Related papers: Generalized damped Milne-Pinney equation and Chiel…

We use the geometric approach to the theory of Lie systems of differential equations in order to study dissipative Ermakov systems. We prove that there is a superposition rule for solutions of such equations. This fact enables us to express…

We introduce a special type of dissipative Ermakov-Pinney equations of the form v_{\zeta \zeta}+g(v)v_{\zeta}+h(v)=0, where h(v)=h_0(v)+cv^{-3} and the nonlinear dissipation g(v) is based on the corresponding Chiellini integrable Abel…

Using the only admissible rank-two realisations of the Lie algebra of the affine group in one dimension in terms of the Lie algebra of Lie symmetries of the Ermakov-Pinney (EP) equation, some classes of second order nonlinear ordinary…

The connection between the method of comparison equations (generalized WKB method) and the Ermakov-Pinney equation is established. A perturbative scheme of solution of the generalized Ermakov-Pinney equation is developed and is applied to…

The work considers the damped Pinney equation, defined as the model arising when a linear in velocity damping term is included in the Pinney equation. In the general case the resulting equation does not admit Lie point symmetries or is…

In [7], a new iterative method for solving linear system of equations was presented which can be considered as a modification of the Gauss-Seidel method. Then in [4] a different approach, say 2D-DSPM, and more effective one was introduced.…

We employ the method of nonlocal generalized Sundman transformations to formulate the linearization problem for equations of the generalized Li\'enard type and show that they may be mapped to equations of the dissipative…

We discuss several approaches to generalized solutions of problems describing the motion of inviscid fluids. We propose a new concept of dissipative solution to the compressible Euler system based on a careful analysis of possible…

We consider the Cauchy problem for a damped Euler-Maxwell system with no ionic background. For smooth enough data satisfying suitable so-called dispersive conditions, we establish the global in time existence and uniqueness of a strong…

A superposition rule for two solutions of a Milne--Pinney equation is derived.

A class of exact solutions is obtained for the Li\'{e}nard type ordinary non-linear differential equation. As a first step in our study the second order Li\'{e}nard type equation is transformed into a second kind Abel type first order…

We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the…

We present the bilinear forms of the (continuous) Painlev\'e equations obtained from the continuous limit of the analogous expresssions for the discrete ones. The advantage of this method is that it leads to very symmetrical results. A new…

We prove the meromorphy of solutions for a wide class of ordinary differential equations. These equations are given by invariant manifolds of non-linear partial differential equations integrable by the inverse scattering method. Some higher…

We give a unified proof for the well-posedness of a class of linear half-space equations with general incoming data and construct a Galerkin method to numerically resolve this type of equations in a systematic way. Our main strategy in both…

In this paper we propose and solve a generalization of the Bernoulli Differential Equation, by means of a generalized fractional derivative. First we prove a generalization of Gronwall's inequality, which is useful for studying the…

In this article we investigate the existence of a solution to a semilinear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one…

We find a two-component generalization of the integrable case of rdDym equation. The reductions of this system include the general rdDym equation, the Boyer-Finley equation, and the deformed Boyer-Finley equation. Also we find a B\"acklund…

The generalized symmetry method is applied to a class of completely discrete equations including the Adler-Bobenko-Suris list. Assuming the existence of a generalized symmetry, we derive a few integrability conditions suitable for testing…

A systematic study of the discrete second order projective system is presented, complemented by the integrability analysis of the associated multilinear mapping. Moreover, we show how we can obtain third order integrable equations as the…