Related papers: Integrable Abel equations and Vein's Abel equation
We present a multi-parameter non-constant-invariant class of Abel ordinary differential equations with the following remarkable features. This one class is shown to unify, that is, contain as particular cases, all the integrable classes…
By topological arguments, we prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions of a class of perturbed nonlinear integral equations. These type of integral equations arise, for example,…
A geometric approach is used to study the Abel first order differential equation of the first kind. The approach is based on the recently developed theory of quasi-Lie systems which allows us to characterise some particular examples of…
In this article we give evaluations of certain series of hyperbolic functions, using Jacobi elliptic functions theory. We also define some new functions that enable us to give characterization of not solvable class of series.
The Abel differential equations play a significant role in various fields of mathematics and applied sciences and are classified into two types: the first kind and the second kind. A novel derivative condition for the general solution of…
This paper study the two--phase problem for the forward-backward parabolic equation with diffusion function of cubic type. Existence and uniqueness for these kind of problems were obtained in literature in the case in which the phases are…
In this paper, we establish the existence of solutions for a particular class of degenerate hyperbolic equations. Following this, we approximate these degenerate equations by employing a sequence of uniformly hyperbolic equations. Notably,…
Various many-body models are treated, which describe $N$ points confined to move on a plane circle. Their Newtonian equations of motion ("accelerations equal forces") are integrable, i. e. they allow the explicit exhibition of $N$ constants…
The well-known analytical solution of Burgers' equation is extended to curvilinear coordinate systems in three-dimensions by a method which is much simpler and more suitable to practical applications than that previously used. The results…
The present work is the first of a serie of two papers, in which we analyse the higher variational equations associated to natural Hamiltonian systems, in their attempt to give Galois obstruction to their integrability. We show that the…
By means of topological methods, we provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of perturbed Hammerstein integral equations. In order to illustrate our theoretical…
This paper is concerned with the numerical solution for a class of nonlinear weakly singular Volterra integral equation of the first kind. The existence and uniqueness issue of the nonlinear Abel integral equations is studied completely. An…
We define noncommutative binary forms. Using the typical representation of Hermite we prove the fundamental theorem of algebra and we derive a noncommutative Cardano formula for cubic forms. We define quantized elliptic and hyperelliptic…
In this work we have obtained Maxwell-type equations for a compressible fluid which sources are functions of velocity and vorticity. A correlation function and the dispersion relation were analyzed as function of the Reynolds number. A…
We derive solvability conditions and closed-form solution for the Weber type integral equation, related to the familiar Weber-Orr integral transforms and the old Weber-Titchmarsh problem (posed in Proc. Lond. Math. Soc. 22 (2) (1924),…
We propose a novel approach to tackle integrability problem for evolutionary differential-difference equations (D$\Delta$Es) on free associative algebras, also referred to as nonabelian D$\Delta$Es. This approach enables us to derive…
A class of differentiable solutions is proved for the isentropic Euler equations in two and three space dimensions. The solutions are explicitly given in terms of solutions to inviscid Burgers equations, and several directions of…
A version of non-Abelian monopole equations is explored through dimensional reductions, with often the addition of algebraic conditions. On zero curvature spaces, spinor related extensions of integrable systems have been generated, and…
Three comparison criteria for the Abel equation of 1es kind are proved. The results obtained are used to obtain global solvability criteria and some criteria of existence of closed solutions for the mentioned equation. The results obtained…
The general solutions with free variable to the second-kind Abel equation, a nonlinear ordinary differential equation that has remained unsolved for nearly two centuries, are presented for the first time by using elementary quadrature…