二维抛物型方程简单实用的显格式

提出了一个解二维抛的型方程初边值问题的简单实用的显格式，证明了其截断误差阶是Ｏ，稳定性条件是α＋β≠１／２且ｍａｘ｛α，β｝≤１／４，其中，α＝α．Δｔ／Δｘ＾２．β＝α．Δｔ／Δｙ＾２。     

Symmetry reduction and exact solutions of the (3+1)-dimensional Zakharovben Kuznetsov equation

By means of the classical method, we investigate the (3+1)-dimensional Zakharov-Kuznetsov equation. The symmetry group of the (3+1)-dimensional Zakharov-Kuznetsov equation is studied first and the theorem of group invariant solutions is constructed. Then using the associated vector fields of the obtained symmetry, we give the one-, two-, and three-parameter optimal systems of group-invariant solutions. Based on the optimal system, we derive the reductions and some new solutions of the (3+1)-dimensional Zakharov-Kuznetsov equation.     

A robust numerical model for premixed flames with high density ratios based on new pressure correction and IMEX schemes

In this study we present a model for the interaction of premixed flames with obstacles in a channel flow. Although the flow equations are solved with Direct Numerical Simulation using a low Mach number approximation, the resolution used in the computation is limited (∼1 mm) hence the inner structure of the flame and the chemical scales are not solved. The species equations are substituted with a source term in the energy equation that simulates a one-step global reaction. A level set method is applied to track the position of the flame and its zero level is used to activate the source term in the energy equation only at the flame front. An immersed boundary method reproduces the geometry of the obstacles. The main contribution of the paper is represented by the proposed numerical approach: an IMEX (implicit–explicit) Runge–Kutta scheme is used for the time integration of the energy equation and a new pressure correction algorithm is introduced for the time integration of the momentum equations. The approach presented here allows to calculate flames which produce high density ratios between burnt and unburnt regions. The model is verified by simulating first simple solutions for one- and two-dimensional flames. At last, the experiments performed by Masri and Ibrahim with square and rectangular bodies are calculated.     

Bifurcations of exact travelling wave solutions for the generalized R-K-L equation

Using the method of dynamical systems for the the generalized Radhakrishnan, Kundu, Lakshmanan equation, the existence of soliton solutions, uncountably infinite many periodic wave solutions and unbounded wave solution are obtained. Exact explicit parametric representations of the above travelling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.     

A spectral interpretation for the explicit formula in the theory of prime numbers

A spectral interpretation for the poles and zeros of the L-function of algebraic number fields is given by Meyer. As Meyer works with Schwartz spaces which are not Hilbert spaces, the information on the location of zeros of the L-function is lost. In 1999, A. Connes gave a spectral interpretation for the critical zeros the Riemann zeta function. He works with Hilbert spaces. In this paper, we show that a variant of Connes’ trace formula is essentially equal to the explicit formula of A. Weil.     

New explicit solutions for the Klein-Gordon equation with cubic nonlinearity

In this paper, we investigate Klein-Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Explicit peakon and solitary wave solutions for the modified Fornberg-Whitham equation

In this paper, the modified Fornberg-Whitham equation is studied by using the bifurcation theory and the method of phase portraits analysis. In some parametric conditions, some peakons and solitary waves are found and their exact parametric representations in explicit form are obtained.     

A note on a three-point partial difference equation with variable coefficients

The explicit solution of four-point linear partial difference equations, provided with variable coefficients and with boundary conditions including the independent variables, was found in a previous note. In addition, the note explained the procedure to be used in case of boundary conditions also including the dependent variables. The aim of this note is to determine the explicit solution of a three-point equation of the above-mentioned second type, encountered in the study of differential difference equations with the method of the steps.