On the Approximate Solution of Integro-Differential Equations with a Degenerate Coefficient
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RICAL METHODS
On the Approximate Solution of Integro-Differential Equations with a Degenerate Coefficient N. S. Gabbasov1∗ 1
Naberezhnye Chelny Institute (Branch), Kazan (Volga Region) Federal University, Naberezhnye Chelny, Tatarstan, 423812 Russia e-mail: ∗ [email protected]
Received December 26, 2019; revised December 26, 2019; accepted May 14, 2020
Abstract—We study a linear integro-differential equation of the third kind with a coefficient having power-order zeros. To solve this equation approximately in the space of generalized functions, we propose and justify a generalized version of the subdomain method based on special Kantorovich polynomials. DOI: 10.1134/S0012266120090128
In the present paper, we study the linear integro-differential equation of the third kind 1
q p Z Y X mi Ax ≡ x(t) (t − ti ) + Kj (t, s)x(j) (s) ds = y(t), i=1
(1)
j=0 −1
where t ∈ I ≡ [−1, 1], q, p ∈ N are given, ti ∈ (−1, 1) and mi ∈ N (i = 1, . . . , q) are given numbers, Kj (j = 0, . . . , p) and y are known “smooth” functions, and x is the unknown function. Studying such equations is undoubtedly of interest from the viewpoint of both theory (in particular, Eq. (1) is a generalization of some classes of linear integral equations of the Fredholm type) and applications. Equations of this kind arise in a number of problems in the neutron transport theory, elasticity, and particle scattering theory (see, e.g., [1, 2] and the bibliography in [1]), the theory of differential equations of mixed type [3], as well as the theory of some loaded integro-differential equations [4]. As a rule, these integro-differential equations of the third kind can be solved exactly only in exceptionally rare special cases; therefore, developing methods for their approximate solution with an appropriate theoretical justification is especially topical. Some results in this direction were obtained in [5–7], where the author proposed and justified special direct methods for solving Eq. (1) in some type V space of generalized functions generated by the “Hadamard finite part of the integral” functional. The papers [8–10] deal with some polynomial and spline methods for solving Eq. (1) in the space D of generalized functions based on the “Dirac delta function” functional. In the present paper, we propose a generalized version of the subdomain method tailored to suit the approximate solution of the integro-differential equation (1) of the third kind in a space of the type D. We mainly focus on justifying the method in the sense of the monograph [11, Ch. 1]. Namely, we prove a theorem on the existence and uniqueness of a solution of the corresponding approximate equation, establish an estimate for the error produced by the approximate solution, and prove the convergence of the sequence of approximate solutions to the exact solution in the space of generalized functions. We also study the stability and conditioning of the approximate equations. 1. TEST AND GENERALIZED FUNCTION SPACES Let C ≡ C(I) be the Banach space of continuous functions on I with the
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