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Handbook of mathematics for engineers and scienteists part 123

Handbook of mathematics for engineers and scienteists part 123. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 822 Integral Equations Example 2. Consider the equation y x jfX 1 K y t dt f x . 16.2.5.21 In accordance with the method of model solutions we consider the following auxiliary equation with power-law right-hand side y x L 1 k xx y t dt xs. Its solution has the form see Example 2 for A -s in Section 16.2.4 Y x s x B s K t t- dt. 1 B s Jo This by means of formula 16.2.5.20 yields the solution of equation 16.2.5.21 for an arbitrary right-hand side y x 1 c i 2ni Jc-i -s f s 1 B s x s ds where f s is the Mellin transform 16.2.5.19 of the function f x . 16.2.6. Successive Approximation Method 16.2.6-1. General scheme. 1 . Consider a Volterra integral equation of the second kind y x - I K x t y t dt f x . 16.2.6.1 Assume that f x is continuous on the interval a b and the kernel K x t is continuous for a x b and a t x. Let us seek the solution by the successive approximation method. To this end we set y x f x Vn x 16.2.6.2 where the pn x are determined by the formulas V1 x i K x t f t dt J a V2 x y K x t V1 t dt y K2 x t f t dt V3 x y K x t v2 t dt y K3 x t f t dt etc. Here x Kn x t K x z Kn-1 z t dz 16.2.6.3 a where n 2 3 . and we have the relations K1 x t K x t and Kn x t 0 for t x. The functions Kn x t given by formulas 16.2.6.3 are called iterated kernels. These kernels satisfy the relation Kn x t i Km x s Kn-m s t ds 16.2.6.4 a where m is an arbitrary positive integer less than n. 16.2. Linear Integral Equations of the Second Kind with Variable Integration Limit 823 2 . The successive approximations can be implemented in a more general scheme px yn x f x y K x t yn-i t dt n 1 2 . 16.2.6.5 where the function y0 x is continuous on the interval a b . The functions y1 x y2 x . which are obtained from 16.2.6.5 are also continuous on a b . Under the assumptions adopted in Item 1 for f x and K x t the sequence yn x converges as n oo to the continuous solution y x of the integral equation. A successful choice of the zeroth approximation y0 x can result in a rapid convergence