TAILIEUCHUNG - Handbook of mathematics for engineers and scienteists part 196

Handbook of mathematics for engineers and scienteists part 196. 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. | . Higher-Order Equations 1333 6 . The Boussinesq equation is solved by the inverse scattering method. Any rapidly decaying function F F x y t as x œ and satisfying simultaneously the two linear equations uf S - ÿ Ew 0 generates a solution of the Boussinesq equation in the form d w 12 K x x t where K x y t is a solution of the linear Gel fand-Levitan-Marchenko integral equation rœ K x y t F x y t K x s t F s y t ds 0. Time t appears here as a parameter. dw d dw d d2 w d2 w dy dx dx dy dx2 dy2 There is a two-dimensional stationary equation of motion of a viscous incompressible fluid it is obtained from the Navier-Stokes equation by the introduction of the stream function w. 1 . Suppose w x y is a solution of the equation in question. Then the functions wi -w y x W2 w Cix C2 Ciy C3 C4 w3 w x cos a y sin a -x sin a y cos a where C1 . C4 and a are arbitrary constants are also solutions of the equation. 2 . Any solution of the Poisson equation Aw C is also a solution of the original equation these are inviscid solutions . 3 . Solutions in the form of a one-variable function or the sum of functions with different arguments w y Ciy3 C2y2 C3y C4 w x y C1x2 C2x C3y2 C4y C5 w x y C1 exp -Ay C2y2 C3y C4 vAx w x y C1 exp Ax - vAx C2 exp Ay vAy C3 w x y C1 exp Ax vAx C2 exp -Ay vAy C3 where C1 . C5 and A are arbitrary constants. 4 . Generalized separable solutions w x y A kx Ay 3 B kx Ay 2 C kx Ay D w x y Ae X y kx B y kx 2 C y kx vA k2 1 x D w x y 6vx y A -1 A y A 3 B y A -1 C y A -2 D 1334 Nonlinear Mathematical Physics Equations w x y Ax B e Xy vAx C w x y A sinh 3x B cosh 3x e Xy V 3 A2 x C A w x y A sin 3x B cos 3x e Xy V A2 - 32 x C A w x y AeXy l3x BeYX v y y 3 Y x C y where A B C D k 3 and A are arbitrary constants. 5 . Generalized separable solution linear in x w x y F y x G y 1 where the functions F F y and G G y are determined by the autonomous system of fourth-order ordinary differential equations F Fyy - FFÿÿy V yy 2 G y Fyy - FG yy G yy. 3 Equation 2 has the .

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