二阶变系数线性微分方程的几个可积类型

<正> 文[1]给出了一些变系数线性微分方程的可积类型,[2、3]利用“连锁”法也给出了一些变系数线性微分方程的可积类型,且包含了[1]的结果.本文继续讨论一般的二阶变系数线性微分方程     

Recent Progress in the $L_p$ Theory for Elliptic and Parabolic Equations with Discontinuous Coefficients

In this paper, we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients. We begin with an approach given by N. V. Krylov to parabolic equations in the whole space with $\rm{VMO}_x$ coefficients. We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions, weighted $L_p$estimates with Muckenhoupt ($A_p$) weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations.     

Solution by quadratures of certain ordinary differential equations

This paper discusses methods that are applicable in the solution by quadratures of linear second‐order differential equations with variable coefficients. These same techniques, when applied to equations with constant coefficients, produce an extremely useful method in the teachingof ordinary differential equations.     

Analysis of Caughley model with diffusion from mathematical ecology

We provide an analysis in function spaces of the nonlinear semigroup generated by the Caughley model with varied diffusion from mathematical ecology. The global long time asymptotic dynamics of the system of equations are well posed in the sense of an attractor. The behaviour of this attractor in small diffusion coefficients is studied. Two limit problems depending on the stability of the spatial domain in diffusion coefficients are obtained. An adequate scaling of the space variable yields a diffusion coefficients dependent spatial domain. The limit model equations are defined in the complete space of the domain and its diffusion coefficients are unitary. If the domain does not change with the diffusion coefficients, we obtain as a limit problem the system of equations with zero diffusion coefficients and no boundary conditions. The family of attractors in small diffusion coefficients is proved in the Hausdroff semidistance of sets to converge in the uniform topology of continuous functions.     

The exact solutions and the relevant constraint conditions for two nonlinear Schrödinger equations with variable coefficients

Two nonlinear Schrödinger equations with variable coefficients are researched, and the various exact solutions (including the bright and dark solitary waves) of the nonlinear Schrödinger equations are obtained with the aid of a subsidiary elliptic-like equation (sub-ODEs for short), at the same time, the constraint conditions which the coefficients of the nonlinear Schrödinger equations with variable coefficients satisfy are presented. The exact solutions and the constraint conditions are helpful in the application of the nonlinear Schrödinger equations with variable coefficients studied in this paper.     

Chebyshev polynomial approximation for high‐order partial differential equations with complicated conditions

In this article, a new method is presented for the solution of high‐order linear partial differential equations (PDEs) with variable coefficients under the most general conditions. The method is based on the approximation by the truncated double Chebyshev series. PDE and conditions are transformed into the matrix equations, which corresponds to a system of linear algebraic equations with the unknown Chebyshev coefficients, via Chebyshev collocation points. Combining these matrix equations and then solving the system yields the Chebyshev coefficients of the solution function. Some numerical results are included to demonstrate the validity and applicability of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On the oscillation of some linear difference equations with periodic coefficients

Some linear difference equations with periodic coefficients (not necessarily nonnegative) are considered. Necessary conditions and sufficient conditions for the oscillation of the solutions are established. Conditions under which all nonoscillatory solutions tend to zero at ∞ are also presented. The results obtained are the discrete analogues of the oscillation results for some linear delay differential equations with periodic coefficients, which were given earlier by the second author [Oscillations of some delay differential equations with periodic coefficients, J. Math. Anal. Appl. 162 (1991) 452–475].     

Asymptotic Expansions and Influence Coefficients for Edge-Loaded Conical Shells

The equations of linear elasticity for rotationally symmetric deformations are expanded using a small parameter related to the thickness to radius of curvature ratio of the shell to obtain the classical thin shell equations of conical shells as a first approximation. These classical equations with variable coefficients permit further asymptotic expansions in the cases of steep as well as shallow cones, yielding systems of equations with constant coefficients. Solutions of these equations are used to compute the influence coefficients relating edge loads and edge displacements.     

A note on geometric conditions for boundary control of wave equations with variable coefficients

The analytical condition given by Wyler for boundary stabilization of wave equations with variable coefficients is compared with the geometrical condition derived by Yao in terms of the Riemannian geometry method for exact controllability of wave equations with variable coefficients. It is shown that these two conditions are equivalent.     

Numerical solution of high-order differential equations by using periodized Shannon wavelets

In this paper, periodized Shannon wavelets are applied as basis functions in solution of the high-order ordinary differential equations and eigenvalue problem. The first periodized Shannon wavelets are defined. The second the connection coefficients of periodized Shannon wavelets are related by a simple variable transformation to the Cattani connection coefficients. Finally, collocation method is used for solving the high-order ordinary differential equations and eigenvalue problem. Some equations are solved in order to find out advantage of such choice of the basis functions.     

Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation

Transformations of the measure of orthogonality for orthogonal polynomials, namely Freud transformations, are considered. Jacobi matrix of the recurrence coefficients of orthogonal polynomials possesses an isospectral deformation under these transformations. Dynamics of the coefficients are described by generalized Toda equations. The classical Toda lattice equations are the simplest special case of dynamics of the coefficients under the Freud transformation of the measure of orthogonality. Also dynamics of Hankel determinants, its minors and other notions corresponding to the orthogonal polynomials are studied.     

Stability of solutions to differential equations with several variable delays. I

We consider a class of scalar linear differential equations with several variable delays and constant coefficients. A family of equations of the class is defined by coefficients and maximum admissible values of delays. We obtain conditions that are necessary and sufficient for the stability of solutions to all equations of the family. It is ascertained that the conditions are determined entirely by properties of the solution to the initial problem for an autonomous equation that belongs to the family. Some alternatives of required conditions are obtained in the form of estimates for solutions to autonomous equations in a finite interval.     

高阶变系数线性微分方程的一些新的可积类型

借助双变换—未知函数的变换和自变量的变换,将几类高阶变系数线性微分方程化为相应的常系数线性微分方程,从而顺利求得它们的通解,得到了变系数线性微分方程新的可积类型,所得结果极大地推广了著名的Euler方程及前人的一些的工作,并给出了相应的实例加以佐证.     

一类高阶整函数系数线性微分方程解的超级

本文利用值分布理论,对高阶整函数系数线性微分方程(其中存在两个系数的级相等且最大)的解的复振荡性质进行了研究,得到了方程解的超级的精确估计.     

一类高阶线性变系数常微分方程的通解

利用升阶法研究了一类高阶线性变系数常微分方程,给出了齐次方程的通解公式,并讨论了非齐次方程待定的特解.     

Special vekua equations with singular coefficients

Special equations of Vekua-type with singular coefficients are considered. As a first step we study the influence of the coefficients of model equations on the choice of the function spaces for its solutions and on the boundary conditions. As an application we sketch the consideration of boundary value problems for Vekua equations with variable coefficients having a strong singularity at z =0     

A Hermite collocation method for the approximate solutions of high‐order linear Fredholm integro‐differential equations

In this study, a Hermite matrix method is presented to solve high‐order linear Fredholm integro‐differential equations with variable coefficients under the mixed conditions in terms of the Hermite polynomials. The proposed method converts the equation and its conditions to matrix equations, which correspond to a system of linear algebraic equations with unknown Hermite coefficients, by means of collocation points on a finite interval. Then, by solving the matrix equation, the Hermite coefficients and the polynomial approach are obtained. Also, examples that illustrate the pertinent features of the method are presented; the accuracy of the solutions and the error analysis are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1707–1721, 2011     

具有变系数的偶数阶中立型差分方程的振动性

考虑了一类具有变系数的偶数阶中立型差分方程的振动性,通过建立一个比较定理,获得一些一类具有变系数的偶数阶中立型差分方程的振动性的充分条件.     

Frobenius integrable decompositions for ninth-order partial differential equations of specific polynomial type

Frobenius integrable decompositions are presented for a kind of ninth-order partial differential equations of specific polynomial type. Two classes of such partial differential equations possessing Frobenius integrable decompositions are connected with two rational Bäcklund transformations of dependent variables. The presented partial differential equations are of constant coefficients, and the corresponding Frobenius integrable ordinary differential equations possess higher-order nonlinearity. The proposed method can be also easily extended to the study of partial differential equations with variable coefficients.     

Divergent Type Quasilinear Dirichlet Problem with Singularities

A quasilinear equation of divergent type with singular data and singular coefficients is approximated by a net of equations of the same type with enough regular coefficients and data. Solutions of the net of equations are obtained by the classical methods. Known a priory estimates are improved so that a net of solutions can be considered as a solution in an appropriate algebra of generalized functions.