一类两参数半线性奇摄动问题解的渐近性态

本文研究了一类两参数半线性奇摄动问题的基本模型.利用奇摄动方法,对该问题解的结构在两个小参数相互关联的三种不同情形下作了讨论.得到了该问题在三种不同情形下的渐近解并证明了在三种情形下解的结构与渐近性态.     

一类四阶方程两参数的奇摄动问题

研究了一类四阶方程两参数的奇摄动问题.利用奇摄动方法,对该同题的解的结构在两个小参数相互关联的三种不同情形下作了全面的彻底的讨论.得到了该问题在三种不同情形下的渐近解;证明了在三种情形下完全不同的解的结构与极限性态;给出了比较完善的结果.     

一类两参数非线性奇摄动边值问题解的渐近性态

本文研究了一类两参数非线性奇摄动边值问题的基本模型.利用奇摄动方法,对该问题解的结构在两个小参数相互关联的三种不同情形下作了讨论,得到了该问题的渐近解并证明了在三种情形下不同的解的结构与渐近性态.     

两参数非局部非线性反应扩散Robin问题的渐近解

研究了一类两参数非局部反应扩散奇摄动Robin问题.利用奇摄动方法,对该问题解的结构在两个小参数相互关联的情形下作了讨论.得到了该问题的渐近解.     

一类双参数奇摄动非线性反应扩散方程

本文研究了一类双参数非线性反应扩散奇摄动问题的模型.利用奇摄动方法,对该问题解的结构在两个小参数相互关联的情形下作了讨论.得到了该问题的渐近解,由解的展开式看出本问题的解同时具有初始层和边界层.     

Poincaré方程的线性差耦合系统

该文对Poincare方程的线性差耦合系统的同相解及反相解进行研究,得到了同相解稳定的参数区域,并在对角线性差耦合的情形下.对其反相解的存在性及稳定性进行了完整的分析,改进了文[1]的结果.     

有限理性下参数最优化问题解的稳定性

主要研究有限理性下参数最优化问题解的稳定性. 即在两类扰动即目标函数及可行集二者, 目标函数、可行集及参数三者分别同时发生扰动的情形下, 对参数最优化问题引入一个抽象的理性函数, 分别建立了参数最优化问题的有限理性模型M, 运用``通有'的方法, 得到了上述两种扰动情形下相应的有限理性模型M的结构稳定性及对\varepsilon-平衡(解)的鲁棒性, 即有限理性下绝大多数的参数最优化问题的解都 是稳定的, 并以一个例子说明所得的稳定性结果均是正确的.     

一类四阶半线性方程的奇摄动解

讨论了一类四阶半线性方程奇摄动边值问题.利用上下解方法,研究了边值问题解的存在性和渐近性态.指出了在该文的情形下具有两参数的原奇摄动问题的解只有一个边界层.     

Ambrosetti-Prodi问题-常微分方程组情形

本文研究常微分方程组情形的Ambrosetti-Prodi型问题.在非线性项超线性,凸性等条件下,得出随着参数的变化,问题无解,有唯一解,至少有两解的结论.     

板和扁壳大挠度问题摄动参数的最小二乘法选择

本文提出了在用摄动法求解板和扁壳轴对称大挠度问题时,确定摄动参数的最小二乘方法.计算了圆板情形的算例,与准确解和其它摄动解做了比较.结果表明,本文解答较其它摄动解有更高的精确度.     

关于两参数常微分方程非线性边值问题的奇异摄动

本文应用改进的多重尺度法及特殊的对多参数问题的讨论方法，考察含两个小参数的常微分方程的非线性边值问题，证明了解的存在性，作出了渐近解，并应用较简单的方法估计了余项。     

关于含双参数的非线性常微分方程的奇异摄动*

本文应用微分不等式方法研究含双小参数的非线性常微分方程的边值问题,作出渐近解并对余项进行估计.     

Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations

We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg–deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained.     

Transmutations and irregular singular points

We consider transumtations for a class of problems in partial differential equations where the underlying equation, involving two assignable parameters, is an associated ordinary differential equation with an irregular singular point. An integral formula for the solution of this associated problem, valid for negative values of a timelike variable t, permits relating the solution of the problems in partial differential equations to be bounded or slow groth solutions of generalized heat problems. Applications of the formulas are made to Cauchy and boundary type problems.     

双参数半线性高阶椭圆型方程的奇摄动解

讨论了一类具有双参数的半线性高阶椭圆型方程边值问题.利用微分不等式理论,研究了边值问题解的存在性和渐近性态.     

Dependence of solutions and eigenvalues of measure differential equations on measures

It is well known that solutions of ordinary differential equations are continuously dependent on finite-dimensional parameters in equations. In this paper we study the dependence of solutions and eigenvalues of second-order linear measure differential equations on measures as an infinitely dimensional parameter. We will provide two fundamental results, which are the continuity and continuous Fréchet differentiability in measures when the weak^? topology and the norm topology of total variations for measures are considered respectively. In some sense the continuity result obtained in this paper is the strongest one. As an application, we will give a natural, simple explanation to extremal problems of eigenvalues of Sturm–Liouville operators with integrable potentials.     

A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem

In this paper, a class of singularly perturbed elliptic partial differential equations posed on a rectangular domain is studied. The differential equation contains two singular perturbation parameters. The solutions of these singularly perturbed problems are decomposed into a sum of regular, boundary layer and corner layer components. Parameter-explicit bounds on the derivatives of each of these components are derived. A numerical algorithm based on an upwind finite difference operator and a tensor product of piecewise-uniform Shishkin meshes is analysed. Parameter-uniform asymptotic error bounds for the numerical approximations are established.     

一类含参数分数阶微分方程边值问题正解的性质研究

研究了一类含参数的分数阶微分方程边值问题,主要运用锥上的不动点定理及混合单调算子特征值问题的性质得出了正解关于参数的性质:存在唯一性、单调性、连续性以及极限性质.最后举例说明了结果的可行性.     

Lie group symmetry analysis for granular media stress equations

The Airy stress function, although frequently employed in classical linear elasticity, does not receive similar usage for granular media problems. For plane strain quasi-static deformations of a cohesionless Coulomb-Mohr granular solid, a single nonlinear partial differential equation is formulated for the Airy stress function by combining the equilibrium equations with the yield condition. This has certain advantages from the usual approach, in which two stress invariants and a stress angle are introduced, and a system of two partial differential equations is needed to describe the flow. In the present study, the symmetry analysis of differential equations is utilised for our single partial differential equation, and by computing an optimal system of one-dimensional Lie algebras, a complete set of group-invariant solutions is derived. By this it is meant that any group-invariant solution of the governing partial differential equation (provided it can be derived via the classical symmetries method) may be obtained as a member of this set by a suitable group transformation. For general values of the parameters (angle of internal friction ? and gravity g) it is found there are three distinct classes of solutions which correspond to granular flows considered previously in the literature. For the two limiting cases of high angle of internal friction and zero gravity, the governing partial differential equation admit larger families of Lie point symmetries, and from these symmetries, further solutions are derived, many of which are new. Furthermore, the majority of these solutions are exact, which is rare for granular flow, especially in the case of gravity driven flows.     

Fractional order iterative functional differential equations with parameter

In this paper, we study a fractional order iterative functional differential equation with parameter. Some theorems to prove existence of the iterative series solutions are presented under some natural conditions. Unfortunately, uniqueness results can not be obtained since the solution operator is not Lipschitz continuous but only Hölder continuous. Meanwhile, data dependence results of solutions and parameters provide possible way to describe the error estimates between explicit and approximative solutions for such problems. We also make some examples to illustrate our results. Finally, we conclude with some possible extensions to general parametrized iterative fractional functional differential equations.