Handbook of mathematics for engineers and scienteists part 194. 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. | . Elliptic Equations 1319 1 . Solutions 1 r 2 A2 B2 w x y -ln ----------- 2 P aP Ax By C 2 w x y 1 In -22A2 B2 --- p L aP sinh2 Ax By C . w x y 1 In i--22 a2 B2 - p L aP cosh2 Ax By C 1 T 2 A2 B2 w x y -ô ln 2 s 77 P aP cos2 Ax By C w x y 1 In P if aP 0 if aP 0 if aP 0 if aP 0 8C .2 ln T A 2 A B 2 CI aP P ln x A y B C I where A B and C are arbitrary constants. The first four solutions are of traveling-wave type and the last one is a radial symmetric solution with center at the point -A -B . 2 . Functional separable solutions w x y - ln P C1eky aP cos kx C2 2k 1 2k2 B2 - A2 w x y ôln T -77--77 7 7------ P aP A cosh kx C1 B sin ky C2 2 1 2k2 A2 B2 w x y 7ln r4 .z -77T F---7---7777 P aP A sinh kx C1 B cos ky C2 2 where A B C1 C2 and k are arbitrary constants x and y can be swapped to give another three solutions . 3 . General solution z x 2 1 - 2aph z h z w x y - pln . -. where h h z is an arbitrary analytic holomorphic function of the complex variable z x iy with nonzero derivative and the bar over a symbol denotes the complex conjugate. 4. 2 p ae w bel0w_ 1 . Traveling-wave solutions 1 w x y - ln P w x y -P In aP C3 exp C1 x C2y aP C1 C2 C2 C2 1 4C3 C2 C2 2 aP Va2P2 bP C2 C2 . C 2 ---------7 2---------- sin C1x C2y C3 C2 C2 exp -C1x - C2y where C1 C2 and C3 are arbitrary constants. 2 . For other exact solutions of this equation see equation with f w ae w be2 3w. 1320 Nonlinear Mathematical Physics Equations 5. 2. aw ln 3w . dx2 dy2 v 7 1 . Solutions w x y 1 exp 3 4a y b 1 a x A 2 w x y 1 exp 3 A x B 2 Aa - 4 A2 x B y C 4 a - A y C 2 where A B and C are arbitrary constants. 2 . There are exact solutions of the following forms w x y F z z Ax By w x y G r r J x Ci 2 y C2 2 w x y f x g y . 6. a sin w . dx2 dy2 V 7 1 . Functional separable solution for a 3 1 z x z . . zcosh F cos A z . z sin A z . w x y 4 arctan cot A-hG F 2 x - By G 2 y Bx where A and B are arbitrary constants. 2 . Functional separable solution generalizes the solution of Item 1 4 w x y
