In this article, a kind of nonlinear wave model with the Caputo fractional derivative is solved by an efficient algorithm, which is formulated by combining a time second-order shifted convolution quadrature (SCQ) formula in time and a mixed element method in space. The stability of numerical scheme is derived, and an optimal error result for unknown functions which include an original function and two auxiliary functions are proven. Further, the numerical tests are conducted to confirm the theoretical results.

In this article, the time-fractional generalized Rosenau-RLW-Burgers equation is numerically solved, where the generalized BDF2- $\theta$ is used to discretize the temporal direction, and the mixed finite element method is applied to the spatial direction. The stability of the fully discrete scheme is proven. Finally, the effectiveness of the numerical scheme is verified through some numerical examples, and the singularity of nonsmooth solutions in the initial time layer is effectively resolved by adding the correction term.

In this article, a physics-informed neural network based on the time difference method is developed to solve one-dimensional (1D) and two-dimensional (2D) nonlinear time distributed-order models. The FBN- $\theta$, which is constructed by combining the fractional second order backward difference formula (BDF2) with the fractional Newton-Gregory formula, where a second-order composite numerical integral formula is used to approximate the distributed-order derivative, and the time direction at time $t_{n+\frac{1}{2}}$ is approximated by making use of the Crank-Nicolson scheme. Selecting the hyperbolic tangent function as the activation function, we construct a multi-output neural network to obtain the numerical solution, which is constrained by the time discrete formula and boundary conditions. Automatic differentiation technology is developed to calculate the spatial partial derivatives. Numerical results are provided to confirm the effectiveness and feasibility of the proposed method and illustrate that compared with the single output neural network, using the multi-output neural network can effectively improve the accuracy of the predicted solution and save a lot of computing time.

In this paper, a two-grid alternating direction implicit (ADI) finite element (FE) method based on the weighted and shifted Grünwald difference (WSGD) operator is proposed for solving a two-dimensional nonlinear time distributed-order fractional sub-diffusion equation. The stability and optimal error estimates with second-order convergence rate in spatial direction are obtained. The storage space can be reduced and computing efficiency can be improved in this method. Two numerical examples are provided to verify the theoretical results.

The physics informed neural network (PINN) has achieved significant success in solving evolution partial differential equations (PDEs). For improving the prediction accuracy of the PINN, we developed a new PINN with Taylor series expansion (TPINN). However, the low accuracy problem for the PINN or TPINN may occur in approximating the solution of strongly nonlinear evolution PDEs or even linear wave equations. For solving this issue, we introduced a novel efficient method, called a forward progressive PINN with Taylor series expansion (FP-TPINN), where the formula obtained by the Taylor series expansion was applied to construct extra supervised learning task and the domain decomposition in time was used to further improve the accuracy of our proposed method. We carried out several numerical experiments to demonstrate that the TPINN significantly improved the accuracy of the PINN. Moreover, we used the Korteweg-de Vries (KdV) equation to indicate that the TPINN can achieve higher accuracy than the SPINN, and illustrated that the FP-TPINN performed better than the pre-training PINN (PT-PINN) and the dimension-augmented PINN (DaPINN) by solving the Allen-Cahn equation.

In this article, a second-order time discrete algorithm with a shifted parameter $\theta$ combined with a fast time two-mesh (TT-M) mixed finite element (MFE) scheme was considered to look for the numerical solution of the nonlinear fractional hyperbolic wave model. The second-order backward difference formula including a shifted parameter $\theta$ (BDF2- $\theta$) with the weighted and shifted Grünwald difference (WSGD) approximation for fractional derivative was used to discretize time direction at time $t_{n-\theta }$, the $H^1$-Galerkin MFE method was applied to approximate the spatial direction, and the fast TT-M method was used to save computing time of the developed MFE system. A priori error estimates for the fully discrete TT-M MFE system were analyzed and proved in detail, where the second-order space-time convergence rate in both $L^2$-norm and $H^1$-norm were derived. Detailed numerical algorithms with smooth and weakly regular solutions were provided. Finally, some numerical examples were provided to illustrate the feasibility and effectiveness for our scheme.