Related papers: Explicit solution of second-order delayed discrete…
A simple closed-form solution to the discrete Lyapunov equation (DLE) is established for certain families of matrices. This solution is expressed in terms of the eigen decomposition (ED) for which closed-form solutions are known for all…
A general solution for a second-order linear constant coefficient dynamic equation with polynomial forcing on time scales is given.
Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction 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 establish conditions guaranteeing that all eventually positive increasing solutions of a half-linear delay differential equation are regularly varying and derive precise asymptotic formulae for them. The results here presented are new…
Simple form scalar differential equation with delay and non-linear negative periodic feedback is considered. The existence of slowly oscillating periodic solutions with the same period as the feedback coefficient is shown numerically within…
It is well known that second order homogeneous linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation underlies the Liouville-Green method and many other techniques for…
Direct solution of simultaneous linear equations is regarded to be slow for large systems of equations and requires special treatment to avoid numerical instability. A new method is proposed that addresses the numerical instability without…
Matrix Riccati equations and other nonlinear ordinary differential equations with superposition formulas are, in the case of constant coefficients, shown to have the same exact solutions as their group theoretical discretizations. Explicit…
We present particular solutions for the following important nonlinear second order differential equations: modified Emden, generalized Lienard, convective Fisher, and generalized Burgers-Huxley. For the latter two equations these solutions…
Linearization of coupled second order nonlinear ordinary differential equations (SNODEs) is one of the open and challenging problems in the theory of differential equations. In this paper we describe a simple and straightforward method to…
In this paper we study in a Hilbert space a homogeneous linear second order difference equation with nonconstant and noncommuting operator coefficients. We build its exact resolutive formula consisting in the explicit non-iterative…
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
In this paper, we propose a delayed perturbation of Mittag-Leffler type matrix function, which is an extension of the classical Mittag-Leffler type matrix function and delayed Mittag-Leffler type matrix function. With the help of the…
In this article I present a fast and direct method for solving several types of linear finite difference equations (FDE) with constant coefficients. The method is based on a polynomial form of the translation operator and its inverse, and…
The oscillatory behavior of the solutions to a differential equation with several non-monotone delay arguments and non-negative coefficients is studied. A new sufficient oscillation condition, involving lim sup, is obtained. An example…
A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized…
The paper deals with the following system of nonlinear difference equations \begin{equation*} x_{n+1}=ax_{n}^{2}y_{n}+bx_{n}y_{n}^{2},\ y_{n+1}=cx_{n}^{2}y_{n}+dx_{n}y_{n}^{2},\ n\in \mathbb{N}_{0}, \end{equation*} where the initial values…
Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential…
We present four classes of nonlinear systems which may be considered discrete analogues of the Garnier system. These systems arise as discrete isomonodromic deformations of systems of linear difference equations in which the associated Lax…