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Handbook of mathematics for engineers and scienteists part 112

Handbook of mathematics for engineers and scienteists part 112. 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. | 15.10. Differential Constraints Method 745 Example. From the class of nonlinear heat equations with a source dw d xdw fi f2 w dt d dx one singles out equations that admit invariant manifolds of the form gi w dw- g2 w . dx2 ax The functions f2 w fi w g2 w and gi w are to be determined in the further analysis. Eliminating the second derivative from 15.10.3.1 and 15.10.3.2 we obtain dw dw 2 A w w dt dx J where A w fi w gi w fi w w fi w g2 w f2 w . 15.10.3.1 15.10.3.2 15.10.3.3 15.10.3.4 The condition of invariance of the manifold 15.10.3.2 under equation 15.10.3.1 is obtained by differentiating 15.10.3.2 with respect to t wxxt 2g1WxWxt g1 wX wt g2 wt. The derivatives wxxt wxt and wt should be eliminated from this relation with the help of equations 15.10.3.2 and 15.10.3.3 and those obtained by their differentiation. As a result we get 2 g2 3 a gi a9i a 1 4 g.g. 5 g2 Ag2 -gi - g1 w 2 . A A - A 0. Equating the coefficients of like powers of wx to zero one obtains three equations which for convenience may be written in the form a Agi 2gi AZ Agi 0 4g2 a Agi Ag2 - gi 0 15.io.3.5 a - 2 W gx . The first equation can be satisfied by taking A Agi 0. The corresponding particular solution of system 15.10.3.5 has the form A - A Ag2 gi -A g2 2C1 C 7 15.i0.3.6 where a A w is an arbitrary function. Taking into account 15.10.3.4 we find the functional coefficients of the original equation 15.10.3.1 and the invariant set 15.10.3.2 fi 03 - iw f2 A - fi g2 gi -Ay g2 2Ci -CM-C. 15.i0.3.7 2 J a vIaI J a Equation 15.10.3.2 together with 15.10.3.7 admits the first integral wl 4Ci a 402 Via 2at t -M 15.i0.3.8 A 2 where a t is an arbitrary function. Let us eliminate wl from 15.10.3.3 by means of 15.10.3.8 and substitute the functions a and from 15.10.3.6 to obtain the equation A wt -C2 Via-at t . 15.i0.3.9 Let us dwell on the special case C2 C3 0. Integrating equation 15.10.3.9 and taking into account that At A wt yield A -a t 9 x 15.i0.3.i0 where 9 x is an arbitrary function. Substituting .