On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space

In this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman–Liouville operator, other various types of the Sonin kernels are a Bessel-type function, functions with power-logarithmic singularities at the origin e.t.c. We pay special attention to study kernels close to power type functions. The main our aim is to study the Sonin–Abel equation in the weighted Lebesgue space, the used method allows us to formulate a criterion of existence and uniqueness of the solution and classify a solution, due to the asymptotics of the Jacobi series coefficients of the right-hand side.

Bifurcation of a Kind of Piecewise Smooth Generalized Abel Equation via First and Second Order Analyses

In this paper, we consider the problem of estimating the number of nontrivial limit cycles for a kind of piecewise trigonometrical smooth generalized Abel equation with the separation line [Formula: see text]. Under the first and second order analyses, we show that the first two order Melnikov functions of the equation share a same structure which can be studied by an ECT-system. Furthermore, let [Formula: see text] be the maximum number of nontrivial limit cycles of the equation bifurcating from the periodic annulus up to [Formula: see text]th order analysis. We prove that [Formula: see text] and [Formula: see text] (resp., [Formula: see text] and [Formula: see text]) when [Formula: see text] is even (resp., odd).

Elementary Quadrature for the Riccati equation

Using suitable transformation in combination with a specific Riccati-type equation solvable, the problem of solving Riccati equation can be transformed into that of a quasi-Abel equation of the second kind. By the extended Julia’s integrability condition, the general solutions of Riccati equation in the form of elementary quadrature are obtained, which contains numerous. This method opens up a new prospect for the study of nonlinear differential equations by analytical method.

On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients

In this paper, we continue our study of the Abel equation with the right-hand side belonging to the Lebesgue weighted space. We have improved the previously known result— the existence and uniqueness theorem formulated in terms of the Jacoby series coefficients that gives us an opportunity to find and classify a solution by virtue of an asymptotic of some relation containing the Jacobi series coefficients of the right-hand side. The main results are the following—the conditions imposed on the parameters, under which the Abel equation has a unique solution represented by the series, are formulated; the relationship between the values of the parameters and the solution smoothness is established. The independence between one of the parameters and the smoothness of the solution is proved.

About Solution of the Nonlinear Generalized Abel Integral Equation

As is known, many problems of electronics, nuclear physics, optics and astrophysics, etc. are described by the Abel integral equation of the first kind. In this paper we consider the nonlinear generalized Abel equation and show that its solution can be represented as an integral of a power function. It is shown that the constructed analytical solution and the symbolic solution obtained by means of the computer mathematics system Maple coincides, and their planar and spatial graphs are presented.