TAILIEUCHUNG - Handbook of mathematics for engineers and scienteists part 118

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. | . Nonlinear Sy stems of Partial Differential Equations 787 For hyperbolic systems equation has two different real roots 2 Pi 92 2 P1 - 92 2 4p2 t. For each eigenvalue Am of from we find the associated eigenfunction bm m Am - 92 bm mqi where pm pm u w is an arbitrary function it will be defined later m 1 2. From in view of we obtain two equations for the two roots bm du dt dw dt m 1 2. du m dx bm n The function m in can be determined from the conditions b m _ dnm b m _ 1 du 2 dw On differentiating the first relation with respect to w and the second with respect to u we equate the mixed derivatives Rm uw and Rm wu. In view of we obtain the following linear first-order partial differential equation for pm l m Xm - 92 dfym91 . dw du This equation can be solved by the method of characteristics see Subsection . Assuming that a solution of equation has been obtained any nontrivial solution can be taken and taking into account formulas we find the functions Rm Rm u w from system . Replacing bm and 2 J in by the right-hand sides of we get two equations dR1 _ dR1 n A1 R1 r2 0 dt dx dR V n A2 R1 R2 0 dt dx where Am R1 R2 Am u w m 1 2. The functions R1 and R2 appearing in system are called Riemann invariants. System 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 is independent of R2 then the solution of system 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 rather .

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