应用改进的简单方程法求非线性数学物理方程的精确解

应用改进的简单方程法求得Cahn-Allen方程和Jimbo-Miwa方程的精确解,这些解包括双曲函数解、三角函数解.当对双曲函数解中的参数取特殊值时,可以得到了孤立波解.当对三角函数解中的参数取特殊值时,可以得到对应的周期波函数解.实践证明,简单方程法对于研究非线性数学物理方程具有非常广泛的应用意义.     

Kannappan’s functional equation on semigroups with involution

We express the complex-valued solutions of Kannappan’s functional equation on semigroups with involution in terms of solutions of d’Alembert’s functional equation.     

A family of trigonometric extremals in the problem of reorienting a spherically symmetrical body with minimum energy consumption

The problem of the optimal control of the reorientation of an absolutely rigid, spherically symmetric body is investigated. An integral quadratic functional, which characterizes the total energy consumption, is chosen as the criterion of the efficiency of the manoeuvre. The resultant torque of the applied external forces serves as the control. Application of the formalism of the Pontryagin's maximum principle leads to an analysis of a third-order non-linear vector differential equation, whose general solution is still unknown at the present time. It is shown that this equation has a particular solution described by trigonometric functions of time, which can be used to completely reconstruct the explicit solution for the corresponding extremal rotation. An analogy with the free rotation of a certain axisymmetric body is proposed.     

Rational interpolation and approximate solution of integral equations

In the space of continuous periodic functions, we construct interpolation rational operators, use them to obtain quadrature formulas with positive coefficients which are exact on rational trigonometric functions of order 2n, and suggest an algorithm for an approximate solution of integral equations of the second kind. We estimate the accuracy of the approximate solution via the best trigonometric rational approximations to the kernel and the right-hand side of the integral equation.     

Range projections and the Moore-Penrose inverse in rings with involution

In this article we study the existence of range projections in rings with involution, relating it to the existence of the Moore-Penrose inverse. The results are applied to the solution of the equation xbx = x in rings with involution, extending the results of Greville for matrices. Simpler new proofs are given of the Moore-Penrose invertibility of regular elements in rings with involution, and of the Ljance's formula.     

Van Vleck’s functional equation for the sine

We solve Van Vleck’s functional equation on semigroups with an involution in terms of multiplicative functions.     

Range projections and the Moore–Penrose inverse in rings with involution

In this article we study the existence of range projections in rings with involution, relating it to the existence of the Moore–Penrose inverse. The results are applied to the solution of the equation xbx?=?x in rings with involution, extending the results of Greville for matrices. Simpler new proofs are given of the Moore–Penrose invertibility of regular elements in rings with involution, and of the Ljance's formula.     

三角级数在Burgers-KdV混合型方程中的应用

利用三角级数法将Burgers-KdV混合型方程转化为一组非线性代数方程,进而用待定系数法求解方程组,最后求出了Burgers-KdV混合型方程的精确解.     

非线性发展方程新的显式精确解

借助Ｍａｔｈｅｍａｔｉｃａ系统,采用三角函数法和吴文俊消元法,本文获得了著名的２＋１维ＫＰ方程的若干精确解,其中包括新的精确解和孤波解．在此基础上,进而得到著名ＫｄＶ方程、Ｈｉｒｏｔａ－Ｓａｔｓｕｍａ方程和耦合ＫｄＶ方程的一些精确解．     

An analytical solution for two and three dimensional nonlinear Burgers' equation

This paper derives analytical solutions for the two dimensional and the three dimensional Burgers' equation. The two-dimensional and three-dimensional Burgers' equation are defined in a square and a cubic space domain, respectively, and a particular set of boundary and initial conditions is considered. The analytical solution for the two dimensional Burgers' equation is given by the quotient of two infinite series which involve Bessel, exponential, and trigonometric functions. The analytical solution for the three dimensional Burgers' equation is given by the quotient of two infinite series which involve hypergeometric, exponential, trigonometric and power functions. For both cases, the solutions can describe shock wave phenomena for large Reynolds numbers (R_e ≥ 100), which is useful for testing numerical methods.     

Reduction of the Sharma–Tasso–Olver equation and series solutions

This paper considers series solutions of the Sharma–Tasso–Olver (STO) equation. By using the extended homogenous balance method, we reduce the STO equation to a linear PDE and obtain Bäcklund transformation of it. Furthermore, the self-transformation of solutions for the STO equation is obtained. By the Bäcklund transformation and various series solutions of the PDE, abundant exact solutions of the STO equation are obtained including the multi-solitary wave solution, trigonometric function series solution, rational series solution and solution consisting of the three types of solutions.     

On the stability of some functional equations for generalized hyperbolic functions and for the generalized cosine equation

Two stability results concerning a system of functional equations for generalized hyperbolic and trigonometric functions on the one hand, and a generalized cosine equation on the other hand are presented.     

Trigonometric identities in *-algebras

We deal with two systems of trigonometric equations where unknown functions are defined on a group and take values in a (not-necessarily commutative) complex algebra with involution. Our purpose is to give some extension of sine and cosine functions from the scalar to the vector case.     

Analysis of trigonometric implicit Runge–Kutta methods

Using generalized collocation techniques based on fitting functions that are trigonometric (rather than algebraic as in classical integrators), we develop a new class of multistage, one-step, variable stepsize, and variable coefficients implicit Runge–Kutta methods to solve oscillatory ODE problems. The coefficients of the methods are functions of the frequency and the stepsize. We refer to this class as trigonometric implicit Runge–Kutta (TIRK) methods. They integrate an equation exactly if its solution is a trigonometric polynomial with a known frequency. We characterize the order and A-stability of the methods and establish results similar to that of classical algebraic collocation RK methods.     

The equiconvergence of expansions in eigenfunctions and associated functions of an integral operator with involution

In this paper we study an integral operator with involution. We solve the problem on the exact inversion of this operator, we obtain and study the integro-differential system for the Fredholm resolvent and, finally, we prove the theorem on the equiconvergence of expansions in eigenfunctions and associated functions, in the usual trigonometric system.     

Application of pseudo‐trigonometry to Bessel series and integrals of Bessel functions

In this article, an extension of the Laplace transform of J_n (t) to pseudo‐trigonometric function is discussed. We are seeking elementary functions expressed by Bessel series. It is shown that the result is applicable to the solution of the first‐order differential equation. The expression of modified Bessel integral formulas in pseudo‐trigonometric function is also discussed.     

A trigonometric Galerkin method for volume integral equations arising in TM grating scattering

Transverse magnetic (TM) scattering of an electromagnetic wave from a periodic dielectric diffraction grating can mathematically be described by a volume integral equation.This volume integral equation, however, in general fails to feature a weakly singular integral operator. Nevertheless, after a suitable periodization, the involved integral operator can be efficiently evaluated on trigonometric polynomials using the fast Fourier transform (FFT) and iterative methods can be used to solve the integral equation. Using Fredholm theory, we prove that a trigonometric Galerkin discretization applied to the periodized integral equation converges with optimal order to the solution of the scattering problem. The main advantage of this FFT-based discretization scheme is that the resulting numerical method is particularly easy to implement, avoiding for instance the need to evaluate quasiperiodic Green’s functions.     

Geometric-Dynamic Approach to Billiard Systems: I. Projective Involution of a Billiard,Direct and Inverse Problems

We suggest a geometric-dynamic approach to billiards as a special kind of reversible dynamic system and establish their relation to projective transformations (involutions) in the framework of this approach. We state the direct and inverse problems for billiards and derive equations determining the solutions of these problems in general form. Some simplest billiard involutions are calculated. We establish functional relations between the involution of a billiard, the equation for its boundary, and the field of normals to the boundary. We show how the involution is related to the curvature of the billiard boundary.     

Symmetry reduction, exact solutions and conservation laws of the Sawada-Kotera-Kadomtsev-Petviashvili equation

In this paper, by applying a direct symmetry method, we obtain the symmetry reduction, group invariant solution and many new exact solutions of SK-KP equation, which include Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions and so on. At last, we also give the conservation laws of SK-KP equation.