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Handbook of mathematics for engineers and scienteists part 118

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Handbook of mathematics for engineers and scienteists part 118. 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. | 15.14. Nonlinear Sy stems of Partial Differential Equations 787 For hyperbolic systems 15.14.4.32 equation 15.14.4.35 has two different real roots A1.2 2 Pi 92 2 P1 - 92 2 4p2 t. 15.14.4.36 For each eigenvalue Am of 15.14.4.36 from 15.14.4.34 we find the associated eigenfunction bm m Am - 92 bm mqi 15.14.4.37 where pm pm u w is an arbitrary function it will be defined later m 1 2. From 15.14.4.33 in view of 15.14.4.34 we obtain two equations for the two roots 15.14.4.36 bm du dt dw dt m 1 2. 15.14.4.38 du m dx bm n The function m in 15.14.4.37 can be determined from the conditions b m _ dnm b m _ dR.m 1 du 2 dw 15.14.4.39 On differentiating the first relation 15.14.4.39 with respect to w and the second with respect to u we equate the mixed derivatives Rm uw and Rm wu. In view of 15.14.4.37 we obtain the following linear first-order partial differential equation for pm l m Xm - 92 dfym91 . dw du 15.14.4.40 This equation can be solved by the method of characteristics see Subsection 13.1.1. Assuming that a solution of equation 15.14.4.40 has been obtained any nontrivial solution can be taken and taking into account formulas 15.14.4.37 we find the functions Rm Rm u w from system 15.14.4.39 . Replacing bm and 2 J in 15.14.4.38 by the right-hand sides of 15.14.4.39 we get two equations dR1 _ dR1 n A1 R1 r2 0 dt dx dR V dR.2 n A2 R1 R2 0 dt dx 15.14.4.41 where Am R1 R2 Am u w m 1 2. The functions R1 and R2 appearing in system 15.14.4.41 are called Riemann invariants. System 15.14.4.41 admits two exact solutions R1 C1 x -A2 C1 R t 1 R2 R C2 x - A1 R1 C2 t 2 R1 where Cm are arbitrary constants and m R3-m are arbitrary functions m 1 2 . If the function A1 in 15.14.4.41 is independent of R2 then the solution of system 15.14.4.41 is reduced to successive integration of two quasilinear first-order partial differential equations. 788 Nonlinear Partial Differential Equations Remark 1. Sometimes it is more convenient to use the formulas b m pmp2 b m pm Am - pi 15.14.4.37a rather .

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