difference schemes for two-dimensional time-fractional mixed diffusion and diffusion-wave equation (TFMDDWEs) are presented. Firstly, a Hermite and Newton quadratic interpolation polynomial have been used for time discretization and central quotient has used in spatial direction. The H2N2 finite difference is constructed. Secondly, in order to increase computational efficiency, the sum-of-exponential is used to approximate the kernel function in the fractional-order operator. The fast H2N2 finite difference is obtained. Thirdly, the stability and convergence of two schemes are studied by energy method. When the tolerance error <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> of fast algorithm is sufficiently small, it proves that both of difference schemes are of <inline-formula><tex-math id="M2">\begin{document}$ 3-\beta\; (1<\beta<2) $\end{document}</tex-math></inline-formula> order convergence in time and of second order convergence in space. Finally, numerical results demonstrate the theoretical convergence and effectiveness of the fast algorithm.</p>

A numerical method based on three-dimensional Legendre wavelet method for two-dimensional time-fractional diffusion equation

In this paper, an efficient numerical method is proposed to handle two-dimensional fractional diffusion equations on a finite domain. The proposed method combines the product of Legendre wavelet bases for two spatial dimensions and a time direction. The operational matrix of the proposed method is obtained. Tikhonov regularization is employed to stabilize the system in cases where the final linear system of equations is large. The convergence analysis of the method is studied and some numerical examples are presented to investigate the efficiency and accuracy of the method.

Two-grid methods for nonlinear time fractional diffusion equations by [formula omitted]-Galerkin FEM

In this paper, two efficient two-grid algorithms with L1 scheme are presented for solving two-dimensional nonlinear time fractional diffusion equations. The classical L1 scheme is considered in the time direction, and the two-grid FE method is used to approximate spatial direction. To linearize the discrete equations, the Newton iteration approach and correction technique are applied. The two-grid algorithms reduce the solution of the nonlinear fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithms save total computational cost. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Moreover, the numerical experiment presented further confirms the theoretical results.

Fast O(N) hybrid Laplace transform-finite difference method in solving 2D time fractional diffusion equation

It is time-memory consuming when numerically solving time fractional partial differential equations, as it requires O ( N 2 ) computational cost and O ( M N ) memory complexity with finite difference methods, where, N and M are the total number of time steps and spatial grid points, respectively. To surmount this issue, we develop an efficient hybrid method with O ( N ) computational cost and O ( M ) memory complexity in solving two-dimensional time fractional diffusion equation. The presented method is based on the Laplace transform method and a finite difference scheme. The stability and convergence of the proposed method are analyzed rigorously by the means of the Fourier method. A comparative study drawn from numerical experiments shows that the hybrid method is accurate and reduces the computational cost, memory requirement as well as the CPU time effectively compared to a standard finite difference scheme.

Approximate solution of the multi-term time fractional diffusion and diffusion-wave equations

We develop a numerical scheme for finding the approximate solution for one- and two-dimensional multi-term time fractional diffusion and diffusion-wave equations considering smooth and nonsmooth solutions. The concept of multi-term time fractional derivatives is conventionally defined in the Caputo view point. In the current research, the convergence analysis of Legendre collocation spectral method was carried out. Spectral collocation method is consequently tested on several benchmark examples, to verify the accuracy and to confirm effectiveness of proposed method. The main advantage of the method is that only a small number of shifted Legendre polynomials are required to obtain accurate and efficient results. The numerical results are provided to demonstrate the reliability of our method and also to compare with other previously reported methods in the literature survey.

Convergence analysis of the anisotropic FEM for 2D time fractional variable coefficient diffusion equations on graded meshes

In this paper, an unconditionally stable fully discrete numerical scheme for the two-dimensional (2D) time fractional variable coefficient diffusion equations with non-smooth solutions is constructed and analyzed. The L2-1σ scheme is applied for the discretization of time fractional derivative on graded meshes and anisotropic finite element method (FEM) is employed for the spatial discretization. The unconditional stability and convergence of the proposed scheme are proved rigorously. It is shown that the order O(h2+N−min{rα,2}) can be achieved, where h is the spatial step, N is the number of partition in temporal direction, r is the temporal meshes grading parameter and α is the order of fractional derivative. A numerical example is provided to verify the sharpness of our error analysis.

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 difference 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 confirm the analysis.

ON THE STABILITY CONDITIONS OF 2D TIME FRACTIONAL DIFFUSION EQUATION

The Caputo definition of fractional derivative has been employed for the time derivative for the two-dimensional time-fractional diffusion equation. The stability condition obtained by reformulation the classical multilevel technique on the finite difference scheme. A numerical example gives a good agreement with the theoretical result

Convergence and superconvergence analysis of finite element methods for the time fractional diffusion equation

In this paper, two classes of finite element methods (FEMs) for the two-dimensional time fractional diffusion equation (TFDE) with non-smooth solution are proposed and analyzed. In the temporal direction, we adopt nonuniform L2-1σ method, and in the spatial direction, the conforming bilinear element and the nonconforming modified quasi-Wilson element are utilized. For the proposed conforming and nonconforming FEMs, by using new fractional discrete Grönwall inequalities, the theoretical analysis including the L2-norm error estimates and H1-norm superclose results are given in details. Furthermore, by virtue of the interpolated postprocessing techniques, the global H1-norm superconvergence results are presented. Finally, some numerical results illustrate the correctness of theoretical analysis.

An accurate numerical technique for solving two-dimensional time fractional order diffusion equation

ABSTRACTIn this paper, we consider a numerical technique for solving the two-dimensional fractional order diffusion equation with a time fractional derivative. The proposed technique depends basically on the fact that an expansion of the required approximate solution in a series of shifted Chebyshev polynomials of the first kind in the time and the Sinc function in the space. Then, the expansion coefficients are determined by reducing the problem with the initial conditions to a system of algebraic equations. The fractional derivatives are expressed in the Caputo sense. The numerical results of the proposed technique are compared with other published results to show the efficiency of the presented technique.

Green's function of two-dimensional time-fractional diffusion equation using addition formula of Wright function

ABSTRACTIn this paper, using the Mellin transform of Wright function we derive an addition formula for the Wright function. In some special cases, addition formulas for the Hermite, Bessel and Mittag-Leffler functions are also given and the Green's function of two-dimensional time-fractional diffusion equation is presented in the whole plane.

A novel high-order approximate scheme for two-dimensional time-fractional diffusion equations with variable coefficient

Based on the spatial quasi-Wilson nonconforming finite element method and temporal L2−1σ formula, a fully-discrete approximate scheme is proposed for a two-dimensional time-fractional diffusion equations with variable coefficient on anisotropic meshes. In order to demonstrate the stable analysis and error estimates, several lemmas are provided, which focus on high accuracy about projection and superclose estimate between the interpolation and projection. Unconditionally stable analysis are derived in L2-norm and broken H1-norm. Moreover, convergence result of accuracy O(h2+τ2) and superclose property of accuracy O(h2+τ2) are deduced by combining interpolation with projection, where h andτ are the step sizes in space and time, respectively. And then, the global superconvergence is presented by employing interpolation postprocessing operator. Finally, numerical results are provided to demonstrate the validity of the theoretical analysis.