Iterative Methods of Solving Ambartsumian Equations. Part 2

The Ambartsumian equation, along with its generalizations, is one of the main integral equations of astrophysics, which have found wide application in many areas of physics and technology. The Ambartsumian equation plays an important role in the study of light scattering in media of infinite optical thickness. Nowadays the analytical solution of this equation is not known; therefore, the development of approximate methods for its solution 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, the substantiation of which has been carried out under rather severe conditions. In the previous work of the authors, a spline-collocation method for solving the Ambartsumian equation with zero-order splines was constructed and substantiated. The accuracy of this method is \(O({{N}^{{ - 1}}})\), where N is a number of collocation nodes. It is of considerable interest to construct an iterative method adapted to the smoothness of the coefficients and kernels of the equation. Light scattering in media of finite optical thickness is described by Ambartsumian equation systems, for an approximate solution of which it is necessary to construct and substantiate effective numerical methods. This study is devoted to the construction of such methods. The construction and substantiation of iterative methods for solving systems of Ambartsumian equations is based on a generalization of the continuous method for solving nonlinear operator equations. The method and its generalization are based on the Lyapunov stability theory and are 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. In this work, spline-collocation methods with first-order splines are constructed for solving the Ambartsumian equation and systems of equations, and their justification is given. Model examples that illustrate the effectiveness of the methods are solved. Equations generalizing the classical Ambartsumian equations are considered. To solve them, the computational schemes of spline-collocation methods are constructed and substantiated. The results obtained can be used to solve a number of astrophysical issues.