Abstract
Ambartsumian equation and its generalizations are some of the main integral equations of astrophysics, which have found wide application in many areas of physics and technology. An analytical solution to this equation is currently unknown, and the development of approximate methods is urgent. To solve the Ambartsumian equation, several iterative methods are proposed that are used in solving practical problems. Methods of collocations and mechanical quadratures have also been constructed and substantiated under rather severe conditions. It is of considerable interest to construct an iterative method adapted to the coefficients and kernels of the equation. This paper is devoted to the construction of such method. The construction of the iterative method is based on a continuous method for solving nonlinear operator equations. The method is based on the Lyapunov stability theory and is stable against perturbation of the initial conditions, coefficients, and kernels of the equations being solved. An additional advantage of the continuous method for solving nonlinear operator equations is that its implementation does not require the reversibility of the Gateaux derivative of the nonlinear operator. An iterative method for solving the Ambartsumian equation is constructed and substantiated. Model examples were solved to illustrate the effectiveness of the method. Equations generalizing the classical Ambartsumian equation are considered. To solve them, computational schemes of collocation and mechanical quadrature methods are constructed, which are implemented by a continuous method for solving nonlinear operator equations.
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REFERENCES
V. A. Ambartsumyan, Scientific Works (Akad. Nauk ArmSSR, Erevan, 1960), vol. 1 [in Russian].
V. A. Ambartsumyan, Scientific Works (Akad. Nauk ArmSSR, Erevan, 1960), vol. 2 [in Russian].
S. Chandrasekhar, Radiation Transfer (Clarendon, Oxford, 1950).
V. V. Sobolev, Light Scattering in Planetary Atmospheres (Nauka, Moscow, 1972) [in Russian].
V. V. Ivanov, Radiation Transfer and Celestial Bodies Spectra (Nauka, Moscow, 1969) [in Russian].
V. A. Ambartsumyan, Astron. Zh. 19 (50), 30 (1942).
V. A. Ambartsumyan, Dokl. Akad. Nauk SSSR 38 (8), 257 (1943).
N. B. Engibaryan and A. A. Arutyunyan, Mat. Sb. 97 (1(5)), 35 (1975).
B. N. Engibaryan, Mat. Zamet. 81 (5), 693 (2007). https://doi.org/10.4213/mzm3712
L. G. Arabadzhyan and N. B. Engibaryan, Itogi Nauki Tekh., Ser. Mat. Anal. 22, 175 (1984).
R. Kh. Aminov and B. G. Gabdulkhaev, Dokl. Akad. Nauk SSSR 304 (2), 322 (1989).
J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Dover, New York, 1952).
J. Hadamard, Leconssur la Propagation des Ondes et les Equations de l’Hydrodynamique (Herman, Paris, 1903).
L. A. Chikin, Uch. Zap. Kazansk. Gos. Univ. 113 (10), 57 (1953).
I. V. Boikov, Differ. Equations 48 (9), 1288 (2012). https://doi.org/10.1134/S001226611209008X
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space (Nauka, Moscow, 1970) [in Russian].
K. Dekker and J. G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations (North Holland, Amsterdam, 1984).
S. M. Lozinskii, Zh. Vych. Mat. Mat. Fiz. 13 (2), 457 (1973).
I. V. Boikov, Approximate Solution of Singular Integral Equations (Penzensk. Gos. Univ., Penza, 2004) [in Russian].
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Translated by M. Drozdova
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Boykov, I.V., Shaldaeva, A.A. Iterative Methods of Solving Ambartsumian Equations. Part 1. Tech. Phys. 67, 429–438 (2022). https://doi.org/10.1134/S1063784222070015
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DOI: https://doi.org/10.1134/S1063784222070015