Related papers: Revisiting second-order linear differential equati…
This work aims to obtain a positive, smooth, even, and homoclinic to zero (i.e zero at infinity) solution to a non-autonomous, second-order, nonlinear differential equation involving quadratic growth on the derivative. We apply Galerkin's…
This paper deals with the homogenization of fully nonlinear second order equation with an oscillating Dirichlet boundary data when the operator and boundary data are $\e$-periodic. We will show that the solution $u_\e$ converges to some…
In physics, all dynamical equations that describe fundamental interactions are second order ordinary differential equations in the time derivatives. In the literature, this property is traced back to a result obtained by Ostrogradski in the…
We define the universal exponential extension of an algebraically closed differential field and investigate its properties in the presence of a nice valuation and in connection with linear differential equations. Next we prove normalization…
In a cylinder $\Omega_T=\Omega\times (0,T)\subset \R^{n+1}_+$ we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form \[ Hu =\sum_{i,j=1}^ma_{ij}(x,t) X_iX_ju - \p_tu = 0, \…
The aim of this article is to prove strong convergence results on the difference between the solution to highly oscillatory problems posed in thin domains and its two-scale expansion. We first consider the case of the linear diffusion…
It is proven that second-order vectorial nonlinear differential systems y''=f(y) , possess a continuum of symmetric solutions. They are shown to possess a continuum of even solutions. If f(y) is an odd function of y , then y''=f(y) is shown…
Second order linear non-autonomous differential equations with negative stiffness are considered. Using Chetaev-like (Lyapunov-like) functions, necessary (sufficient) conditions are found for the solutions to be bounded for all initial…
The rich non-linear dynamics of the coupled oscillators (under second harmonic injection) can be leveraged to solve computationally hard problems in combinatorial optimization such as finding the ground state of the Ising Hamiltonian. While…
We establish the existence of non-stationary solutions to a symmetric system of second-order autonomous differential equations. Our technique is based on the equivariant degree theory and involves a novel characterization of orbit types of…
Second-order partial differential equations in non-divergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or…
We study the quantitative homogenization of linear second order elliptic equations in non-divergence form with highly oscillating periodic diffusion coefficients and with large drifts, in the so-called ``centered'' setting where…
This dissertation describes the space of heteroclinic orbits for a class of semilinear parabolic equations, focusing primarily on the case where the nonlinearity is a second degree polynomial with variable coefficients. Along the way, a new…
The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases…
We study the rigidity problem for $(-\alpha)$-homogeneous solutions to the two-dimensional incompressible stationary Euler equations in sector-type domains $\Omega_{a, b, \theta_0}:= \{(r,\theta): a<r<b, \ 0<\theta<\theta_0\}$, where…
In this article we study solutions to second order linear difference equations with variable coefficients. Under mild conditions we provide closed form solutions using finite continued fraction representations. The proof of the results are…
We explore the Hyers-Ulam stability of perturbations for a homogeneous linear differential system with $2\times 2$ constant coefficient matrix. New necessary and sufficient conditions for the linear system to be Hyers-Ulam stable are…
In this paper, we analyze a second-order differential equation with a piecewise constant argument and reflection coupled to periodic boundary conditions. Our main contribution is the construction of the related Green's function and a…
In the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace…
Some properties of global solution of scalar Riccati equation are studied. On the basis of these properties using the Whiburn's and Leighton - Nehary's theorems some oscillatory and criteria are proved for second order linear systems of…