Related papers: Exact solution of nonlinear ordinary differential …
A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
The paper presents a new method for finding first integrals of ordinary difference equations which do not possess Lagrangians, nor Hamiltonians. As an example we solve a third order nonlinear ordinary differential equation and its invariant…
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…
Finite difference schemes are here solved by means of a linear matrix equation. The theoretical study of the related algebraic system is exposed, and enables us to minimize the error due to a finite difference approximation.
We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose…
Various traditional numerical methods for solving initial value problems of differential equations often produce local solutions near the initial value point, despite the problems having larger interval solutions. Even current popular…
The nonlinear eigenvalue problem of a class of second order semi-transcendental differential equations is studied. A nonlinear eigenvalue is defined as the initial condition which gives rise a separatrix solution. A semi-transcendental…
In this paper, we consider the initial value problem for some nonlinear second-order ODEs of Duffing type. We study the large time behavior of the solutions to this problem, from both the perspectives of mathematical and numerical analysis.…
We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in \cite{cstv}, and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of…
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…
Purely numerical methods do not always provide an accurate way to find all the global solutions to nonlinear ODE on infinite intervals. For example, finite-difference methods fail to capture the asymptotic behavior of solutions, which might…
A nonlinear partial differential equation is a nonlinear relationship between an unknown function and how it changes due to two or more input variables. A numerical method reduces such an equation to arithmetic for quick visualization, but…
For the first time, Schr\"odinger equations with cubic and more complex nonlinearities containing the unknown function with constant delay are analyzed. The physical considerations that can lead to the appearance of a delay in such…
We report a new analytical method for exact solution of homogeneous linear ordinary differential equations with arbitrary order and variable coefficients. The method is based on the definition of jump transfer matrices and their extension…
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
This work introduces a methodology to solve ordinary differential equations using the Schur decomposition of the linear representation of the differential equation. This is done by first transforming the system into an upper triangular…
We define a proper differential sequence of ordinary differential equations and introduce a method to derive an alternative sequence of integrals for such a sequence. We describe some general properties which are illustrated by several…
The real numbers, it is taught at universities, correspond to our idea of a continuum, although the hyperreal numbers are located ``in between'' the real numbers. The number $x + dx$, where $dx$ should be an infinitesimal number and $x$…
We establish the existence of weak solutions of a nonlinear radiation-type boundary value problem for elliptic equation on divergence form with discontinuous leading coefficient. Quantitative estimates play a crucial role on the real…