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In this article, we are going to look at the requirements regarding a monotone function f ∈ ℝ →ℝ ≥0, and regarding the sets of natural numbers ( A i ) i = 1 ∞ ⊆ d m n ( f ) \left( {{A_i}} \right)_{i = 1}^\infty \subseteq dmn\left( f \right) , which requirements are sufficient for the asymptotic ∑ n ∈ A N P ( n ) ≤ N θ f ( n ) ∼ ρ ( 1 / θ ) ∑ n ∈ A N f ( n ) \sum\limits_{\matrix{{n \in {A_N}} \hfill \cr {P\left( n \right) \le {N^\theta }} \hfill \cr } } {f\left( n \right) \sim \rho \left( {1/\theta } \right)\sum\limits_{n \in {A_N}} {f\left( n \right)} } to hold, where N is a positive integer, θ ∈ (0, 1) is a constant, P(n) denotes the largest prime factor of n, and ρ is the Dickman function.

The mathematical model makes it possible to study the phenomenon as a whole, to predict its development, to make quantitative estimates of changes occurring in the system over time. There is a large number of analytical and numerical me-thods for solving ordinary differential equations and their systems, by which one can simulate the dynamics of oscillatory processes. The number of physical problems that are solved with the help of differential equations of the third or fourth order is constantly growing. Such problems arise where there are uneven transitions from one physical characteristic to another, including questions of quantum mechanics, astrophysics, and the theory of elasticity. Unfortunately, for many practically important questions, the problems described by differential equations are very complex, and their exact solution is difficult or impossible. Since these solutions describe those or other natural processes, for the researcher important information is about them. Therefore, getting the exact solution in an analytical form is an important and actual problem. A new method for solving linear homogeneous differential equations of the third and fourth orders with variable coefficients was constructed, which allowed us to obtain an analytic solution throughout the field of equation determination. The proposed method reduces the order of the differential equation and is a generalization of Abel's formula for the case of linear differential equations of the third and fourth orders with variable coefficients. The developed method is based on the use of the properties of determinant Vronsky. The article gives the conditions for which determinant can be expressed through elementary functions. The conditions that ensure the possibility of using the proposed method are explored and substantiated. A recurrence formula has been found to obtain a partial solution that simplifies the process of solving these equations.

We construct an explicit representation of the solution of a multidimensional Abel integral equation of the second kind with partial fractional integrals in terms of the Wright function.

In this paper, we study the contact geometry of second-order ordinary differential equations that are quadratic in the highest derivative (the so-called quadratic Abel equations). Namely, we realize each quadratic Abel equation as the kernel of some nonlinear differential operator. This operator is defined by a quadratic form on the Cartan distribution in the 1-jet space. This observation makes it possible to establish a one-to-one correspondence between quadratic Abel equations and quadratic forms on Cartan distribution. Using this realization, we construct a contact-invariant {e}-structure associated with a nondegenerate Abel equation (i.e., the basis of vector fields that is invariant under contact transformations). Finally, in terms of this {e}-structure we solve the problem of contact equivalence of nondegenerate Abel equations

We consider an algorithm for constructing auto-Backlund transformations for finitedimensional Hamiltonian systems whose integration reduces to the inversion of the Abel map. In this case, using equations of motion, one can construct Abel differential equations and identify the sought Backlund transformation with the well-known equivalence relation between the roots of the Abel polynomial. As examples, we construct Backlund transformations for the Lagrange top, Kowalevski top, and Goryachev–Chaplygin top, which are related to hyperelliptic curves of genera 1 and 2, as well as for the Goryachev and Dullin–Matveev systems, which are related to trigonal curves in the plane.

Given m ? N, m ? 1, and a Sheffer matrix S = [sn,k]n,k?0, we obtain the exponential generating series for the coefficients (a+(m+1)n a+mn)-1 sa+(m+1)n,a+mn. Then, by using this series, we obtain two general combinatorial identities, and their specialization to r-Stirling, r-Lah and r-idempotent numbers. In particular, using this approach, we recover two well known binomial identities, namely Gould's identity and Hagen-Rothe's identity. Moreover, we generalize these results obtaining an exchange identity for a cross sequence (or for two Sheffer sequences) and an Abel-like identity for a cross sequence (or for an s-Appell sequence). We also obtain some new Sheffer matrices.

Using a corollary to Karamata's main theorem [Math. Z. 32 (1930), 319-320], we prove that if a slowly decreasing sequence of real numbers is Abel summable, then it is convergent in the ordinary sense. © 2016, Proyecciones Journal of Mathematics.

In this paper, we use a generalization of the Riemann---Liouville fractional integral for interval-valued functions to study a theory of the interval Abel integral equation (IAIE). Our aim is to clarify under which suitable conditions the IAIE is solvable. The theory is illustrated by solving some examples.

New and known properties of the resolvent of the kernel of linear Abel integral equations of the form \begin{equation} \tag{A$_\lambda$} x(t) = f(t) - \lambda \int_0^t (t - s)^{q-1}x(s)\,ds, \end{equation} where $\lambda > 0$ and $q \in (0,1)$, are assembled and derived here. First, a priori bounds on potential solutions of the resolvent equation \begin{equation} \tag{R$_\lambda $} R(t) = {\lambda }t^{q-1} - {\lambda }\int_0^t (t - s)^{q-1}R(s)\,ds \end{equation} are obtained. Second, it is proven - using these bounds, Banach's contraction mapping principle, new continuation and translation results, and Schaefer's fixed point theorem - that (R$_\lambda $) has a unique continuous solution on $(0,\infty)$, which is called the resolvent in the literature and denoted here by $R(t)$. Third, both known and new properties of $R(t)$ are derived. Fourth, $R(t)$ is shown to be completely monotone and the unique continuous solution of the initial value problem of fractional order $q$: \begin{equation} \tag{I$_\lambda $} D^{q}x(t) = -\lambda \Gamma(q) x(t), \qquad \lim_{t \to 0^+} t^{1-q}x(t) = \lambda , \end{equation} where $D^{q}$ denotes the Riemann-Liouville fractional differential operator. Finally, the resolvent integral function $\int_0^t R(s)\,ds$ is shown to be the unique continuous solution of an integral equation closely related to (R$_\lambda$). Closed-form expressions for it and $R(t)$ are derived.

For the Abel integral equation with a continuous solution and approximately defined right-hand side, a method of regularization without restrictions on the parameter of this equation is proposed.

This paper gives the definition and analysis of Abel dynamic equations on a general time scale. As such, the results contain as special cases results for classical Abel differential equations and results for new Abel difference equations. By using appropriate transformations, expressions of Abel dynamic equations of second kind are derived on the general time scale. This also leads to a specific class of Abel dynamic equations of first kind. Finally, the canonical Abel dynamic equation is defined and examined.

The main aim of the present work is to propose a new and simple algorithm for Abel integral equation, namely homotopy perturbation transform method (HPTM). The homotopy perturbation transform method is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. Abel’s integral equation occurs in the mathematical modeling of several models in physics, astrophysics, solid mechanics and applied sciences. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical examples are given to illustrate the accuracy and stability of this method.

Consider the scalar differential equation x ′ = ∑ i = 0 m a i ( t ) x n i , where a i ( t ) are T-periodic analytic functions, and 1 ≤ n i ≤ n . For any polynomial Q ( x ) = x n 0 − ∑ i = 1 m α i x n i , the equation can be written as x ′ = a 0 Q ( x ) + R ( t , x ) . Let W be the Wronskian of Q and R with respect to x, and Q ˜ , W ˜ the previous polynomials after removing multiplicity of roots and solutions of the differential equation. We prove that if the vector field defined by the differential equation is “transversal” at every point of Q ˜ ( x ) = 0 or W ˜ ( t , x ) = 0 then the number of limit cycles (isolated periodic solutions in the set of periodic solutions) of the differential equation is at most 3 n − 1 .

We study the geometry of cubic Abel differential equations on one-dimensional real curve. We prove that such an equation is the kernel of some nonlinear differential operator. This operator is defined by a cubic on the Cartan distribution on the space of 1-jets. Based on this observation, we construct a contact-invariant {e}-structure associated with a non-degenerate Abel equation and obtain contact classification of such equations.

We consider the polynomial system dxdt=−y−p(x,y), dydt=x+q(x,y) where p,q are homogeneous polynomials of degree n⩾2. By finding integrable forms of the system, we obtain several new sets of center conditions valid for general values of n. We also find a general class of centers valid for even values of n which is based on certain parity properties of a related differential equation. We further give several conditions which are valid specifically for n=4,5. In addition, we demonstrate that the system can be transformed to an Abel equation having rational coefficients and show how the solution to the system leads to the solution of the corresponding Abel equation.

In this paper, a numerical method for solving Abel integral equation is presented. We first convert the integral equation to the algebraic equation. Then we find numerical inversion of Laplace transform by power series. At last the Pade approximants is effectively used to improve the convergence rate and accuracy of the computed series. The method is described and illustrated with numerical examples. The results reveal that the method is accurate and easy to implement.

In this study, the homotopy analysis method is used for solving the Abel differential equation with fractional order within the Caputo sense. Stabilityand convergence of the proposed approach is investigated. The numerical results demonstrate that the homotopy analysis method is accurate and readily implemented.

We discuss some outcomes of an umbral generalization of the Abel identity. First we prove that a concise proof of the Lagrange inversion formula can be deduced from it. Second, we show that the whole class of Sheffer sequences, if manipulated to an umbral level, coincides with the subclass of Abel polynomials. Finally, we apply these techniques to obtain explicit formulae for some classical polynomial sequences, even in non Sheffer cases (Chebyshev and Gegenbauer polynomials).

El títol de la versió pre-print de l'article és: The solution of the composition conjecture for Abel equations Trigonometric Abel differential equations appear when one studies the number of limit cycles and the center-focus problem for certain families of planar polynomial systems. Inside trigonometric Abel equations there is a class of centers, the composition centers, that have been widely studied during these last years. We fully characterize this type of centers. They are given by the couples of trigonometric polynomials for which all the generalized moments vanish and also coincide with the strongly persistent centers. This result solves the so called Composition Conjecture for trigonometric Abel differential equations. We also prove the equivalent version of this result for Abel equations with polynomial coefficients.

At the time when Legendre published his treatise [128], Abel, a young Norwegian mathematician, was beginning to publish a huge generalization of the addition theorems for elliptic functions. This generalization dealt with all the integrals of the form: 9.1 y being an arbitrary algebraic function of the variable x. Those integrals were later called “abelian transcendents” by Jacobi in [107]. In the sequel, we use the simpler name of abelian integrals.

We consider properties and center conditions for plane polynomial systems of the formsx˙=-y-p1(x,y)-p2(x,y),y˙=x+q1(x,y)+q2(x,y)wherep1,q1andp2,q2are polynomials of degreesnand2n-1, respectively, for integersn≥2. We restrict our attention to those systems for whichyp2(x,y)+xq2(x,y)=0. In this case the system can be transformed to a trigonometric Abel equation which is similar in form to the one obtained for homogeneous systems(p2=q2=0). From this we show that any center condition of a homogeneous system for a givenncan be transformed to a center condition of the corresponding generalized cubic system and we use a similar idea to obtain center conditions for several other related systems. As in the case of the homogeneous system, these systems can also be transformed to Abel equations having rational coefficients and we briefly discuss an application of this to a particular Abel equation.

A direct almost Bernstein operational matrix of integration is used to propose a stable algorithm for numerical inversion of the generalized Abel integral equation. The applicability of the earlier proposed methods was restricted to the numerical inversion of a part of the generalized Abel integral equation. The method is quite accurate and stable as illustrated by applying it to intensity data with and without random noise to invert and compare it with the known analytical inverse. Thus it is a good method for applying to experimental intensities distorted by noise.

Agraïments: The second author is partially supported by NSFC-10831003 and by AGAUR grant number 2009PIV00064. We prove that a family of functions defined through some definite integrals forms an extended complete Chebyshev system. The key point of our proof consists of reducing the study of certain Wronskians to the Gram determinants of a suitable set of new functions. Our result is then applied to give upper bounds for the number of isolated periodic solutions of some perturbed Abel equations.

We discuss a multiple nonlinear Abel type integral equation. The basic results provide criteria for the existence of nontrivial everywhere positive solutions. They are expressed in terms of the generalized Osgood condition. The global behavior of the solution, especially the conditions when it experiences blow-up, is also considered.

This paper presents a numerical method for solving Abel’s integral equation as singular Volterra integral equations. In the proposed method, the functions in Abel’s integral equation are approximated based on Bernstein polynomials (BPs) and therefore, the solving of Abel’s integral equation is reduced to the solving of linear algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

In this paper we prove an identity involving the Lambert W function which is derived through the solution of an instance of the Abel differential equation of the first kind. For x ∈ ℜ, the Lambert W function is defined as the principle branch of the function that satisfies the following equation Lambert W (x) e LambertW(x) = x. The Abel Dif dy(t) ferential Equation of the First Kind is defined as . The identity is proved through a connection of the solution of a special case of this type of differential equations with the solutions of a third degree polynomial equation.

New results are proved to estimate the number of periodic solutions of a differential equation of Abel type by using a modification of a technique introduced by Ilyashenko. The main tool is an estimate on the number of zeros of a holomorphic function. A concrete example is analyzed but the results are presented to make the method flexible and applicable to other equations.

We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied by the author earlier and present two classes of special functions, namely, ultraexponential and infralogarithm f-type functions. As a result of present investigation, we obtain general solution of the Abel equation α(f(x))=α(x)+1 under some conditions on a real function f and prove a new completely different uniqueness theorem for the Abel equation stating that the infralogarithm f-type function is its unique solution. We also show that the infralogarithm f-type function is an essentially unique solution of the Abel equation. Similar theorems are proved for the ultraexponential f-type functions and their functional equation β(x)=f(β(x−1)) which can be considered as dual to the Abel equation. We also solve certain problem being unsolved before, study some properties of two considered functional equations and some relations between them. Запропоновано узагальненi форми ультраекспоненцiальних та iнфралогарифмiчних функцiй, що були введенi i вивченi автором ранiше, та наведено два класи спецiальних функцiй — ультраекспоненцiального та iнфралогарифмiчного f-типу. В результатi дослiджень отримано загальний розв’язок рiвняння Абеля α(f(x))=α(x)+1 за певних умов для реальної функцiї f i доведено нову цiлком iншу теорему єдиностi для рiвняння Абеля з твердженням про те, що функцiя iнфралогарифмiчного f-типу є єдиним розв’язком цього рiвняння. Також показано, що функцiя iнфралогарифмiчного f-типу є суттєво єдиним розв’язком рiвняння Абеля. Подiбнi теореми доведено для функцiй ультраекспоненцiального f-типу та їх функцiонального рiвняння β(x)=f(β(x−1)), яке можна вважати дуальним для рiвняння Абеля. Також розв’язано задачу, що не була розв’язана до теперiшнього часу, вивчено властивостi двох розглядуваних функцiональних рiвнянь та деякi спiввiдношення мiж ними.

Many problems in physics like reconstruction of the radially distributed emissivity from the line-of-sight projected intensity, the 3-D image reconstruction from cone beam projections in computerized tomography, etc. lead naturally, in the case of radial symmetry, to the study of Abel’s type integral equation. Obtaining the physically relevant quantity from the measured one requires, therefore the inversion of the Abel’s integral equation. The aim of this letter is to present a user friendly algorithm to invert generalized Abel integral equation by using homotopy perturbation method. The stability of the algorithm is analysed. The validity and applicability of this powerful technique is illustrated through various particular cases which demonstrate its efficiency and simplicity in solving these types of integral equations.

This paper presents a new approximate method of Abel differential equation. By using the shifted Chebyshev expansion of the unknown function, Abel differential equation is approximately transformed to a system of nonlinear equations for the unknown coefficients. A desired solution can be determined by solving the resulting nonlinear system. This method gives a simple and closed form of approximate solution of Abel differential equation. The solution is calculated in the form of a series with easily computable components. The numerical results show the effectiveness of the method for this type of equation. Comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate.

An implicit series solution admitted by a boundary value problem modelled by a Lane-Emden equation of the second kind is obtained. The boundary value problem was derived by Frank-Kamenetskii to model the steady temperature in a vessel in which a thermal explosion is taking place. The Lane-Emden equation is reduced to an autonomous second-order ordinary differential equation by means of a coordinate transformation. The autonomous second-order ordinary differential equation is reduced to a first-order Abel equation. A power series solution of the first-order Abel equation is obtained. The power series solution of the Abel equation is transformed into an implicit series solution of the original Lane-Emden equation satisfying the boundary conditions of the original problem. We show that the implicit power series solution is valid for values of the dimensionless Frank-Kamenetskii parameter @d

A direct almost Bernstein operational matrix of integration is used to propose a stable algorithm for numerical inversion of the generalized Abel integral equation. The applicability of the earlier proposed methods was restricted to the numerical inversion of a part of the generalized Abel integral equation. The method is quite accurate and stable as illustrated by applying it to intensity data with and without random noise to invert and compare it with the known analytical inverse. Thus it is a good method for applying to experimental intensities distorted by noise.

For a Banach space X let 𝔄 be the set of continuous linear operators A : X → X with ‖A‖ < 1, I be the identity operator and 𝔄 c ≔ {A ∈ 𝔄 : ‖I – A‖ ≤ c(1 – ‖A‖)}, where c ≥ 1 is a constant. Let, moreover, (xk ) k≥0 be a sequence in X such that the series converges and ƒ : 𝔄 ∪ {I} → X be the mapping defined by the equality It is shown that ƒ is continuous on 𝔄 and for every c ≥ 1 the restriction of ƒ to 𝔄 c ∪ {I} is continuous at I.

In this work algebro-geometric conditions to have a certain first integral for an Abel differential equation are given. These conditions establish a bridge with classical Galois theory because we transform the differential problem of finding a first integral for an Abel equation into an algebraic problem.

We discuss a nonlinear Abel equation on the half-line ( − ∞ , c ) , c > 0 . The basic results provide criteria for the existence of nontrivial everywhere positive solutions. They are expressed in terms of the generalized Osgood condition.

This paper deals with a new method to determine the dependence of the electrical conductivity of metals or semiconductors on temperature. It is based on the fact that the current–voltage relationship is easily measurable. This inverse problem is solved by the classical Abel integral equation.

By using appropriate transformations in combination with specific Abel equations solvable in closed form containing arbitrary functions, an implicit solution as well as the associated sufficient condition are derived for certain differential equations of the Abel class of the first kind.

In Cheb-Terrab and Roche (Comput Phys Commun 130(1–2):204–231, 2000) a classification of the Abel equations known as solvable in the literature was presented. In this paper, we show that all the integrable rational Abel differential equations that appear in Cheb-Terrab and Roche (Comput Phys Commun 130(1–2):204–231, 2000) and consequently in Cheb-Terrab and Roche (Eur J Appl Math 14(2):217–229, 2003) can be reduced to a Riccati differential equation or to a first-order linear differential equation through a change with a rational map. The change is given explicitly for each class. Moreover, we have found a unified way to find the rational map from the knowledge of the explicitly first integral.

This article considers the analytical approximation of limit cycles that may appear in Abel equations written in the normal form. The procedure uses an iterative approach that takes advantage of the contraction mapping theorem. Thus, the obtained sequence exhibits uniform convergence to the target periodic solution. The effectiveness of the technique is illustrated through the approximation of an unstable limit cycle that appears in an Abel equation arising in a tracking control problem that affects an elementary, second-order bilinear power converter.

This paper presents a new, stable, approximate inversion of Abel integral equation. By using the Taylor expansion of the unknown function, Abel equation is approximately transformed to a system of linear equations for the unknown function together with its derivatives. A desired solution can be determined by solving the resulting system according to Cramer’s rule. This method gives a simple and closed form of approximate Abel inversion, which can be performed by symbolic computation. The nth-order approximation is exact for a polynomial of degree up to n. Abel integral equation is approximately expressed in terms of integrals of input data; so the suggested approach is stable for experimental data with random noise. An error analysis of this approach is given. Finally, several numerical examples are given to illustrate the accuracy and stability of this method.

We prove that the first-order nonlinear differential system governing the plastic spin response in simple shear (Dafalias’s equations) is reduced to an equivalent equation of the Abel normal form by means of admissible functional transformations. In similar Abel equations result also the original and the generalized Volterra differential systems, describing the problem of two populations conflicting with one another. The above reduced Abel equations do not admit exact analytic solutions in terms of known (tabulated) functions, since only very special cases of these types of equations can be analytically solved in parametric form. We provide a mathematical solution methodology leading to the construction of exact implicit analytic solutions of the above-mentioned type of equations. Since the plastic spin nonlinear differential system results in a special unsolvable form of Abel’s equation of the normal form, we perform the exact implicit analytic solution of this system too.