Abel Differential Equations Research Articles

Relativistic dissipative cosmological models and abel differential equation

We analyze the second-order strongly nonlinear differential equation describing the relativistic evolution of a causal dissipative cosmological fluid in a conformally flat space-time. By means of appropriate transformations, the evolution equation can be reduced to a first-order second-type Abel differential equation. The general solution of the Abel equation is obtained in an exact form and, as a consequence, it is shown that the general solution of the second-order evolution differential equation can be represented in an exact parametric form.

Integrable cases of gravitating static isothermal fluid spheres

It is shown that different approaches toward the solution of the Einstein equations for a static spherically symmetric perfect fluid with a γ-law equation of state lead to an Abel differential equation of the second kind. Its only integrable cases at present are flat space–time, de Sitter solution and its Buchdahl transform, Einstein static universe, and the Klein–Tolman solution.

New method for generating general solution of Abel differential equation

A new method for generating a general solution of the nonlinear first kind Abel type differential equation from a particular one is presented.

On the decomposition method for the approximate solution of nonlinear ordinary differential equations

The study of linear and nonlinear differential equations constitute a main topic in the undergraduate curriculum. Several standard methods are normally employed for the solution of such problems. In this paper, the decomposition method is introduced as a powerful alternative scheme for solving ordinary differential equations. The technique is well adapted for obtaining and identifying the closed form solution or for getting an approximate solution comparable to existing numerical schemes. The solution obtained is in the form of a power series with elegantly computable terms. An essential feature of this technique is its rapid convergence and the high degree of accuracy by which it approximates a solution; it is shown that only few iterations are sufficient to obtain accurate approximation to the exact solution. The technique is implemented for well-known nonlinear differential equations that are usually taught at the undergraduate level. The method is illustrated by considering the Abel, Bernoulli and Ricatti differential equations. Most of the symbolic and numerical computations have been performed using the computer algebra system Maple.

Solutions generating technique for Abel-type nonlinear ordinary differential equations

A second-order nonlinear differential equation describing the dynamics of a bulk viscous fluid filled general relativistic Friedmann-Robertson-Walker Universe is considered. By means of appropriate transformations, the evolution equation of the space-time is transformed into a third-order polynomial Abel differential equation. With the use of some particular solutions of this equation, repeated classes of exact solutions are generated.

Center and composition conditions for Abel differential equation, and rational curves

We consider the Abel Equation $\r'= p(\t)\r^2 + q(\t)\r^3$ (*) with $ p(\t)$, $q(\t)$ - polynomials in $\sin \t$, $\cos \t$. The center problem for this equation (which is closely related to the classical center problem for polynomial vector fields on the plane) is to find conditions on $p$ and $q$ under which all the solutions $\r(\t)$ of this equation are periodic, i.e. $\r(0)=\r(2 \pi)$ for all initial values $\r(0)$. We consider the equation (*) as an equation on the complex plane $\frac{dy}{dz}=p(z)y^2 + q(z) y^3$ (**) with $p$, $q$ -- Laurent polynomials. Then the center condition is that its solution $y(z)$ is a univalued function along the circle $|z|=1$. We study the behavior of the equation (**) under mappings of the complex plane onto Riemann Surfaces. This approach relates the center problem to the algebra of rational functions under composition and to the geometry of rational curves. We obtain the sufficient conditions for the center in the form $\int_{|z|=1} P^i Q^j dP=0$ with $P=\int p$, $Q=\int q$.

Isochronous Centers in Planar Polynomial Systems

Two algorithms are given for finding conditions for a critical point to be an isochronous center. The first is based on a systematic search for a transformation to the simple harmonic oscillator and as an example is used to find conditions for an isochronous center in the Kukles system; the second algorithm is specific to systems with homogeneous nonlinearities and is based on a connection with an Abel differential equation. General properties of systems with isochronous centers are also considered and results on Lienard and Hamiltonian systems are deduced; a close connection is demonstrated between isochronous centers and complex centers.

Almost periodic solutions for Abel equations

Using the Liapunov function method, the existence of almost periodic solutions of a scalar differential equation is discussed The results for the scalar differential equation are then applied to prove the existence and stability of almost periodic solutions of Abel differential equations We obtain several interesting results which improve the results due to Chongyou [1] and Dongpin [2].

Limit Cycles for a Class of Abel Equations

The number of solutions of the Abel differential equation ${{dx(t)} /{dt}} = A(t)x(t)^3 + B(t)x(t)^2 + C(t)x(t)$ satisfying the condition $x(0) = x(1)$ is studied, under the hypothesis that either $A(t)$ or $B(t)$ does not change sign for $t \in [0,1]$. The main result obtained is that there are either infinitely many or at most three such solutions. This result is also applied to control the maximum number of limit cycles for some planar polynomial vector fields with homogeneous nonlinearities.

A new solution of Schmutzer's projective relativistic cosmology

Schmutzer's cosmological model on the basis of his projective unified field theory leads to a system of differential equations equivalent to a set of highly complicated Abel differential equations [1, 2]. This paper is concerned with the properties of this coupled system of differential equations. The main results of this paper are as follows, (i) We have found the explicit general solution of the cosmological equations for the caseγ=±5/2,ɛ= ±1. (ii) The procedure of solving the general caseγ arbitraryɛ=0 is easily within this formalism, (iii) After having found the solution withγ arbitrary,ɛ=±1, one gets the corresponding solutionγ= -γ, ɛ=±1. (iv) In general the coupled system of differential equations describing Schmutzer's cosmological model is equivalent to a single generalized Emden differential equation and it is identical to Emden's differential equation for special values of the parameterγ.

Cosmology on the basis of a new five-dimensional projective unified field theory. I. Dust universe in general

The previously developed five-dimensional projective unified field theory is applied to a cosmological model generalising the Friedmann model by taking into account the occurring scalar field. The system of differential equations leading to an Abel differential equation is treated in detail. For special cases, exact solutions have been found.

Simultaneous solutions of a system of Abel equations and differential equations with several deviations

Simultaneous solutions of a system of Abel equations and differential equations with several deviations

On the forced response of a class of Van der Pol-Duffing oscillator

A first integral in the form of the well-known Abel's differential equation is obtained for the governing differential equation of a class of Van der Pol-Duffing oscillators with or without external inputs to the system.

Functions Which Satisfy Abel’s Differential Equation

(4) 4)13 + (P23 + 033 30)10203 = 1. The addition formulae and other properties have been given in recent times by Silberstein [1], Oniga [2] and Bruwier [3], [4], while Mikusinski [5], [6] and Poli [7], [8] have studied the corresponding third order circular functions. Earlier workers in this field were Appell [9], Glaisher [10] and Villarceau [11]. It is the object of this note to show that the solution of a special type of Abel's nonlinear differential equation of the first kind ([12], [13], [14]), viz., dy = fo(x) + fi(X)y + f2(X)y2 + f3(x)y3 dx X)3 can be expressed explicitly in terms of combinations of 4)-functions and the

Shock Waves and Abel's Differential Equation

AbstractIt ist shown that the ordinary non‐linear second order differential equation of the phenomenological theory of the one‐dimensional steady shock wave in heat conducting viscous gases can be transformed into the Abelian differential equation of the second class. Among the known special solutions of Abel's equation is one applicable to the shock wave and leads to a hitherto unknown closed form solution.