二阶非线性奇摄动微分方程的渐近解

研究了二阶非线性奇摄动微分方程的边值问题.利用匹配原则和微分不等式原理,得到一阶非线性问题的渐近解,进而得到二阶奇摄动问题的解的渐近估计.     

On stability of zero solution of an essentially nonlinear second-order differential equation

Small periodic (with respect to time) perturbations of an essentially nonlinear differential equation of the second order are studied. It is supposed that the restoring force of the unperturbed equation contains both a conservative and a dissipative part. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. The peculiarity of the considered problem is that the frequency of oscillations is an infinitely small function of the amplitude. The stability problem for the zero solution is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method to compute it. Liapunov’s approach cannot be applied to nonautonomous perturbations (in particular, to periodic ones), because it is based on the possibility to exclude the time variable from the system. Modifying Lyapunov’s method, the following results were obtained. “Action–angle” variables are introduced. A polynomial transformation of the action variable, providing a possibility to compute Lyapunov’s constant, is presented. In the general case, the structure of the polynomial transformation is studied. It turns out that the “length” of the polynomial is a periodic function of the exponent of the conservative part of the restoring force in the unperturbed equation. The least period is equal to four.     

Global attractivity in a nonlinear second-order difference equation

In this paper we obtain a global attractivity result for the positive equilibrium of a nonlinear second-order difference equation of the form x_n+1 = f(x_n, x_n+1), n = 0, 1, ? The result applies to the difference equation x_n+1 =A+bx_n/A+_n?1, n = 0, 1, ? Where a, b, A ? (0, ∞). © 1996 John Wiley & Sons, Inc.     

一类二阶非线性差分方程的全局吸引性

考虑二阶非线性差分方程xn＋1=a＋bxn/A＋xn-1,n=0,1,2,….证明了当条件a,b,A∈（0,∞）成立时方程的唯一正平衡点x^-=（b-A＋√（（b-A）2＋4a））～（1/2））/2是方程的所有正解的一个全局吸引子,所得推论证明了由Kocic和Ladas提出的一个猜想是正确的.     

Positive periodic solution for second-order nonlinear differential equation with singularity of attractive type

This paper is devoted to investigate the following second-order nonlinear differential equation with singularity of attractive type $$ x''-a(t)x=f(t,x)+e(t), $$ where the nonlinear term $f$ has a singularity at the origin. By using the Green''s function of the linear differential equation with constant coefficient and Schauder''s fixed point theorem, we establish some existence results of positive periodic solutions.     

An asymptotic solution of a second-order linear differential equation

This paper discusses an asymptotic formula for solutions of a second-order linear differential equation. The asymptotic formula will enable us to provide information about the distribution of eigenvalues for the case of nonexistence of continuous spectrum in a singular Sturm-Liouville type boundary value problem. The result can be regarded as a partial generalization of those obtained by E. C. Titchmarsh and C. G. C. Pitts.     

Asymptotic behavior of solutions of a second-order nonlinear differential equation

Asymptotic properties of proper solutions of a certain class of essentially nonlinear binomial differential equations of the second order are investigated.     

On a nonlinear distributed order fractional differential equation

We study the existence and the uniqueness of mild and classical solutions for a class of equations of the form . Such equations arise in distributed derivatives models of viscoelasticity and system identification theory. We also formulate a variational principle for a more general equation based on a method of doubling of variables for such equations.     

Numerical solution of a nonlinear integro-differential equation

Galerkin's method with appropriate discretization in time is considered for approximating the solution of the nonlinear integro-differential equation $\text{u}{}_{\mathrm{t}}\text{(x, t) = ∝}{}_0{}^{\mathrm{t}}\text{a(t ? τ)}\text{?}\text{?x}\text{σ(u}{}_{\mathrm{x}}\text{(x, τ)) dτ + f(x, t)}$, 0 < x < 1, 0 < t < T.An error estimate in a suitable norm will be derived for the difference u ? u^h between the exact solution u and the approximant u^h. It turns out that the rate of convergence of u^h to u as h → 0 is optimal. This result was confirmed by the numerical experiments.     

Existence of a solution to a nonlinear equation

Equation (−Δ+k²)u+f(u)=0 in D, u_|∂D=0, where k=const>0 and D⊂R³ is a bounded domain, has a solution if is a continuous function in the region |u|?a, piecewise-continuous in the region |u|?a, with finitely many discontinuity points uj such that f(uj±0) exist, and uf(y)?0 for |u|?a, where a?0 is an arbitrary fixed number.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

On a numerical method for constructing a positive solution of the two-point boundary-value problem for a second-order nonlinear differential equation

An iterative method is proposed for finding an approximation to the positive solution of the two-point boundary-value problem $y'' + c(x)y^m = 0,0 < x < 1,y(0) = y(1) = 0,$ where m = const > 1 and c(x) is a continuous nonnegative function on [0, 1]. The convergence of this method is proved. An error estimate is also obtained.     

Nontrivial solution of a nonlinear second-order three-point boundary value problem

In this paper, for a second-order three-point boundary value problem u" f(t,u)=0, 0＜t＜l,au(0) - bu'(0) = 0, u(1) - αu(η) = 0,where η∈ (0, 1), a, b, α∈ R with a2 b2 ＞ 0, the existence of its nontrivial solution is studied.The conditions on f which guarantee the existence of nontrivial solution are formulated. As an application, some examples to demonstrate the results are given.     

On the solvability of a nonlinear second-order elliptic equation at resonance

We study the existence of solutions of the Neumann problem for semilinear second-order elliptic equations at resonance in which the nonlinear terms may grow superlinearly in one of the directions and , and sublinearly in the other. Solvability results are obtained under assumptions either with or without a Landesman-Lazer condition. The proofs are based on degree-theoretic arguments.

     

Periodic solutions of a nonlinear second-order differential equation with deviating argument

With the help of the coincidence degree continuation theorem, the existence of periodic solutions of a nonlinear second-order differential equation with deviating argument

x^″(t)+f₁(x(t))x^′(t)+f₂(x(t))(x^′(t)⁾²+g(x(t−τ(t)))=0,     

Numerical solution of a nonlinear reaction diffusion equation

In this paper, the authors propose a numerical method to compute the solution of a Cauchy problem with blow-up of the solution. The problem is split in two parts: a hyperbolic problem which is solved by using Hopf and Lax formula and a parabolic problem solved by a backward linearized Euler method in time and a finite element method in space. It is proved that the numerical solution blows up in a finite time as the exact solution and the support of the approximation of a self-similar solution remains bounded. The convergence of the scheme is obtained.