TAILIEUCHUNG - Handbook of mathematics for engineers and scienteists part 106

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. | . Method of Functional Separation of Variables 703 Let us dwell on the case . According to h Ai z Ci 2 A where Ai -C3 3 and A2 -C2 are any numbers. Substituting and into 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 F w AiBieu A2BieT2u where u . 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. It remains to determine t and x . We substitute into the functional differential equation . Taking into account we find Wtt - t 2 - aip3 - b - p x p - Fx 2 F t - 3a 2 - i f x 3a 2 0. Differentiating 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 are arbitrary constants. Eliminating the derivatives from using 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 the analysis is performed in a similar way. Table presents the final results for the cases - . Case 2. Integrating the third and fourth equations in 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 where D1 and D2 are arbitrary constants. In both cases the function F w in equation is arbitrary. The first row in corresponds to the traveling-wave solution w

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