tailieunhanh - Handbook of mathematics for engineers and scienteists part 128

Handbook of mathematics for engineers and scienteists part 128. 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. | . Nonlinear Integral Equations 857 Example 1. Consider the integral equation I y x - t y t dt Axm m -1. Jo Applying the Laplace transform to the equation under consideration with regard to the relation L xm r m 1 p m-1 we obtain y2 p AF rn 1 p m where F m is the gamma function. On extracting the square root of both sides of the equation we obtain y p .111 1 p . Applying the Laplace inversion formula we obtain two solutions to the original integral equation 11 1 1 .111 1 _ r 1 -x 2 y2 x r 1 x 2 . . Method of differentiation for integral equations. Sometimes differentiation possibly multiple of a nonlinear integral equation with subsequent elimination of the integral terms by means of the original equation makes it possible to reduce this equation to a nonlinear ordinary differential equation. Below we briefly list some equations of this type. 1 . The equation ex y x J f t y t dt g x can be reduced by differentiation to the nonlinear first-order equation yX f X y - gX x 0 with the initial condition y a g a . 2 . The equation y x Î x - t f t y t dt g x J a can be reduced by double differentiation with the subsequent elimination of the integral term by using the original equation to the nonlinear second-order equation yXx f x y - gXx x 0. The initial conditions for the function y y x have the form y a g a yX a gX a . 3 . The equation y x Î eX x-t f t y t dt g x a can be reduced by differentiation to the nonlinear first-order equation y x f x y - Xy Xg x -g x x 0. The desired function y y x must satisfy the initial condition y a g a . 858 Integral Equations 4 . Equations of the form y x cosh A x -1 f t y t dt g x a y x sinh A x -1 f t y t dt g x a y x cos A x - t f t y t dt g x a y x i sin A x - t f t y t dt g x a can also be reduced to second-order ordinary differential equations by double differentiation. For these equations see the book by Polyanin and Manzhirov .

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