Numerical simulations for the energy-supercritical nonlinear wave equation

We carry out numerical simulations of the defocusing energy-supercritical nonlinear wave equation for a range of spherically-symmetric initial conditions. We demonstrate numerically that the critical Sobolev norm of solutions remains bounded in time. This lends support to conditional scattering results that have been recently established for nonlinear wave equations.Comment: 28 pages, 13 figures. New references and new cases adde

Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications

In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ ($0 < \alpha < 2$) in hypersingular integral form. The proposed finite difference methods provide a fractional analogue of the central difference schemes to the fractional Laplacian, and as $\alpha \to 2^-$, they collapse to the central difference schemes of the classical Laplace operator $-\Delta$. We prove that our methods are consistent if $u \in C^{\lfloor\alpha\rfloor, \alpha-\lfloor\alpha\rfloor+\epsilon}({\mathbb R}^d)$, and the local truncation error is ${\mathcal O}(h^\epsilon)$, with $\epsilon > 0$ a small constant and $\lfloor \cdot \rfloor$ denoting the floor function. If $u \in C^{2+\lfloor\alpha\rfloor, \alpha-\lfloor\alpha\rfloor+\epsilon}({\mathbb R}^d)$, they can achieve the second order of accuracy for any $\alpha \in (0, 2)$. These results hold for any dimension $d \ge 1$ and thus improve the existing error estimates for the finite difference method of the one-dimensional fractional Laplacian. Extensive numerical experiments are provided and confirm our analytical results. We then apply our method to solve the fractional Poisson problems and the fractional Allen-Cahn equations. Numerical simulations suggest that to achieve the second order of accuracy, the solution of the fractional Poisson problem should {\it at most} satisfy $u \in C^{1,1}({\mathbb R}^d)$. One merit of our methods is that they yield a multilevel Toeplitz stiffness matrix, an appealing property for the development of fast algorithms via the fast Fourier transform (FFT). Our studies of the two- and three-dimensional fractional Allen-Cahn equations demonstrate the efficiency of our methods in solving the high-dimensional fractional problems.Comment: 24 pages, 6 figures, and 6 table

Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ for $\alpha \in (0, 2)$. The main advantage of our method is to easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain the scheme structure and computer implementation unchanged. Moreover, our discretization of the fractional Laplacian results in a symmetric (multilevel) Toeplitz differentiation matrix, which not only saves memory cost in simulations but enables efficient computations via the fast Fourier transforms. The performance of our method in both approximating the fractional Laplacian and solving the fractional Poisson problems was detailedly examined. It shows that our method has an optimal accuracy of ${\mathcal O}(h^2)$ for constant or linear basis functions, while ${\mathcal O}(h^4)$ if quadratic basis functions are used, with $h$ a small mesh size. Note that this accuracy holds for any $\alpha \in (0, 2)$ and can be further increased if higher-degree basis functions are used. If the solution of fractional Poisson problem satisfies $u \in C^{m, l}(\bar{\Omega})$ for $m \in {\mathbb N}$ and $0 < l < 1$, then our method has an accuracy of ${\mathcal O}\big(h^{\min\{m+l,\, 2\}}\big)$ for constant and linear basis functions, while ${\mathcal O}\big(h^{\min\{m+l,\, 4\}}\big)$ for quadratic basis functions. Additionally, our method can be readily applied to study generalized fractional Laplacians with a symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.Comment: 21 pages, 7 figure

Dynamical Laws of the Coupled Gross-Pitaevskii Equations for Spin-1 Bose-Einstein Condensates

In this paper, we derive analytically the dynamical laws of the coupled Gross- Pitaevskii equations (CGPEs) without/with an angular momentum rotation term and an external magnetic field for modelling nonrotating/rotating spin-1 Bose-Eintein condensates. We prove the conservation of the angular momentum expectation when the external trapping potential is radially symmetric in two dimensions and cylindrically symmetric in three dimensions; obtain a system of first order ordinary differential equations (ODEs) governing the dynamics of the density of each component and solve the ODEs analytically in a few cases; derive a second order ODE for the dynamics of the condensate width and show that it is a periodic function without/with a perturbation; construct the analytical solution of the CGPEs when the initial data is chosen as a stationary state with its center- of-mass shifted away from the external trap center. Finally, these dynamical laws are confirmed by the direct numerical simulation results of the CGPEs

A novel and simple spectral method for nonlocal PDEs with the fractional Laplacian

We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$. Numerical analysis and experiments are provided to study its performance. Our method has the same symbol $|\xi|^\alpha$ as the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This {\it unique feature} distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature, which are usually limited to periodic boundary conditions (see Remark \ref{remark0}). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector products. The computational complexity is ${\mathcal O}(2N\log(2N))$, and the memory storage is ${\mathcal O}(N)$ with $N$ the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems

Flavonoids and Pectins

Pectins and flavonoids are two related groups of important secondary metabolites derived from plants. The interaction between pectins and flavonoids can affect their shelf-life stability, functionality, bioavailability, and bioaccessibility. In this chapter, we will concentrate on the current opinions on the flavonoids to understand how to classify this group of secondary metabolites, what biological and pharmacological activities they possess, and how to biosynthesize them in plants. We will then discuss the general strategies for the derivation of these small secondary compounds. The strategies comprise traditional plant extraction, chemical synthesis, and biosynthesis. We will also discuss the advantages and disadvantages of these three production strategies in the derivation of flavonoids and the future research directions in generating health-beneficial flavonoids using the biosynthetic strategy

Dynamic Analysis of the Infinite Plate on Orthotropic Foundation Subjected to Moving Loads

Based on the Kirchhoff thin plate theory and elastodynamics theory, the Kirchhoff small deformation&nbsp;infinite elastic thin plate is adopted to simulate the pavement, and the orthotropic elastic half space is used to&nbsp;simulate subgrade. The mechanical model and dynamic equations in the rectangular coordinate system are&nbsp;established for the infinite elastic plate on orthotropic foundation subjected to moving loads. The integral forms&nbsp;of plane strain dynamic responses are derived by means of Fourier transform and inverse Fourier transform.&nbsp;Numerical examples are conducted on condition that the harmonic vibrating strip load is applied on the plate&nbsp;surface. Studies are conducted to investigate the effect of the soil orthotropic parameters on dynamic response of&nbsp;subgrade and the plate. The results indicate that the anisotropy of the soil has a great influence on the dynamic&nbsp;response of subgrade and pavement interaction, and that dynamic response can be described more accurately by&nbsp;considering the orthogonal anisotropy of foundation

New Requirements for English Majors in the Traditional Chinese Medicine Foreign Trade Industry in the New Era

In recent years, traditional Chinese medicine has shown a unique advantage in the treatment and recovery of infected patients, coupled with the gratifying situation of the import and export of traditional Chinese medicine products, and the vibrant prospects of the traditional Chinese medicine foreign trade market, the current situation of the traditional Chinese medicine foreign trade industry is quite promising. Therefore, the demand for Chinese medicine foreign trade talents by Chinese medicine foreign trade enterprises has greatly risen, which also puts forward new requirements for English majors. This paper analyzes the development trend of Chinese medicine cross-border e-commerce industry and the new requirements for English talents in the current era, discusses the necessary working ability and quality that English professionals engaged in Chinese medicine foreign trade must have in the current context, and expounds the main strategies and ways to improve the professional ability and work quality of Chinese medicine foreign trade professionals in the new era from multiple directions based on the practice of the project. This paper hopes to promote the development of the foreign trade industry of traditional Chinese medicine, and at the same time provide certain references for the cultivation of English professionals, and explore a new direction for the employment of English majors