Related papers: General Solutions of the Abel Differential Equatio…
Pell-Abel equation is a functional equation of the form P^{2}-DQ^{2} = 1, with a given polynomial D free of squares and unknown polynomials P and Q. We show that the space of Pell-Abel equations with the fixed degrees of D and of a…
We give a new method for numerically solving Abel integral equations of first kind. An estimation for the error is obtained. The method is based on approximations of fractional integrals and Caputo derivatives. Using trapezoidal rule and…
In this work we present some classes of models whose the corresponding two coupled first-order nonlinear equations can be put into a linear form, and consequently be solved completely. In these cases the so-called trial orbit method is…
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…
We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficient may depend on space and time. Pointwise convergence of the…
The bivariate difference field provides an algebraic framework for a sequence satisfying a recurrence of order two. Based on this, we focus on sequences satisfying a recurrence of higher order, and consider the multivariate difference…
Line integration of generalized functions is studied. Second order partial differential equations with piecewise continuous and generalized variable coefficients over Cayley-Dickson algebras are investigated. Formulas for integrations of…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering…
We introduce the degenerate Bernoulli numbers of the second kind as a degenerate version of the Bernoulli numbers of the second kind. We derive a family of nonlinear differential equations satisfied by a function closely related to the…
We consider semiclassical orthogonal polynomials on the unit circle associated with a weight function that satisfy a Pearson-type differential equation involving two polynomials of degree at most three. Structure relations and difference…
In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for a hierarchy of integrable difference equations. We also provide an example of such a solution that is related to sequence generated…
The generalized Maxwell equations are considered which include an additional gradient term. Such equations describe massless particles possessing spins one and zero. We find and investigate the matrix formulation of the first order of…
We develop techniques at the interface between differential algebra and model theory to study the following problems of exponential algebraicity: Does a given algebraic differential equation admits an exponentially algebraic solution, that…
I begin from a particular field of generalised Puiseux series and investigate a class of nonlinear differential equations in the field. It is appeared that the main part of differential equation determines solvability and positions of…
In this article we give, for the fist time the solution of the general difference equation of 2-degree. We also give as application the expansion of a continued fraction into series, which was first proved, found in the past by the author.
Einstein equations for several matter sources in Robertson-Walker and Bianchi I type metrics, are shown to reduce to a kind of second order nonlinear ordinary differential equation $\ddot{y}+\alpha f(y)\dot{y}+\beta f(y)\int{f(y) dy}+\gamma…
In this study, a recursive solution technique in conjunction with generalized integrating factors is presented and applied to address first and second order linear differential equations. This approach demonstrates practical utility in…
A universal system of difference equations associated with a hyperelliptic curve is derived constituting the discrete analogue of the Dubrovin equations arising in the theory of finite-gap integration. The parametrisation of the solutions…
We exploit a new theory of duality transformations to construct dual representations of models incompatible with traditional duality transformations. Hence we obtain a solution to the long-standing problem of non-Abelian dualities that…