Tham khảo tài liệu 'handbook ofintegral equations phần 4', ngoại ngữ, ngữ pháp tiếng anh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Kernels Containing Integer Powers of x and t or Rational Functions 11. J x -1 3y t dt f x . Let us remove the modulus in the integrand Ị x - t 3y t dt Ị t - x 3y t dt f x . 1 Differentiating 1 twice yields 6 I x - t y t dt 6 Ị t - x y t dt fXX x . This equation can be rewritten in the form rb . . x -1 y t dt ifXx x . 2 J a Therefore the solution of the integral equation is given by y x 12 y X Xxx x . 3 The right-hand side f x of the equation must satisfy certain conditions. To obtain these conditions one must substitute solution 3 into 1 with x a and x b and into 2 with x a and x b and then integrate the four resulting relations by parts. 12. Ị I x3 -131 y t dt f x . This is a special case of equation with g x x3. 13. Ị xt2 -t31 y t dt f x 0 a b ro. The substitution w t t2y t leads to an equation of the form i x - t w t dt f x . a 14. Ị x2t -t31 y t dt f x . The substitution w t t y t leads to an equation of the form r b x2 - t2 w t dt f x . a 15. J x3 - @t3 y t dt f x 3 0. This is a special case of equation with g x x3 and 3 A3. 1998 by CRC Press LLC b 16. J x - t 2n 1 y t dt f x n 0 1 2 . Solution . 1 The right-hand side f x of the equation must satisfy certain conditions. To obtain these conditions one must substitute solution 1 into the relations t - a 2n 1y t dt f a t - a 2n-ky t dt - -1----fXk 1 a Ja Ja Ak Ak 2n 1 2n . 2n 1 - k k 0 1 . 2n and then integrate the resulting equations by parts. f x y t dt 17. f x . Jo x t The left-hand side of this equation is the Stieltjes transform. 1 . By setting x ez t eT y t e T 2w r f x e z 2g z we obtain an integral equation with difference kernel of the form rTO w r dr n 7s J-TO 2 cosh 2 z - T g z whose solution is given by w z i cosh nu g u eiux du g u i g z e iuz dz. v2n3 J-TO v2n J-TO Reference P. P. Zabreyko A. I. Koshelev et al. 1975 . 2 . Under some assumptions the solution of the original equation can be represented in the form y x nm __ -1 n rx2n 1 f n x l n
