| Summary: | Background. Ambartsumian’s equation and its generalizations are one 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’s 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 article is devoted to the construction of such
method. Materials and methods. 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. Results. An iterative method for
solving the Ambartsumian’s equation is constructed and substantiated. Model examples were
solved to illustrate the effectiveness of the method. Conclusions. Equations generalizing the
classical Ambartsumian’s 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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