The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation

This paper investigates a nonlinear time-fractional mixed sub-diffusion and diffusion-wave equation with delay. The problem is particularly challenging due to its nonlinear nature, the presence of a time delay, and the incorporation of both fractional diffusion and fractional wave terms, introducing computational complexities for numerical analysis. To address this, we transform the model into a new generalized form involving Riemann–Liouville fractional integrals and a Caputo derivative of order α∈(0,1). Subsequently, a temporal second-order linearized difference scheme is presented to approximate the model, and its unconditional stability is rigorously proven based on discrete Gronwall's inequality. To reduce computational and storage costs, we extend the discussion to a fast variant of the proposed difference scheme. The sum of exponentials approach is employed to approximate the kernel function in fractional-order operators, leading to fast difference analogs for the Riemann–Liouville fractional integral and the Caputo derivative. A fast variant of the developed direct method is introduced based on these analogs. Numerical results are provided to validate our theoretical findings and assess the accuracy and efficiency of the difference schemes. 

Temporal second-order difference schemes for the nonlinear time-fractional mixed sub-diffusion and diffusion-wave equation with delay

In this paper, the conforming virtual element method (VEM) is considered to solve the two-dimensional fractional cable equation involving two Riemann–Liouville fractional derivatives. We adopt the Backward Euler Method and the classical scheme for the numerical discrete scheme of the time derivative. Meanwhile, the conforming VEM, which is generated for arbitrary order of accuracy and the arbitrary polygonal meshes, is analysed for the discretization of the spatial direction. Based on the energy projection operator, the fully discrete formula is proved to be unconditionally stable, and the optimal convergence results are derived with regard to the -norm in detail. Finally, some numerical experiments are implemented to verify the theoretical results.

The virtual element method for solving two-dimensional fractional cable equation on general polygonal meshes

&lt;p&gt;In this paper, we introduced the gradient-enhanced fractional physics-informed neural networks (gfPINNs) for solving the forward and inverse problems of the multiterm time-fractional Burger-type equation. The gfPINNs leverage gradient information derived from the residual of the fractional partial differential equation and embed the gradient into the loss function. Since the standard chain rule in integer calculus is invalid in fractional calculus, the automatic differentiation of neural networks does not apply to fractional operators. The automatic differentiation for the integer order operators and the finite difference discretization for the fractional operators were used to construct the residual in the loss function. The numerical results demonstrate the effectiveness of gfPINNs in solving the multiterm time-fractional Burger-type equation. By comparing the experimental results of fractional physics-informed neural networks (fPINNs) and gfPINNs, it can be seen that the training performance of gfPINNs is better than fPINNs.&lt;/p&gt; 

Gradient-enhanced fractional physics-informed neural networks for solving forward and inverse problems of the multiterm time-fractional Burger-type equation

Abstract In this article, we propose a second‐order shifted composite numerical integral formula, which is denoted as the SCNIF2. We transform the nonlinear time fractional wave equation into a partial differential equation with a fractional integral term and use the SCNIF2 in time and the finite element algorithm in space to formulate a fully discrete scheme. In order to decrease the initial error of the numerical scheme, we add some starting parts. In addition, we prove the stability and error estimation of the algorithm. Finally, we illustrate the effect of the starting parts and the accuracy of the numerical scheme through some numerical tests. 

Finite element algorithm with a second‐order shifted composite numerical integral formula for a nonlinear time fractional wave equation

Superconvergence analysis of a two-grid finite element method for nonlinear time-fractional diffusion equations

In this paper, we develop a fast linearized virtual element method (VEM) for the approximation of the nonlinear time-fractional diffusion equations on polygonal meshes. The L1-scheme with graded meshes is used to deal with the non-smooth system, the Newton linearized method is adopted to handle the nonlinear term and VEM is employed to discrete the spatial variable. Then the error splitting approach is used to prove the unconditional optimal error estimate of the fully discrete linearized L1-VEM scheme. In order to reduce the storage and computational cost caused by the nonlocality of the Caputo fractional operator, a fast memory-saving L1-VEM is developed. It is proved that the difference between the solution of the L1-VEM and the fast L1-VEM can be made arbitrarily small and is independently of the sizes of the time and/or space grids. Finally, numerical results are implemented to verify the theoretical results.

A fast linearized virtual element method on graded meshes for nonlinear time-fractional diffusion equations

In this paper, we develop a two-grid method (TGM) based on the FEM for 2D nonlinear time fractional two-term mixed sub-diﬀusion and diﬀusion wave equations. A two-grid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the ﬁne grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin ﬁnite element method for the space derivatives and the ﬁnite diﬀerence scheme for the time Caputo derivative with order α ∈ (1 , 2) and α 1 ∈ (0 , 1). Numerical stability and optimal error estimate O ( h r +1 + H 2 r +2 + τ min { 3 − α, 2 − α 1 } ) in L 2 -norm are presented for two-grid scheme, where t, H and h are the time step size, coarse grid mesh size and ﬁne grid mesh size, respectively. Finally, numerical experiments are provided to conﬁrm our theoretical results and eﬀectiveness of the proposed algorithm.

/pdf/a-two-grid-finite-element-approximation-for-nonlinear-time-3i6haoh6.pdf

A Two-Grid Finite Element Approximation for Nonlinear Time Fractional Two-Term Mixed Sub-Diffusion and Diffusion Wave Equations

In this paper, we develop a two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes. The L1 graded mesh scheme is considered in the time direction, and the VEM is used to approximate spatial direction. The two-grid virtual element algorithm reduces the solution of the nonlinear time 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 algorithm not only saves total computational cost, but also maintains the optimal accuracy. Optimal error estimates are analysed in detail for both the VEM scheme and the corresponding two-grid VEM scheme. Finally, numerical experiments presented confirm the theoretical findings.

A two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes

$$\beta $$-robust virtual element method for distributed order time-fractional reaction-diffusion equation with variable coefficients

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