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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.

##### MSC:
 4.5e+11 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 3e+26 Boundary value problems in the complex plane
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##### References:
 [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)
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