三维Zakharov-Kuznetsov方程解的衰减性

考虑三维Zakharov-Kuznetsov方程的初值问题,证明了该初值问题解的指数衰减性.这个性质与加权Sobolev空间中解的持久性及该问题解的唯一连续性相关.     

一类非线性演化方程初值问题的幂级数解

研究了一类非线性演化方程初值问题.通过不变子空间方法,这类初值问题被约化为常微分方程组的初值问题.这类初值问题是适定的.本文给出了这类初值问题关于时间变量t的幂级数解.     

一类四阶非线性波动方程的初值问题

应用压缩映射原理和延拓定理,在分数次sobolev空间中,证明一类四阶非线性波动方程的初值问题,存在唯一的整体广义解和整体古典解.还给出该初值问题解爆破的充分条件.     

非线性扰动Klein-Gordon方程初值问题的渐近理论

在二维空间中研究一类非线性扰动Klein-Gordon方程初值问题解的渐近理论. 首先利用压缩映象原理,结合一些先验估计式及Bessel函数的收敛性,根据Klein-Gordon方程初值问题的等价积分方程,在二次连续可微空间中得到了初值问题解的适定性;其次,利用扰动方法构造了初值问题的形式近似解,并得到了该形式近似解的渐近合理性;最后给出了所得渐近理论的一个应用,用渐近近似定理分析了一个具体的非线性Klein-Gordon方程初值问题解的渐近近似程度．     

Banach空间中混合型积分-微分方程解的存在性

将微分方程初值问题转化为等价的积分方程,近来此方法被应用于讨论非线性微分方程初值问题解的存在性.利用凸幂凝聚算子的不动点定理,研究了Banach空间中混合型非线性二阶积分-微分方程的初值问题解的存在性.     

矩形薄板瞬态响应的卷积型DQ半解析法

卷积型的Gurtin变分原理是目前在数学上唯一能和动力学初值问题完全等价的变分原理,它完全反映了有关初值问题的全部特征,通过卷积将矩形薄板原始控制方程构造成包含初始条件的新的具有完整初值问题特征的控制方程.对新的控制方程在时间域取解析函数,在空间域采用离散的DQ(differential quadrature)法,从而构造了卷积型DQ半解析法.该方法既可以达到和Gurtin变分原理相同的效果,又避开了Gurtin泛函的繁复,经对矩形薄板的动力响应问题的计算表明,该方法是一种精度好效率高的求解动力响应问题的计算方法.     

Banach空间中脉冲积分-微分方程的初值问题

给出了 Banach空间中一阶线性脉冲积分 -微分方程初值问题解的存在唯一性的一个新证法 ,改进了已有结果 .利用它讨论了一阶非线性脉冲积分 -微分方程初值问题的解 ,所得结果大大推广了已有的相关结果 .     

无限区间上多分数阶微分方程初值问题解的存在与唯一性

本文研究一类无限区间上具有Riemann-Liouville 导数的多分数阶非线性微分方程初值问题,在一类加权函数空间上使用Schauder 不动点定理建立了该问题解的存在性和唯一性结果, 举例说明了定理的应用.     

二阶常微初值问题的时间间断最小二乘有限元法

采用时间间断最小二乘线性有限元方法求解二阶常微分方程初值问题.利用回收技巧及离散Gronwall引理证明了方法的稳定性.通过引入有限元空间上的范数,给出了方法在该范数意义下丰满的误差估计.数值实验验证了理论分析结果.     

Banach空间中一类n阶积分-微分方程初值问题的解

利用一个新的比较定理和乘积空间的锥理论,得出了Banach空间中的一般高阶积分—微分初值问题的解存在性.     

取油管振动方程解的整体不存在性

讨论推广的海底取油管振动方程的初边值问题和初值问题解的整体不存在性,对初边值问题推广了Gmira和Guedda得到的结果,对初值问题的结果是新的.     

关于不可压、无粘流体的Euler方程初值问题的适定性(Ⅰ)

以分层理论为基础,讨论了Euler方程不适定的初值问题以及不适定问题的形式可解性,并给出了某些不适定初值问题存在形式解的条件与计算方法。特别讨论了R4中的超平面{t=0}上初值问题的适定性并给出了存在不唯一解的例证。     

Decay mode solution of nonlinear boundary–initial value problems for the cylindrical (spherical) Boussinesq–Burgers equations

The major target of this paper is to construct new nonlinear boundary–initial value problems for Boussinesq–Burgers Equations, and derive the solutions of these nonlinear boundary–initial value problems by the simplified homogeneous balance method. The nonlinear transformation and its inversion between the Boussinesq–Burgers Equations and the linear heat conduction equation are firstly derived; then a new nonlinear boundary–initial value problem for the Boussinesq–Burgers equations with variable damping on the half infinite straight line is put forward for the first time, and the solution of this nonlinear boundary–initial value problem is obtained, especially, the decay mode solution of nonlinear boundary–initial value problem for the cylindrical (spherical) Boussinesq–Burgers equations is obtained.     

Inverse problem for the diffusion equation in the case of spherical symmetry

The initial boundary value problem for the diffusion equation is considered in the case of spherical symmetry and an unknown initial condition. Additional information used for determining the unknown initial condition is an external volume potential whose density is the Laplace operator applied to the solution of the initial boundary value problem. The uniqueness of the solution of the inverse problem is studied depending on the parameters entering into the boundary conditions. It is shown that the solution of the inverse problem is either unique or not unique up to a one-dimensional linear subspace.     

Inverse problem for the diffusion equation with overdetermination in the form of an external volume potential

An initial-boundary value problem for the diffusion equation with an unknown initial condition is considered. Additional information used for determining the unknown initial condition is an external volume potential whose density is the Laplacian calculated for the solution of the initial-boundary value problem. Uniqueness theorems for the inverse problem are proved in the case when the spatial domain of the initial-boundary value problem is a spherical layer or a parallelepiped.     

A variational method applied to a linear hyperbolic initial boundary value problem

This paper deals with the application of a variational method to a boundary value problem of the wave equation. Starting with an initial boundary value problem (which is given) introduction of a boundary condition at the final time leads to a boundary value problem with one of the initial conditions redundant. This redundant initial condition is used by the trial function of the direct method (of the Ritz type) which is employed to stationarize the variational principle.     

Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries

The exhaustive group classification of a class of variable coefficient generalized KdV equations is presented, which completes and enhances results existing in the literature. Lie symmetries are used for solving an initial and boundary value problem for certain subclasses of the above class. Namely, the found Lie symmetries are applied in order to reduce the initial and boundary value problem for the generalized KdV equations (which are PDEs) to an initial value problem for nonlinear third-order ODEs. The latter problem is solved numerically using the finite difference method. Numerical solutions are computed and the vast parameter space is studied.     

Fractional model and solution for the Black‐Scholes equation

This work presents a new model of the fractional Black‐Scholes equation by using the right fractional derivatives to model the terminal value problem. Through nondimensionalization and variable replacements, we convert the terminal value problem into an initial value problem for a fractional convection diffusion equation. Then the problem is solved by using the Fourier‐Laplace transform. The fundamental solutions of the derived initial value problem are given and simulated and display a slow anomalous diffusion in the fractional case.     

Method of weighted residuals as applied to nonlinear differential equations

A new method for solving a class of nonlinear boundary-value problems is presented. In this method, the nonlinear equation is linearized by guessing an initial solution and using it to evaluate the nonlinear terms. Next, a method of weighted residuals is applied to transform the linearized form of the boundary value problem to an initial value problem. The second (improved) solution is obtained by integrating the initial value problem by a fourth order Runge-Kutta scheme. The entire process is repeated until a desired convergence criterion is achieved.