几种广义非线性发展方程的新解

本文研究了构造了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解的问题.利用三种辅助方程及其新解,获得了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解.这些解由双曲余割函数、双曲正切函数、双曲正割函数、双曲余切函数和余割函数组成.     

Integral representation of solutions to Fuchsian system and Heun's equation

We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard's solution of the sixth Painlevé equation, and to Heun's equation.     

On solutions of a discretized heat equation in discrete Clifford analysis

The main purpose of this paper was to study solutions of the heat equation in the setting of discrete Clifford analysis. More precisely we consider this equation with discrete space and continuous time. Thereby we focus on representations of solutions by means of dual Taylor series expansions. Furthermore we develop a discrete convolution theory, apply it to the inhomogeneous heat equation and construct solutions for the related Cauchy problem by means of heat polynomials.     

Special Monodromy Groups and the Riemann-Hilbert Problem for the Riemann Equation

In this paper, we solve the Riemann-Hilbert problem for the Riemann equation and for the hypergeometric equation. We describe all possible representations of the monodromy of the Riemann equation. We show that if the monodromy of the Riemann equation belongs to SL(2, ℂ), then it can be realized not only by the Riemann equation, but also by the more special Riemann-Sturm-Liouville equation. For the hypergeometric equation, we construct a criterion for its monodromy group to belong to SL(2, ℤ).__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 753–767.Original Russian Text Copyright ©2005 by V. A. Poberezhnyi.     

Geometric Properties and Exact Travelling Wave Solutions for the Generalized Burger-Fisher Equation and the Sharma-Tasso-Olver Equation

In this paper, we study the dynamical behavior and exact parametric representations of the traveling wave solutions for the generalized Burger-Fisher equation and the Sharma-Tasso-Olver equation under different parametric conditions, the exact monotonic and non-monotonic kink wave solutions, two-peak solitary wave solutions, periodic wave solutions, as well as unbounded traveling wave solutions are obtained.     

Numerical stability in the calculation of eigenfrequencies using integral equations

We comment on a phenomenon of instability that appears while computing eigenfrequencies using the integral equation framework. More precisely, it is currently known that the real symmetric matrices are well, and sometimes the best, adapted to numerical treatment. However, we show that this is not the case, if we wish to determine with high accuracy the spectrum of elliptic, and other related operators, using integral representations.     

On the properties of a Cauchy-type problem for an abstract differential equation with fractional derivatives

We study the relationship between the solutions of abstract differential equations with fractional derivatives and their stability with respect to the perturbation by a bounded operator. Besides, we obtain representations for the solution of an inhomogeneous equation and for an equation containing a fractional power of the generator of a cosine operator function.     

The inverse electromagnetic scattering problem by a penetrable cylinder at oblique incidence

In this work, we consider the method of non-linear boundary integral equation for solving numerically the inverse scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder in three dimensions. We consider the indirect method and simple representations for the electric and the magnetic fields in order to derive a system of five integral equations, four on the boundary of the cylinder and one on the unit circle where we measure the far-field pattern of the scattered wave. We solve the system iteratively by linearizing only the far-field equation. Numerical results illustrate the feasibility of the proposed scheme.     

Asymptotic solutions to a discrete airy equation

A nonstandard finite difference scheme for the Airy equation leads to a linear, second—order difference equation for which the theorems of Poincaré and Perron do not apply if asymptotic representations of the solutions are desired. Using the method of dominant balance, a suggested form for the asymptotic solutions is obtained. This relation is then used to construct the required asymptotic representations for the two linearly independent solutions     

Functional Integrals for the Schrodinger Equation on Compact Riemannian Manifolds

In this paper, we represent the solution of the Cauchy problem for the Schrodinger equation on compact Riemannian manifolds in terms of functional integrals with respect to the Wiener measure corresponding to the Brownian motion in a manifold and with respect to the Smolyanov surface measures constructed from the Wiener measure on trajectories in the underlying space. The representation of the solution is obtained for the case of analytic (on some sets) potential and analytic initial condition under certain assumptions on the geometric characteristics of the manifold. In the proof, we use a method due to Doss and the representations via functional integrals of the solution to the Cauchy problem for the heat equation in a compact Riemannian manifold.     

Probabilistic solutions to nonlinear fractional differential equations of generalized Caputo and Riemann–Liouville type

This paper provides well-posedness and integral representations of the solutions to nonlinear equations involving generalized Caputo and Riemann–Liouville type fractional derivatives. As particular cases, we study the linear equation with non constant coefficients and the generalized composite fractional relaxation equation. Our approach relies on the probabilistic representation of the solution to the generalized linear problem recently obtained by the authors. These results encompass some known cases in the context of classical fractional derivatives, as well as their far reaching extensions including various mixed derivatives.     

Moments of the Time of Ruin, Surplus Before Ruin and the Deficit at Ruin in the Erlang（N） Risk Process

In this paper we consider the ＂penalty＂ function in the Erlang（n） risk model. Using the integro- differential equation we established, we obtain the explicit expressions for the moments of Erlang（2） risk model. When the claim size distribution is Light-Tailed and the penalty function is bounded, we obtain the exact representations for the moments of Erlang（n） risk model.     

On a Solution of the Quaternion Matrix Equation X - A X B = C and Its Application

This paper first studies the solution of a complex matrix equation X - AXB = C, obtains an explicit solution of the equation by means of characteristic polynomial, and then studies the quaternion matrix equation X - A X B = C, characterizes the existence of a solution to the matrix equation, and derives closed-form solutions of the matrix equation in explicit forms by means of real representations of quaternion matrices. This paper also gives an application to the complex matrix equation X - AXB =C.     

Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation

Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 252–264, May, 2008.     

彩虹障碍期权的定价问题

采用偏微分方程方法研究了彩虹障碍期权的定价问题,推导出它满足的偏微分方程,通过求解这个偏微分方程得出了八种彩虹障碍期权的定价公式及四个看涨——看跌平价公式.     

脉冲时滞抛物型方程解的振动准则

本文研究含脉冲的时滞抛物型方程解的振动性,在两类不同边界条件下得到了若干解的振动准则．     

Surfaces of revolution associated with the kink-type solutions of the SIdV equation

In this paper, we study the evolution scenarios of surfaces of revolution associated with the kink-type solutions of an integrable equation, which is called the SIdV equation because of its scale-invariant property and relationship with the Korteweg-de Vries equation, where the kink-type solutions play the role of a metric. We put forward two kinds of evolution scenarios for surfaces of revolution associated with two types of kink-type metric (solution) and we study the key properties of these surfaces.     

Equivalence of three kinds of well-posed-ness of discontinuous Riemann-Hilbert problem for elliptic complex equation in multiply connected domains

In this article, we first introduce the general linear elliptic complex equation of first order with certain conditions, and then propose discontinuous Riemann-Hilbert problem and some kinds of modified well-posed-ness for the complex equation. Then we verify the equivalence of three kinds of well-posed-ness. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.     

特征为2的有限域上对称矩阵方程解的计数公式及q超几何级数表达

设Fq是有q＝2t个元的有限域.本文利用Fq上奇异辛几何和奇异伪辛几何理论,给出当A,C是Fq上对称矩阵时,Fq上适合XAXT＝C的解存在的充要条件以及秩k的解X和解X的个数的明显公式,并且用q超几何级数简化表达解数公式．     

Exact traveling wave solutions and dynamical behavior for the (n+1)-dimensional multiple sine-Gordon equation

Using the methods of dynamical systems for the (n 1)-dimensional multiple sine-Gordon equation, the existences of uncountably infinite many periodic wave solutions and breaking bounded wave solutions axe obtained. For the double sine-Gordon equation, the exact explicit parametric representations of the bounded traveling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.