zbMATH — the first resource for mathematics

A solution method for the linear Chandrasekhar equation. (English) Zbl 1110.45002
The authors consider the linearized Chandrasekhar-Ambarzumyan equation: \[ \begin{aligned} A\varphi(x):= & \varphi(x)-x\int_0^1\frac{\varphi(x)k(s) +k(x)\varphi(s)}{s+x}\,ds\\ = & c(x)\varphi(x)-xk(x)\int_0^1\frac{\varphi(s)\,ds}{s +x}=g(x),\qquad 0<x<1\, .\end{aligned} \] Due to the multiplier \(x\) in front of the integral with fixed singularity, \(A=cI+T\), where \(T\) is a compact operator and the coefficient is peculiar: \[ c(x):=1-x\int_0^1\frac{k(s)}{s+x}\,ds. \] A Stiltjes-type transform reduces the equation to a boundary value problem for harmonic functions \(\Phi(x)\) in the upper half-plane, solved in closed form. Under additional constraints a solution \(\Phi(x)\) extends holomorphically in the lower half-plane slit along a straight line segment and the solution of the original problem is recovered as the values of \(\Phi(x)\) on the lips of this slit. The authors present explicit criteria for the Fredholm property of \(A\) and formulae for the complete set of solutions.

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
30E25 Boundary value problems in the complex plane
Full Text: DOI
[1] Chandrasekhar, Bulletin of the American Mathematical Society 53 pp 641– (1947)
[2] Radiative Transfer. Clarendon Press: Oxford, 1950. · Zbl 0037.43201
[3] The Mathematics of Radiative Transfer. Cambridge University Press: Cambridge, MA, 1960.
[4] Analytische Lösungen der Auto-, Kreuz- und Tripel-Korrelationsgleichungen und verwandter Integral- und Integrodifferentialgleichungen. Sitzungsberichte der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Naturwissenschaftliche Klasse, Band 129, Heft 6, 2004; 35.
[5] On nonlinear integral equations which play a rôle in the theory of Wiener–Hopf equations I, II. In Topics in Differential and Integral Equations and Operator Theory, Operator Theory and Applications, (eds), vol. 7. Birkhäuser: Basel, 1983; 173–242. · doi:10.1007/978-3-0348-5416-0_3
[6] Hively, SIAM Journal on Mathematical Analysis 9 pp 787– (1978) · Zbl 0388.45004
[7] Kelly, SIAM Journal on Mathematical Analysis 10 pp 844– (1979)
[8] Legget, SIAM Journal on Mathematical Analysis 7 pp 542– (1976)
[9] Legget, Journal of Mathematical Analysis and Applications 57 pp 462– (1977)
[10] Seikkala, Acta Universitatis Ouluen A 144 pp 1– (1983)
[11] Solutions by a series of the Chandrasekhar H-equation. Proceedings of the Summer School in Numerical Analysis, Neittanmäki P (ed.), Jyväskylä, 1985; 291–298.
[12] Albrecht, ZAMM Journal for Applied Mathematics and Mechanics 70 pp t588– (1990)
[13] Integral Equations with Fixed Singularities. Teubner: Leipzig, 1979.
[14] . Topics in Hardy Classes and Univalent Functions. Birkäuser: Basel, 1994. · Zbl 0816.30001 · doi:10.1007/978-3-0348-8520-1
[15] Boundary Value Problems. Pergamon Press: Oxford, 1966.
[16] Singular Integral Equations. Noordhoff: Groningen, 1953.
[17] Berkovič, Izvestiija Vusov Matematika (Kazan) 1 pp 3– (1966)
[18] von Wolfersdorf, ZAA Journal for Analysis and its Applications 22 pp 647– (2003)
[19] Engibaryan, Mathematics of the USSR-Sbornik 26 pp 35– (1975)
[20] Engibaryan, Mathematics of the USSR-Sbornik 52 pp 121– (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.