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We revisit the results on admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements…
For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a…
Linearization problem of ordinary differential equations by a new set of tangent transformations is considered in the paper. This set of transformations allows one to extend the set of transformations applied for the linearization problem.…
In this paper we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations (ODEs) of arbitrary order related to linear ODEs through nonlocal relations. The procedure makes use…
This paper develops a characterisation of when solutions of forced second order linear differential equations converge to the zero solution of the asymptotically stable and unforced second order equation, or when the solution is bounded,…
In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.
This is an introduction to the theory of disconjugacy for a second order linear differential equation. We give new proofs of some of basic results and obtain new sufficient conditions for disconjugacy (in particular, on the whole real…
System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of…
In this paper, we discuss a systematic and self consistent procedure to factorize a rather general class of coupled nonlinear ordinary differential equations (ODEs), namely coupled quadratic and mixed Li\'enard type equations, which include…
The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the…
The question of defining unique, generally applicable constrained second, and higher-order, derivatives is investigated. It is shown that second-order constrained derivatives obtained via two successive constrained differentiations provide…
This paper is centred on solving differential equations by symmetry groups for first order ODEs and is in response to Starrett (2007). It also explores the possibility of averting the assumptions by Olver (2000) that, in practice finding…
The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn…
We derive the general conditions for fully-nonlinear symmetry-integrable second-order evolution equations and their first-order recursion operators. We then apply the established Propositions to find links between a class of fully-nonlinear…
We represent an algorithm reducing a big class of systems of ($M+1$)-dimensional nonlinear partial differential equations (PDEs) to the systems of $M$-dimensional first order PDEs. Thus, we integrate the original system with respect to only…
In the first part of planned series of papers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in two independent variables with constant parameters are given. The main goal here is…
Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A…
In this paper, we present new techniques for solving a large variety of partial differential equations. The proposed method reduces the PDEs to first order differential equations known as classical equations such as Bernoulli, Ricatti and…
We explore a nonlocal connection between certain linear and nonlinear ordinary differential equations (ODEs), representing physically important oscillator systems, and identify a class of integrable nonlinear ODEs of any order. We also…
We propose new linear combinations of compositions of a basic second-order scheme with appropriately chosen coefficients to construct higher order numerical integrators for differential equations. They can be considered as a generalization…