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Symmetry Analysis of Abel's Equation
Studies in Applied Mathematics, 1998A solution algorithm for Abel's equation and some generalizations based on a nontrivial Lie symmetry of a particular kind, i.e., so‐called structure‐preserving symmetry, is described. For the existence of such a symmetry a criterion in terms of the coefficients of the so‐called rational normal form of the given equation is derived. If it is affirmative,
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On A Functional Equation of Abel
Results in Mathematics, 1994The authors determine the general solution of \(\psi (x + y) = g(xy) + h(x - y)\), for \(\psi, g,h : K \to G\), where \(K\) is a field and \(G\) is an Abelian group, which was first treated by Abel in his 1823 manuscript.
Chung, J. K. +4 more
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1995
The following Volterra integral equation of the first kind is due to Abel (1823): $$g(x) = \int\limits_a^x {\frac{{f(y)}} {{\sqrt {x - y} }}dy\;for\;x \geqslant a}$$ (6.1.1) . Since the denominator \(\sqrt {x - y} \) has a zero at y=x, the integral in (1) is to be understood in the improper sense (cf.
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The following Volterra integral equation of the first kind is due to Abel (1823): $$g(x) = \int\limits_a^x {\frac{{f(y)}} {{\sqrt {x - y} }}dy\;for\;x \geqslant a}$$ (6.1.1) . Since the denominator \(\sqrt {x - y} \) has a zero at y=x, the integral in (1) is to be understood in the improper sense (cf.
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Parametric Centers for Trigonometric Abel Equations
Journal of Dynamics and Differential Equations, 2008his article is devoted to one-dimensional perturbative theory on R × S1. There is a recursive formula for the successive obstructions to parametric center at any order of the perturbation parameter. The first obstruction is studied by means of complex analysis techniques.
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A functional equation of abel revisited
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1994The equation \(g(x) + g(y) = h(xf(y) + yf(x))\) \((f,g,h\) unknown) was solved by \textit{N. H. Abel} [J. Reine Angew. Math. 21, 386-394 (1827)] under differentiability suppositions. It also belonged to those functional equations which D. Hilbert in 1900, in the second part of the fifth of his famous unsolved problems, proposed for solution under ...
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Functions Which Satisfy Abel’s Differential Equation
SIAM Journal on Applied Mathematics, 1967(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
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Regularization of an Abel equation
Integral Equations and Operator Theory, 1997A class of Abel equations in the space \(C_0[0,1]\) of continuous functions with compact supports are replaced with their finite difference approximations on a uniform grid and then regularized by the standard Tikhonov method. Estimates for regularized and discretized finite-dimensional solutions of the initial value problem are presented in the case ...
Wang, Ping, Zheng, Kewang
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Group iteration for Abel’s functional equation
Nonlinear Analysis: Hybrid Systems, 2007Some group-theoretic iteration technique is developed for an investigation of the generalized Abel functional equation \[ \alpha(f(x))= g(\alpha(x)). \] The technique is based on a change of a variable in intervals between adjacent fixed points of the given function \(f\).
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Abel Equations and Generalizations
2009In this chapter, Abel’s equation, exponential iteration, associative and commutative equations, trigonometric equations, and systems of equations are treated. Generalizations and connections to information measures are treated. Hilbert, in his famous address to the International Congress of Mathematicians held in Paris in 1900 [372], posed many ...
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