Handbook of mathematics for engineers and scienteists part 193

Handbook of mathematics for engineers and scienteists part 193. 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. | 1312 Nonlinear Mathematical Physics Equations 5 . Solution A B and C are arbitrary constants w x t z exp i Axt - 3 A2t3 Bt C z x - At2 where the function z is determined by the ordinary differential equation zz f W - Az Bty 0. 6 . Solutions w x t j exp iQ x t x t x C2 i f C1t -1 2 dt C3 V c 11 4t j where C1 C2 and C3 are arbitrary real constants. 7 . Solution r i z z dx w x t u x exp i x t p x t C1t C2 C3 J u2 x where C1 C2 and C3 are arbitrary real constants and the function u u x is determined by the autonomous ordinary differential equation u x - C1u - C u -3 f u u 0. 8 . There is an exact solution of the form w x t u z exp iAt i z z kx At where A k and A are arbitrary real constants. . Hyperbolic Equations . Nonlinear Wave Equations of the Form a -V f w dt dX d2w d2w n 1 dt2 dx2 aw w 1 . Traveling-wave solutions for a 0 w x t w x t 2b sinh2 z 1 1-n a n 1 2b cosh2 z - a n 1 J 1 1-n z 2y a 1 - n x sinh C1 1 coshC1 C2 if b n 1 0 z ịỵ ã 1 - n x sinh C1 t coshC1 C2 if b n 1 0 2b cos2 z ã n 1 w x t where C1 and C2 are arbitrary constants. 2 . Traveling-wave solutions for a 0 and b n 1 0 j_ V z 2 Via 1 - n x sinh Ci t cosh Ci C2. 2 3 . For a 0 there is a self-similar solution of the form w t 1-n F z where z x t. 4 . For other exact solutions of this equation see equation with f w aw bwn. . Hyperbolic Equations 1313 2. d2w 9t2 d2w dx2 awn bw2n-1. Solutions w x. t 1 a n x sinh Ci t cosh Ci C2 2 - 2 n 1 2an 1 w x. t 4a 1 - n 2 t C1 2 - x C2 2 14 L an where C1 and C2 are arbitrary constants. 3. d2w 9t2 2 d2w a 2 dx2 be13 . 1 . Traveling-wave solutions 2 B2 - a2 A2 w x. t 4- In 3 w x. t 4- In 3 w x. t 4- In 3 w x. t 4- In ----s------------- 3 b3 cos2 Ax Bt C _ where A. B. and C are arbitrary constants. 2 . Functional separable solutions w x. t 1 l 82 3 w x. t - ln C1eXx 3 w x. t - ln C1eXx 3 w x. t -3 In C1eaXt w x. t - ln C1eaXt 3 b3 Ax Bt C 2J 2 a2A2 - B2 _ b 3 cosh2 Ax Bt C _ 2 B2 - a2 A2 b3 sinh2 Ax Bt C 2 B2 - a2 A2 I- 2 ln x A 2 - a2 t B 2 C .