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We tersely review a recently introduced technique to identify systems of two nonlinearly-coupled Ordinary Di{\S}erential Equations (ODEs) solvable by algebraic operations; and we report some specifc examples of this kind, namely systems of…
This note considers fairly general quasi-homogeneous systems of first-order nonlinear ODEs and homogeneous systems of second-order nonlinear ODEs that contain arbitrary functions of several arguments. It presents several exact solutions to…
We prove results on solvability of nonlinear elliptic partial differential systems of principle type of second order. They are consequences of existence of non-radial solutions for nonlinear partial differential systems of Poisson type. As…
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…
Nonlinear ordinary differential equations can rarely be solved analytically. Koopman operator theory provides a way to solve nonlinear systems by mapping nonlinear dynamics to a linear space using eigenfunctions. Unfortunately, finding such…
This article is dedicated to the proof of the existence of classical solutions for a class of non-linear integral variational problems. Those problems are involved in nonlocal image and signal processing.
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.…
Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to…
In this article, we construct novel explicit solutions for nonlinear Schr\"odinger systems with spatially inhomogeneous nonlinearity by means of the Lie symmetry method. We focus the attention to solutions with non-trivial phase, which have…
In this paper we study the general group classification of systems of linear second-order ordinary differential equations inspired from earlier works and recent results on the group classification of such systems. Some interesting results…
Spectral method related to Lame equation with finite-gap potential is used to study the optical cascading equations. These equations are known not to be integrable by inverse scattering method. Due to "partial integrability" two-gap…
We unearth the interconnection between various analytical methods which are widely used in the current literature to identify integrable nonlinear dynamical systems described by third-order nonlinear ordinary differentiable equations…
In this work, we propose a parallel-in-time solver for linear and nonlinear ordinary differential equations. The approach is based on an efficient multilevel solver of the Schur complement related to a multilevel time partition. For linear…
New problem is studied that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. Method is discussed to construct nonlinear ordinary differential equations with exact solutions. Main…
In this work, we establish a connection between the extended Prelle-Singer procedure with other widely used analytical methods to identify integrable systems in the case of $n^{th}$-order nonlinear ordinary differential equations (ODEs). By…
A system of inhomogeneous second-order difference equations with linear parts given by noncommutative matrix coefficients are considered. Closed form of its solution is derived by means of newly defined delayed matrix sine/cosine using the…
Dynamic power system models are instrumental in real-time stability, monitoring, and control. Such models are traditionally posed as systems of nonlinear differential algebraic equations (DAEs): the dynamical part models generator…
Some interesting (periodic!) solutions of certain systems of $4$ nonlinear Ordinary Differential Equations $dx_{n}\left( t\right) /dt=P_{2}^{\left( n\right) }\left[ x_{m}\left( t\right) \right] /\left[ x_{1}\left( t\right) +x_{2}\left(…
We consider a class of linear ODEs of second order with variable coefficients and construct its Lie algebra of Lie group of equivalence transformations. Further we find invariants and differential invariants of this Lie algebra and by using…
We prove in this article the well posedness of non - linear Ordinary Differential Equations (ODE) of first and second order in Orlicz spaces with unbounded domain of definition.