Abel Differential Equations Research Articles

Analytical solutions of virus propagation model in blockchain networks

The main goal of this paper is to find analytical solutions of a system of nonlinear ordinary differential equations arising in the virus propagation in blockchain networks. The presented method reduces the problem to an Abel differential equation of the first kind and solve it directly.

Exact solution of the Susceptible–Exposed–Infectious–Recovered–Deceased (SEIRD) epidemic model

An exact solution of an initial value problem for the Susceptible–Exposed–Infectious–Recovered–Deceased (SEIRD) epidemic model is derived, and various properties of the exact solution are obtained. It is shown that the parametric form of the exact solution satisfies some linear differential system including a positive solution of an Abel differential equation of the second kind. In this paper Abel differential equations play an important role in establishing the exact solution of the SEIRD differential system, in particular the number of infected individuals can be represented in a simple form by using a positive solution of an initial value problem for an Abel differential equation. Uniqueness of positive solutions of an initial value problem to SEIRD differential system is also investigated, and it is shown that the exact solution is a unique solution in the class of positive solutions.

Charged dust in Einstein–Gauss–Bonnet models

We investigate the influence of the higher order curvature terms on the static configuration of a charged dust distribution in EGB gravity. The EGB field equations for such a fluid are generated in higher dimensions. The governing equation can be written as an Abel differential equation of the second kind, or a second order linear differential equation. Exact solutions are found to these equations in terms of special functions, series and polynomials. The Abel differential equation of the second kind is reducible to a canonical differential equation; three new families of solutions are found by constraining the coefficients of the canonical equation. The charged dust model is shown to be physically well behaved in a region at the centre, and dust spheres can be generated. The higher order curvature terms influence the dynamics of charged dust and the gravitational behaviour which is distinct from general relativity.

Conditions to the existence of center for Abel equations

Consider the generalized Abel equation x′(t)=f(t)xm(t)+g(t)xn(t), with m>n≥2, a>0 is a constant and f and g are continuous functions. Observe that when n=2 and m=3 we have an Abel differential equation. We obtain conditions on f and g coefficients of Abel equation that ensure the existence of a center in x=0.

Charged fluids in higher order gravity

We generate the field equations for a charged gravitating perfect fluid in Einstein–Gauss–Bonnet gravity for all spacetime dimensions. The spacetime is static and spherically symmetric which gives rise to the charged condition of pressure isotropy that is an Abel differential equation of the second kind. We show that this equation can be reduced to a canonical differential equation that is first order and nonlinear in nature, in higher dimensions. The canonical form admits an exact solution generating algorithm, yielding implicit solutions in general, by choosing one of the potentials and the electromagnetic field. An exact solution to the canonical equation is found that reduces to the neutral model found earlier. In addition, three new classes of solutions arise without specifying the gravitational potentials and the electromagnetic field; instead constraints are placed on the canonical differential equation. This is due to the fact that the presence of the electromagnetic field allows for a greater degree of freedom, and there is no correspondence with neutral matter. Other classes of exact solutions are presented in terms of elementary and special functions (the Heun confluent functions) when the canonical form cannot be applied.

Fractional Abel Differential Equation Application in Edge Detection: FADEED

There is a long history of perturbed implementations of Abel differential equations in dynamics, linear systems with stochasticity, modeling approaches, and linear algebra. Numerous research has been undertaken, the bulk of which have focused on Abel differential equation techniques and the use of the Abel differential equation using the variation iteration method. Edges define boundaries and are hence of primary relevance in image processing. Edge detection removes unnecessary data, noise, and frequencies from a picture while maintaining crucial structural aspects. To extract edges based on canny, the suggested technique employs fractional Abel differential equation (FADE) logic. In this case, a picture is used with the windowing technique and is submitted to a series of Abel differential equations. The purpose of this research is to assess the performance of a FADE system in edge detection. The results produced by the linear canny operator are compared to those obtained with pictures with substantial contrast variation. This experimental investigation will provide a faster procedure and a better output image than the current methods. The main goal of the proposed technique is to use the edge points and the information they collect for edge identification, resulting in faster and more accurate results. For all of the experimental experiments, MATLAB software was employed.

Existence of exact solution of the Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model

Exact solutions of the SEIR epidemic model are derived, and various properties of solutions are obtained directly from the exact solution. In this paper Abel differential equations play an important role in establishing the exact solution of SEIR differential system, in particular the number of infected individuals can be represented in a simple form by using a positive solution of an Abel differential equation. It is shown that the parametric form of the exact solution satisfies some linear differential system including a positive solution of an Abel differential equation.

Isotropic Perfect Fluids in Modified Gravity

We generate the Einstein–Gauss–Bonnet field equations in higher dimensions for a spherically symmetric static spacetime. The matter distribution is a neutral fluid with isotropic pressure. The condition of isotropic pressure, an Abel differential equation of the second kind, is transformed to a first order nonlinear canonical differential equation. This provides a mechanism to generate exact solutions systematically in higher dimensions. Our solution generating algorithm is a different approach from those considered earlier. We show that a specific choice of one potential leads to a new solution for the second potential for all spacetime dimensions. Several other families of exact solutions to the condition of pressure isotropy are found for all spacetime dimensions. Earlier results are regained from our treatments. The difference with general relativity is highlighted in our study.

Stability analysis of Abel's equation of the first kind

<abstract><p>This paper established sufficient conditions for the stability of Abel's differential equation of the first kind. These conditions explicate the impact of the asymptotic behaviors exhibited by the time-varying coefficients on the overall stability of the system. More precisely, we studied the positivity, the continuation and the boundedness of solutions. Additionally, we investigated the attractivity, the asymptotic stability, the uniform stability and the instability of the system. The results were scrutinized by numerical simulations.</p></abstract>

Rational limit cycles of Abel differential equations

We study the number of rational limit cycles of the Abel equation x ′ = A ( t ) x 3 + B ( t ) x 2 , where A ( t ) and B ( t ) are real trigonometric polynomials. We show that this number is at most the degree of A ( t ) plus one.

Rational solutions of Abel differential equations

We study the rational solutions of the Abel equation x′=A(t)x3+B(t)x2 where A and B∈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 bound on the number of rational solutions is obtained. Moreover, we prove that if there are more than (deg⁡(A)+1)/2 rational solutions then the equation admits a Darboux first integral.

Corrigendum: On the Abel differential equations of third kind

<p style='text-indent:20px;'>In this paper, using the Poincaré compactification technique we classify the topological phase portraits of a special kind of quadratic differential system, the Abel quadratic equations of third kind. In [<xref ref-type="bibr" rid="b1">1</xref>] where such investigation was presented for the first time some phase portraits were not correct and some were missing. Here we provide the complete list of non equivalent phase portraits that the Abel quadratic equations of third kind can exhibit and the bifurcation diagram of a <inline-formula><tex-math id="M1">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>-parametric subfamily of it.</p>

ON THE NUMBER OF NONTRIVIAL RATIONAL SOLUTIONS FOR ABEL EQUATIONS

In this paper, a systematic algorithm is provided to determine the sharp upper bound on the number of nontrivial rational solutions for the Abel differential equations $dy/dx=f_m(x)y^2+g_n(x)y^3$, where $f_m(x)$ and $g_n(x)$ are real polynomials of degree $m$ and $n$ respectively. As an application, we present a thorough study for an important case, $(m,n)=(4,9$).

Center problem for generic degenerate vector fields

We generalize the method of construction of an integrating factor for Abel differential equations, developed in Briskin et al. (1998), for any generic monodromic singularity. Here generic means that the vector field has not characteristic directions in the quasi-homogeneous leading term in certain coordinates. We apply this method to some degenerate differential systems.

An attractive numerical algorithm for solving nonlinear Caputo\u2013Fabrizio fractional Abel differential equation in a Hilbert space

Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space mathcal{H}^{2}[a,b]. We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.

A cubic nonlinear population growth model for single species: theory, an explicit\u2013implicit solution algorithm and applications

In this paper, we extend existing population growth models and propose a model based on a nonlinear cubic differential equation that reveals itself as a special subclass of Abel differential equations of first kind. We first summarize properties of the time-continuous problem formulation. We state the boundedness, global existence, and uniqueness of solutions for all times. Proofs of these properties are thoroughly given in the Appendix to this paper. Subsequently, we develop an explicit–implicit time-discrete numerical solution algorithm for our time-continuous population growth model and show that many properties of the time-continuous case transfer to our numerical explicit–implicit time-discrete solution scheme. We provide numerical examples to illustrate different behaviors of our proposed model. Furthermore, we compare our explicit–implicit discretization scheme to the classical Eulerian discretization. The latter violates the nonnegativity constraints on population sizes, whereas we prove and illustrate that our explicit–implicit discretization algorithm preserves this constraint. Finally, we describe a parameter estimation approach to apply our algorithm to two different real-world data sets.

Some open problems in low dimensional dynamical systems

The aim of this paper is to share with the mathematical community a list of 33 problems that I have found along the years in my research. I believe that it is worth to think about them and, hopefully, solve some of the problems or make some substantial progress. Many of them are about planar differential equations but there are also questions about other mathematical aspects: Abel differential equations, difference equations, global asymptotic stability, geometrical questions, problems involving polynomials and some recreational problems with a dynamical component.

On the Solution of Periodic Abel's Differential Equations of the First Kind

On the Solution of Periodic Abel's Differential Equations of the First Kind

THE NUMBER OF RATIONAL SOLUTIONS OF ABEL EQUATIONS

<abstract> In this paper, we study rational solutions of the Abel differential equations $ dy/dx = f_m(x)y^2+g_n(x)y^3 $, where $ f_m(x) $ and $ g_n(x) $ are real polynomials of degree $ m $ and $ n $ respectively. The main result of the paper is as follows: We give a systematic upper bound on the number of the nontrivial rational solutions of such equations in all these cases. Then we prove that these upper bounds can be reached in most cases. Finally, we present some examples of Abel equations having exactly $ i $ nontrivial rational solutions, where $ 1\leq i\leq 5 $. </abstract>

Upper bounds of limit cycles in Abel differential equations with invariant curves

New criteria are established for upper bounds on the number of limit cycles of periodic Abel differential equations having two periodic invariant curves, one of them bounded. The criteria are applied to obtain upper bounds of either zero or one limit cycle for planar differential systems.