Two-dimensional global (BiGlobal) stability in the region of the minor axis is investigated in the case of hypersonic elliptic cones with major-minor axis ratios of $2:1$ and $3:1$ at Mach number 6.0, and the BiGlobal-$e^N$ method is proposed to predict the transition location of the boundary layer. Matrix-free BiGlobal stability analysis is used to find unstable modes, including the Y-mode and the Z-mode. The growth rates in the streamwise-frequency plane for these modes are obtained. The $N_{\max\_{all}}$ factor is proposed, which represents the maximum amplification factor that all BiGlobal unstable modes can reach. Using a comparison of the $N_{\max\_{all}}$ factor with the transition location measured in a wind tunnel experiment for the $2:1$ elliptic cone, the transition prediction criterion is determined, i.e., $N_{tr} = 8.6.$ In the transition position, the amplification factors of several modes reach a level close to 8.6, which implies that none of them has the absolute superiority sufficient to cause the transition itself. Finally, the BiGlobal-$e^N$ method is employed to predict the transition location in the region of the minor axis of the $3:1$ elliptical cone. It is found that a larger major-minor axis ratio leads to stronger instability and an earlier transition.

In this paper, we develop a cell-average based neural network (CANN) method for solving the Hunter-Saxton equation with its zero-viscosity and zero-dispersion limits. Motivated from the finite volume schemes, the cell-average based neural network method is constructed based on the finite volume integrals of the original PDEs. Supervised training is designed to learn the solution average difference between two neighboring time steps. The training data set is generated by the cell average based on a single initial value of the given PDE. The training process employs multiple time levels of cell averages to maintain stability and control temporal accumulation errors. After being well trained based on appropriate meshes, this method can be utilized like a regular explicit finite volume method to evolve the solution under large time steps. Furthermore, it can be applied to solve different type of initial value problems without retraining the neural network. In order to validate the capability and robustness of the CANN method, we also utilize it to deal with the corrupted learning data which is generated from the Gaussian white noise. Several numerical examples of different types of Hunter-Saxton equations are presented to demonstrate the effectiveness, accuracy, capability, and robustness of the proposed method.

We present a primal-dual discontinuous Galerkin finite element method for a type of ill-posed elliptic Cauchy problem. It is shown that the discrete problem attains a unique solution, if the solution of the ill-posed elliptic Cauchy problems is unique. An optimal error estimate is obtained in a $H^1$-like norm. Numerical experiments are provided to demonstrate the efficiency of the proposed method.

This paper proposes the memristor-memristor (M-M) and the memristor-gradient descent (M-GD) neural networks based on the classical back propagation neural network. The presented models are employed to predict the compressive strength of high-performance hydraulic concrete (HPC), and are tested by well-fitting and accurate predictions with the experimental data. The developed algorithms are also evaluated through comparisons with the classical learning algorithms including the gradient descent method, the gradient descent with momentum, the gradient descent with adaptive learning rate, the elastic gradient descent, and the Levenberg-Marquardt algorithm. It is observed that the established M-GD generally outperforms the classical algorithms and M-M. The constructed M-M neural network has a quite high convergence speed, and the strength prediction error induced by it can roughly satisfy the demands in construction engineering. This work extends the nonlinear memristor to a brand-new field, and provides an effective methodology for forecasting the compressive strength prediction of HPC. 

In this paper, we consider modified Galerkin and iterated modified Galerkin methods for solving a class of two point boundary value problems. The methods are applied after constructing the equivalent derivative dependent Fredholm-Hammerstein integral equations to the boundary value problem. Existence and convergence of the approximate solutions to the actual solution is discussed and the rates of convergence are obtained. Superconvergence results for the approximate and iterated approximate solutions of piecewise polynomial based modified Galerkin method in infinity norm are given. We have also established that iterated modified Galerkin approximation improves over the modified Galerkin solution. Numerical examples are presented to illustrate the theoretical results.

In this paper a nonpolynomial Jacobi spectral-collocation (NJSC) method for the second kind Fredholm integral equations (FIEs) with weakly singular kernel $|s−t|^{−\gamma}$ is proposed. By dividing the integral interval symmetrically into two parts and applying the NJSC method symmetrically to the two weakly singular FIEs respectively, the mild singularities of the interval endpoints can be captured and the exponential convergence can be obtained. A detailed $L^∞$ convergence analysis of the numerical solution is derived. The NJSC method is then extended to two dimensional case and similar exponential convergence results are obtained for low regular solutions. Numerical examples are presented to demonstrate the efficiency of the proposed method.

Numerical approximation of the Ericksen-Leslie system with variable density is considered in this paper. The spherical constraint condition of the orientation field is preserved by using polar coordinates to reformulate the system. The equivalent new system is computationally cheaper because the vector function of the orientation field is replaced by a scalar function. An iteration penalty method is applied to construct a numerical scheme so that stability is improved. We first prove that the scheme is uniquely solvable and unconditionally stable in energy. Then we show that this scheme is of first-order convergence rate by rigorous error estimation. Finally, some numerical simulations are performed to illustrate the accuracy and effectiveness of the scheme.

In this paper, a new high-resolution WENO-MIM scheme that can achieve optimal accuracy at high-order critical points is developed. The nonlinear weight function of the new scheme can be obtained by adding a freely adjustable term with a parameter $λ$ to the mapping function of the WENO-IM scheme. A sufficient condition shows the shock-capturing ability of WENO-MIM will be enhanced with the increase of $λ.$ The parameter $λ=467$ obtained from experience can guarantee the new scheme achieves high resolution. Numerical example results show that the present scheme can achieve optimal accuracy at high-order critical points and perform significantly better than other WENO schemes in highly efficient computing of various compressible fluid problems.