Solutions of the equation \documentclass{article}\pagestyle{empty}\begin{document}$$ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $$\end{document}. Exact solutions of some partial differential equations in mechanics and physics

The solutions of the equation $ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $, for L? a linear operator are derived. Different forms for L? whether it is time independent or time dependent and self-commutative (or not) at different times are considered separately. By using the results obtained, exact solutions of some partial differential equations are found for the first time.     

Periodic solutions of the differential delay equation\dot x(t) = - f(x(t - 1))

For an odd function 1/2(x) defined only on a finite interval, this paper deals with the existence of periodic solutions and the number of simple periodic solutions of the differential delay equation (DDE) . By use of the method of qualitative analysis combined with the constructing of special solutions a series of interesting results are obtained on these problems.Supported by the National Natural Science Foundation of China.     

Properties of solutions ofu″+g(t) u 2n−1 =0, II

Summary Part I [7] of this paper appeared in this journal in 1962. In Sections 1, 2 of this paper we consider the equation of the title wheng(t)>0. In Section 1 we examine the strength of the hypothesis of continuity ofg(t) in known theorems. In Section 2 we determine the combinations of boundedness and oscillation that may occur for solutions of this equation. Section 3 is devoted to properties of solutions of the equation wheng(t)<0. In general, differentiability ofg(t) is not required in the paper.     

Existence of Solutions for a Four-p oint Boundary Value Problem with a p(t)-Laplacian

This paper deals with the existence of solutions to the p(t)-Laplacian equation with four-point boundary conditions. It is shown, by Leray-Schauder fixed point theorem and degree method, that under sui...     

微分方程x+g(x)=p(T)的调和解

通过估计方程x+g(x)=p(t)之解绕原点一周的时间,讨论了周期解存在的条件及调和解的存在唯一性。     

The stability of RDDE\ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) + q_2 (t)x(t - r(t)) = O

In this paper, the author considered the stability of zero solution of linear RDDE $$\begin{gathered} \ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) + q_2 (t)x(t - r(t)) = O, \hfill \\ \ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) = O \hfill \\ \end{gathered} $$ using Liapunov-Razumikhin functional and transformations and obtained some sufficient conditions for the stability of Eqs.(1) and (2). These results are suitable both for boundedp _i (t),q _i (t) andr(t).i=1,2.     

Continuability,boundness and asymptotic behavior of solutions ofx″ +q(t)f(x)=r(t)

Summary In addition to obtaining sufficient conditions for continuability of solutions of x″ + q(t)f(x)=r(t), some sufficient conditions and some necessary and sufficient conditions for boundedness are obtained. The asymptotic behavior of solutions is studied through examination of r(t)/q(t) as t → ∞. Supported by Mississippi State University Biological and Physical Sciences Research Institute. Entrata in Redazione il 6 febbraio 1973.     

Existence of exactlyn+1 simple 4-periodic solutions of the differential delay equation\dot x(t) = - f(x(t - 1))

By discussing the zeros of periodic solutions we give in this paper a criterion for the existence of exactlyn+1 simple 4-periodic solutions of the differential delay equation Supported by the Chinese National Foundation for Natural Sciences.     

An explicit form of the differential equationsy″=q(t)y with periodic solutions

Summary This paper contains an explicit form of a differential equation of the type y″=q(t)y, all solutions of which are periodic or half-periodic. Entrata in Redazione il 30 giugno 1969.     

On the equationx″+f(x)h(x′)x′+g(x)=e(t)

Summary We consider the equation x″+f(x)h(x′)x′+g(x)=e(t) in which f, g, and h are continuous, e is sectionally continuous and absolutely integrable, h(u)>0, xg(x)>0 if x ≠ 0, and f(x)≥0. Necessary and sufficient conditions are given for boundedness of all solutions and their derivatives. When f(0)>(0) we give necessary and sufficient conditions for all solutions and their derivatives to converge to zero. Entrata in Redazione il 14 giugno 1969.     

Solution of the operator equationi∈dy/dt=A(t)y on intervals containing turning points

We obtain formal solutions of the equation i∈ dy/dt=A(t)y in the form of complete asymptotic expansions as ∈→0 on intervals containing parabolic or hyperbolic turning points. The highest orders of the power series in ∈ for the formal solutions are studied in detail. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 357–371, March, 2000     

广义的带导数非线性薛定谔方程的有理解

广义带导数非线性薛定谔方程是与Kaup-Newell谱问题相联系的一个非线性发展方程,方程可在合适的条件方程下,利用Wronsiki技巧,寻找广义双Wronsikian形式的一般解,进而得到其孤子解和有理解.     

ON HERMITIAN POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATION X-A^＊X^-2 A=I

The Hermitian positive definite solutions of the matrix equation X-A^＊X^-2 A=I are studied. A theorem for existence of solutions is given for every complex matrix A. A solution in case A is normal is given. The basic fixed point iterations for the equation are discussed in detail. Some convergence conditions of the basic fixed point iterations to approximate the solutions to the equation are given.     

The expanding behavior of positive set H u (t) of a degenerate parabolic equation

This work studies the expanding behavior of the positive set of solutions and the continuous dependence on the nonlinearity for a degenerate parabolic partial differential equation u_t=Df(u){u_{t}=\Delta\phi(u)} . Our objective is to give an explicit expression of speed of propagation of the solution and to show that the solution continuously depends on the nonlinearity of the equation.     

Bounds on the solution of the time-varying linear matrix differential equation P(t) = AH(t) P(t) + P(t) A(t) + Q(t)

^** Email: pagilla{at}ceat.okstate.edu We derive upper and lower bounds for the trace of the solutionof the time-varying linear matrix differential equation (t)= A^H(t) P(t) + P(t) A(t) + Q(t). A practical numerical exampleis given to verify the bounds. The bounds obtained are usefulsince the considered equation is encountered in a number ofapplications in systems and control theory.     

A Liapunov functional for a matrix neutral difference-differential equation with one delay

For the matrix neutral difference-differential equation $\text{x}\text{?}\text{(t) + A}\text{x}\text{?}\text{(t ? τ)  Bx(t) + Cx(t ? τ)}$ we construct a quadratic Liapunov functional which gives necessary and sufficient conditions for the asymptotic stability of the solutions of that equation. We consider a difference equation approximation of the difference-differential equation, and for this difference equation we construct a Liapunov function from which we obtain the desired Liapunov functional by an appropriate limiting process. The Liapunov functional thus obtained gives the best possible estimate for the rates of growth or decay of the solutions of the matrix neutral difference-differential equation. The results obtained are natural generalizations of previous results obtained for a matrix retarded difference-differential equation with one delay.     

Asymptotic behavior of solutions to the damped quasilinear equation ∂2∂t2 u(x,t) + γ∂u∂t (x,t) − ∂∂xσ ∂u∂x (x,t)) = 0

Asymptotic lower bounds for the L² norms of solutions of initial-boundary value problems associated with the equation of the title are derived for a simple case in which the equation fails to exhibit strict hyperbolicity. It is shown that in such cases it can be expected that the norm of a solution will be bounded away from zero as t → +∞ even as the damping factor γ becomes infinitely large.     

The Matrix Equation A_(mn)X_(ns)B_(st)=Cover an Arbitrary Skew Field

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方程w"-w+f(t,w)=O的Dirichlet边值问题的正解存在性与多解性

考察了下列常微分方程的Dirichlet边值问题的正解[w″(t)-w(t) f(t,w(t))=0,0≤t≤1 w(0)=w(1)=0建立了n正解的存在性，其中n是一个任意的自然数。     

x''''=g(x,t)+h(t)型方程的拓扑动力系统与Kurzweil-Henstock积分

本文借助Sell等人建立的局部动力系统理论,利用Kurzweil-Henstock积分,建立了x′=g(x,t)+h(t)型非自治微分方程的拓扑动力系统,为进一步讨论这种方程解的渐近行为作了基础性的工作.本文的工作也是Sell等人工作中有关结论的推广.