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Handbook of mathematics for engineers and scienteists part 126. 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. | 16.4. Linear Integral Equations of the Second Kind with Constant Limits of Integration 843 On applying the Mellin transform to equation 16.4.6.15 and taking into account the fact that the integral with such a kernel is transformed into the product by the rule see Subsection 11.3.2 M 1 Q t y t dt Q s y s I Jo t 7 t 7 J we obtain the following equation for the transform y s y s - Q s y s f s . The solution of this equation is given by the formula A f s i s -_ Q- 16.4.6.18 On applying the Mellin inversion formula to equation 16.4.6.18 we obtain the solution of the original integral equation 1 c ix f s L SQ ds 164 9 This solution can also be represented via the resolvent in the form y x f x i 1N fX f t dt 16.4.6.20 Jo t v t 7 where we have used the notation N x M-1 N s N s Q . 16.4.6.21 1 - Q s under the application of this analytical method of solution the following technical difficulties can occur a in the calculation of the transform for a given kernel K x and b in the calculation of the solution for the known transform y s . To find the corresponding integrals tables of direct and inverse Mellin transforms are applied e.g. see Sections T3.5 and T3.6 . In many cases the relationship between the Mellin transform and the Fourier and Laplace transforms is first used M f x s F f ex is L f ex -s L f e x s 16.4.6.22 and then tables of direct and inverse Fourier transforms and Laplace transforms are applied see Sections T3.1-T3.4 . 16.4.6-3. Equation with the kernel K x t t3Q xt on the semiaxis. Consider the following equation on the semiaxis y x - t3Q xt y t dt f x . 0 16.4.6.23 844 Integral Equations To solve this equation we apply the Mellin transform. On multiplying equation 16.4.6.23 by xs-1 and integrating with respect to x from zero to infinity we obtain i y x xs-1 dx - i y t t dt i Q xt xs-1 dx i f x xs-1 dx. 16.4.6.24 Jo Jo Jo Jo Let us make the change of variables z xt. We finally obtain y s - Q s y f t s dt f s . 16.4.6.25 o Taking into account the relation r y
