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Regularisation of Abel's integral equation
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1987SynopsisConsider the Abel integral operatorwhere 0 < α < 1. Supposeuis inH1(0, 1) ofH1-norm ≦E, andfis an element ofL2(0, 1) such that ∥Au–f∥L−2< ε. We give a regularised approximate solutionuβ(f) of the equationwhich satisfiesand can be computed simply by performing some integrations.
Dang Dinh Hai, Dang Dinh Ang
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Systems of Generalized Abel Equations
SIAM Journal on Mathematical Analysis, 1979Certain mixed boundary value problems arising in the classical theory of elasticity lead to the solution of certain systems of generalized Abel integral equgtions. A method is presented where these systems are reduced to uncoupled pairs of Riemann boundary value problems. Closed form solutions are obtained.
Lowengrub, M., Walton, J.
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Algorithmic solution of Abel’s equation
Computing, 1998zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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On the Center Problem for Generalized Abel Equations
Acta Mathematica Sinica, English Series, 2023Consider on the cylinder the differential equation \[ \frac{{dr}}{{d\theta}} = p(\theta) r^n + q(\theta) r^l \tag{1}, \] where \(p\) and \(q\) are continuous periodic functions with period \(2\pi\), \(n\) and \(l\) are natural numbers satisfying \(l>n\).
Liu, Chang Jian, Wang, Shao Qing
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Aequationes mathematicae, 2012
Let \(I\) be an open real interval and \(\mathcal{S}\) be a nonempty subset of all increasing bijections of \(I\) and \(\lambda:\mathcal{S}\rightarrow\mathbb R\). The authors investigate the functional equation \[ \phi\bigl(f(x)\bigr)=\phi(x)+\lambda(f)\qquad(f\in\mathcal{S},x\in I) \] in the case when the group \(G\) generated by \(\mathcal{S}\) is ...
Farzadfard, Hojjat, Robati, B. Khani
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Let \(I\) be an open real interval and \(\mathcal{S}\) be a nonempty subset of all increasing bijections of \(I\) and \(\lambda:\mathcal{S}\rightarrow\mathbb R\). The authors investigate the functional equation \[ \phi\bigl(f(x)\bigr)=\phi(x)+\lambda(f)\qquad(f\in\mathcal{S},x\in I) \] in the case when the group \(G\) generated by \(\mathcal{S}\) is ...
Farzadfard, Hojjat, Robati, B. Khani
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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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Journal of Quantitative Spectroscopy and Radiative Transfer, 1987
Abstract An explicit solution is offered of Abel's equation. This equation arises often during broadside-on observation of a plasma of cylindrical symmetry.
J.B. Tatum, W.A. Jaworski
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Abstract An explicit solution is offered of Abel's equation. This equation arises often during broadside-on observation of a plasma of cylindrical symmetry.
J.B. Tatum, W.A. Jaworski
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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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Stability Results for Abel Equation
1985The author presents some estimates of solutions to Abel type integral equations. The problem of solving an Abel type integral equation is an ill-posed problem in the sense of Hadamard. The author uses these estimates to restore the stability of solutions whose first-order or fractional derivative is a priori bounded in \(L^ p\).
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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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