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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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2003
Generalized Abel equations have the form $$ \phi \left( {F\left( x \right)} \right) = g\left( {x,\phi \left( x \right)} \right) $$ (3.0.1) where F : M → M is a given mapping, g(x, y) is a given function of x ∈ M, y∈ℝ and φ(x) is a solution. The Abel, Schroder and cohomological equations are particular cases of (3.0.1).
Genrich Belitskii, Vadim Tkachenko
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Generalized Abel equations have the form $$ \phi \left( {F\left( x \right)} \right) = g\left( {x,\phi \left( x \right)} \right) $$ (3.0.1) where F : M → M is a given mapping, g(x, y) is a given function of x ∈ M, y∈ℝ and φ(x) is a solution. The Abel, Schroder and cohomological equations are particular cases of (3.0.1).
Genrich Belitskii, Vadim Tkachenko
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Interval Abel integral equation
Soft Computing, 2016The authors study the solvability of the interval Abel integral equation of the form: \[ \frac{1}{\Gamma(\alpha)} \int\limits_a^t (t-s)^{\alpha-1} X(s) \text{d}s = Y(t) ,\, t \in [a, b] \tag{1} \] where \(\alpha \in (0, 1)\), \(K\) is the set of any nonempty compact intervals of the real line \(\mathbb{R}\), \(Y(\cdot) : [a, b] \to K\) being a given ...
Lupulescu, Vasile, Van Hoa, Ngo
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1990
Even though they have a rather specialized structure, Abel equations form an important class of integral equations in applications. This happens because completely independent problems lead to the solution of such equations. After an initial survey of Abel integral equations, this chapter focuses on the numerical solution of these equations when the ...
R. S. Anderssen, F. R. Hoog
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Even though they have a rather specialized structure, Abel equations form an important class of integral equations in applications. This happens because completely independent problems lead to the solution of such equations. After an initial survey of Abel integral equations, this chapter focuses on the numerical solution of these equations when the ...
R. S. Anderssen, F. R. Hoog
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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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Algorithmic solution of Abel’s equation
Computing, 1998zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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2012
In this chapter we study the structure of the set of curves M associated with real polynomials of degree n by means of the Chebyshev correspondence.
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In this chapter we study the structure of the set of curves M associated with real polynomials of degree n by means of the Chebyshev correspondence.
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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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