Construction of Functional Polynomials for Solutions of Integrodifferential Equations
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CONSTRUCTION OF FUNCTIONAL POLYNOMIALS FOR SOLUTIONS OF INTEGRODIFFERENTIAL EQUATIONS V. A. Litvinov
UDC 519.6
The integrodifferential equations of mathematical physics are objects of research, and construction of interpolating polynomials to obtain approximate solutions of such equations is the subject of investigation. This paper lays out a technique for constructing approximate expressions for functionals on solutions of integrodifferential equations which are an analog of the Hermite polynomial used to interpolate functions. In the example of the diffusion equation, it is shown that the use of such basis solutions allows a substantial increase in the accuracy of the approximate representation of the functionals in comparison to the first approximation of perturbation theory with practically the same computational costs. Keywords: differential equations, integral equations, integration, numerical methods, Hermite polynomials.
Differential, integral, and integrodifferential equations number among the main tools of theoretical research into physical phenomena and processes. One of the elements of such studies is a comparison of the results of theoretical calculations with experimentally measured quantities which are functionals of parameters entering into the equation and describing properties of the medium or the nature of the interaction of particles and radiation with the medium. A significant part of the comparison of the results of theoretical calculations with the results of experimental observations is carried out with the aim of obtaining information about the state of the medium or the nature of the interaction of particles and radiation with the medium, the so-called inverse problem. Here even the presence of an analytical solution of the equation describing the process under study is not a direct instrument for solving the inverse problem since it may not contain, and in the majority of cases does not contain, an explicit form of the dependence of the functional on the parameters of the problem. In a number of cases, it is possible to obtain an approximate estimate of the value of an unknown parameter of the model by using the first approximation of perturbation theory, in fact linearizing the dependence of the solution describing the process on the unknown parameter. Here, the range of variation of the unknown parameter in which the examined dependence is nearly linear can be small in comparison with the region of acceptable uncertainty. It is possible to expand the range of approximate description by increasing the number of basis solutions. Here, in contrast to the traditional first approximation of perturbation theory, it is possible already with two basis solutions to construct a third-degree polynomial with comparable computational expenses to solve the same equations.
DESCRIPTION OF THE METHOD A linear approximation of the dependence of readings of a functional of some parameter characterizing the interaction of particles (radiation) with a medium can be constructed with the help of parametri
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