相关论文: Comparison criteria for the Abel equation of 1st k…
In this paper we use the comparison method for investigation of first order polynomial differential equations. We prove two comparison criteria for these equations. The proved criteria we use to obtain some global solvability criteria for…
Three comparison criteria are obtained for second order Riccati equations. On the basis of these criteria some global existence theorems are proved mentioned equations. The results obtained are used to derive a non oscillation criterion for…
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…
We suggest an approach for description of integrable cases of the Abel equations. It is based on increasing of the order of equations up to the second one and using equivalence transformations for the corresponding second-order ordinary…
This paper is dedicated to present an exact solution for a nonlinear differential equation so-called Abel equation. This equation was known as one of the group of unsolvable differential equations. The present method is applicable for any…
The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation $x\mapsto x^{-1}$, and there…
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…
A solution of the Abel equation $\dot{x}=A(t)x^3+B(t)x^2$ such that $x(0)=x(1)$ is called a periodic orbit of the equation. Our main result proves that if there exist two real numbers $a$ and $b$ such that the function $aA(t)+bB(t)$ is not…
We study the rational solutions of the Abel equation $x'=A(t)x^3+B(t)x^2$ where $A,B\in C[t]$. We prove that if $deg(A)$ is even or $deg(B)>(deg(A)-1)/2$ then the equation has at most two rational solutions. For any other case, an upper…
Convex solutions $A,B,I,J$ of four Abel equations are numerically studied. We do not know exact formulas for any of these functions, but conjecture that $A,B$ and $I,J$ are closely related. [Corrigendum at end.]
We first reformulate and expand with several novel findings some of the basic results in the integrability of Abel equations. Next, these results are applied to Vein's Abel equation whose solutions are expressed in terms of the third order…
We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence.
The Chiellini integrability condition of the first order first kind Abel equation $dy/dx=f(x)y^2+g(x)y^3$ is extended to the case of the general Abel equation of the form $dy/dx=a(x)+b(x)y+f(x)y^{\alpha -1}+g(x)y^{\alpha}$, where $\alpha…
There are several versions of Bell's inequalities, proved in different contexts, using different sets of assumptions. The discussions of their experimental violation often disregard some required assumptions and use loose formulations of…
Proposed is a new formal approach for solution of extreme multi-criteria problems transforming them into single-criterion mathematical models, without any additional information. Transforming rules are based on comparison standards and…
We present a criterion that provides an easy sufficient condition in order that a collection of Abelian integrals has the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be…
The systematic method for the conversion of first class constraints to the equivalent set of Abelian one based on the Dirac equivalence transformation is developed. The representation for the corresponding matrix performing this…
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…
The problem of linearization by point transformations is solved for equations in the generalized Riccati and Abel chain of order not exceeding the fourth. It is shown in particular that nonlinear third order and fourth order equations from…
This paper deals with the comparison of two common types of equivalence groups of differential equations, and this gives rise to a number of results presented in the form of theorems. It is shown in particular that one type can be…