tailieunhanh - Handbook of mathematics for engineers and scienteists part 198

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 198', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Nonlinear Systems of Two Second-Order Equations 1347 du d2u u dt dx2 w dw d2 w u d W d This system is a special case of system with b a and hence it admits the above solutions given in Items 1 -5 . In addition it has some interesting properties and other solutions which are given below. Suppose u u x t w w x t is a solution of the system. Then the functions u1 Au x C1 t C2 u2 exp Xx aX2t u x 2aXt t w1 Aw x Ci t C2 w2 exp Xx aX2t w x 2aXt t . where A C1 C2 and A are arbitrary constants are also solutions of these equations. 6 . Point-source solution 2 x2 u P T 2 f X w where the functions w w t and t are determined by the autonomous system of ordinary differential equations . 1 w v - T ac - 2a l stt . 7 . Functional separable solution u exp kxt 3 ak2t3 - At y 2 w exp kxt 3ak2t3 - At z where k and A are arbitrary constants and the functions y y and z z are determined by the autonomous system of ordinary differential equations ay A - k y yf y z 0 az A - k z zg y z 0. 8 . Let k be a root of the algebraic transcendental equation f k g k . 1 Solution u kext 0 w ext 0 A f k where the function 0 0 x t satisfies the linear heat equation 5 adj 9 . Periodic solution u Ak exp - x sin ßx - 2aßgt B w A exp - x sin ßx - 2aßgt B ß k2 1 f k a where A B and a are arbitrary constants and k is a root of the algebraic transcendental equation 1 . 1348 Systems of Partial Differential Equations 10 . Solution u Q t exp gW dt 0 x t w exp J g t dt 0 x t where the function p Q t is described by the separable first-order nonlinear ordinary differential equation Wt - 9 V V 2 and the function 0 0 x t satisfies the linear heat equation --s To the particular solution k const of equation 2 there corresponds the solution given in Item 8 . The general solution of equation 2 is written out in implicit form as dtp f - g l t c . 11 . The transformation u a1 U b1W w a2 U b2W where an and bn are arbitrary constants n 1 2 leads to an equation of similar form for U and W. du d2u u dt dx2