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Handbook of mathematics for engineers and scienteists part 106. 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. | 15.6. Method of Functional Separation of Variables 703 Let us dwell on the case 15.6.3.22b . According to 15.6.3.23 h Ai z Ci 2 A 15.6.3.25 where Ai -C3 3 and A2 -C2 are any numbers. Substituting 15.6.3.22b and 15.6.3.25 into 15.6.3.24 yields w Bi In z Ci B2 F AiBi z Ci A2Bi . z Ci 2 Eliminating z we arrive at the explicit form of the right-hand side of equation 15.6.3.16 F w AiBieu A2BieT2u where u . 15.6.3.26 B1 For simplicity we set Ci 0 Bi i and B2 0 and denote Ai a and A2 b. Thus we have w z ln z F w aew be 2 g z -i z h z az2 b z. 15.6.3.27 It remains to determine t and x . We substitute 15.6.3.27 into the functional differential equation 15.6.3.18 . Taking into account 15.6.3.17 we find Wtt - t 2 - aip3 - b - p x p - Fx 2 F t - 3a 2 - i f x 3a 2 0. 15.6.3.28 Differentiating 15.6.3.28 with respect to t and x yields the separable equation ttt - 6aipipt y x - F xx 6aFFx 0 whose solution is determined by the ordinary differential equations ttt - 6a t A t Fxxx 6aFFx AFx where A is the separation constant. Each equation can be integrated twice thus resulting in 2 2a 3 Aif Ci C2 156 3 29 Fx -2a 3 AF C3F C4 where Ci.C4 are arbitrary constants. Eliminating the derivatives from 15.6.3.28 using 15.6.3.29 we find that the arbitrary constants are related by C3 -Ci and C4 C2 b. So the functions t and f x are determined by the first-order nonlinear autonomous equations t 2 2a Ac2 Ci C2 Fx -2aF AF - CiF C2 b. The solutions of these equations are expressed in terms of elliptic functions. For the other cases in 15.6.3.22 the analysis is performed in a similar way. Table 15.6 presents the final results for the cases 15.6.3.22a - 15.6.3.22e . Case 2. Integrating the third and fourth equations in 15.6.3.21 yields ÿ s Bt Di p B - C t D2 if A 0 -1 At Di 2 - B p 1- Ax D2 2 B . C if A 0 4 A A 4A A 15.6.3.30 where D1 and D2 are arbitrary constants. In both cases the function F w in equation 15.6.3.16 is arbitrary. The first row in 15.6.3.30 corresponds to the traveling-wave solution w
