Related papers: Interconnections between various analytic approach…
In this paper, we explore the embedding of nonlinear dynamical systems into linear ordinary differential equations (ODEs) via the Carleman linearization method. Under dissipative conditions, numerous previous works have established rigorous…
In [1], we have presented the theoretical background for finding the Elementary Invariants for a 3D system of first order rational differential equations (1ODEs). We have also provided an algorithm to find such Invariants. Here we introduce…
We identify contact transformations which linearize the given equations in the Riccati and Abel chains of nonlinear scalar and coupled ordinary differential equations to the same order. The identified contact transformations are not of…
We examine the reductions of the order of certain third- and second-order nonlinear equations with arbitrary nonlinearity through their symmetries and some appropriate transformations. We use the folding transformation which enables one to…
A characterization of the symmetry algebra of the $n$th order ordinary differential equations (ODEs) with maximal symmetry and all third order linearizable ODEs is given. This is used to show that such an algebra $\mathfrak{g}$ determines…
We consider the phase-integral method applied to an arbitrary linear ordinary second-order differential equation with non-analytical coefficients. We propose a universal technique based on the Frobenius method which allows to obtain new…
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…
We study second order and third order linear differential equations with analytic coefficients under the viewpoint of finding formal solutions and studying their convergence. We address some untouched aspects of Frobenius methods for second…
The solution of systems of non-autonomous linear ordinary differential equations is crucial in a variety of applications, such us nuclear magnetic resonance spectroscopy. A new method with spectral accuracy has been recently introduced in…
We perform a classification of third order integrable systems of evolution equations with respect to higher symmetries. Applying it, we consider polynomial systems that are 0-homogeneous under a suitable weighting of variables with main…
Viewing optimization methods as numerical integrators for ordinary differential equations (ODEs) provides a thought-provoking modern framework for studying accelerated first-order optimizers. In this literature, acceleration is often…
In this work, we introduce a method based on piecewise polynomial interpolation to enclose rigorously solutions of nonlinear ODEs. Using a technique which we call a priori bootstrap, we transform the problem of solving the ODE into one of…
Inverse problem or parameter estimation of ordinary differential equations (ODEs), the iterative process of minimizing the mismatch between model-predicted and experimental states by tuning the parameter values within an optimization…
In present paper we propose seemingly new method for finding solutions of some types of nonlinear PDEs in closed form. The method is based on decomposition of nonlinear operators on sequence of operators of lower orders. It is shown that…
Recently, a remarkable correspondence has been unveiled between a certain class of ordinary linear differential equations (ODE) and integrable models. In the first part of the report, we survey the results concerning the 2nd order…
For nonlinear reduced-order models, especially for those with non-polynomial nonlinearities, the computational complexity still depends on the dimension of the original dynamical system. As a result, the reduced-order model loses its…
This work develops a framework to discover relations between the components of the solution to a given initial-value problem for a first-order system of ordinary differential equations. This is done by using sparse identification techniques…
We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find…
Symmetry, which describes invariance, is an eternal concern in mathematics and physics, especially in the investigation of solutions to the partial differential equation (PDE). A PDE's nonlocally related PDE systems provide excellent…
A generalization of the reduction technique for ODEs recently introduced by Gao and Liu is given. It is shown that the use of algebraic methods allows the extension of the procedure to much more general flows, as well as the derivation of…