tailieunhanh - Handbook of mathematics for engineers and scienteists 78part

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists 78part', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Second-Order Nonlinear Differential Equations 507 2 . We now consider an equation of the general form eyXX F x y y x subject to boundary conditions . For the leading term of the outer expansion y y0 x we have the equation F x yo yO 0. In the general case when using the method of matched asymptotic expansions the position of the boundary layer and the form of the inner extended variable have to be determined in the course of the solution of the problem. First we assume that the boundary layer is located near the left boundary. In we make a change of variable z x 5 e and rewrite the equation as 52 _ 1 yzz yF z y -y . The function 5 5 e is selected so that the right-hand side of equation has a nonzero limit value as e - 0 provided that z y and y z are of the order of 1. Example 5. For F x y y x -kxxy x y where 0 A 1 the substitution z x 5 e brings equation to x 52 Vzz kz yz y. In order that the right-hand side of this equation has a nonzero limit value as e 0 one has to set 51 A e 1 or 51 a e const where const is any positive number. It follows that 5 e 1 A. The leading asymptotic term of the inner expansion in the boundary layer y y0 z is determined by the equation y0 kzxy 0 0 where the prime denotes differentiation with respect to z. If the position of the boundary layer is selected incorrectly the outer and inner expansions cannot be matched. In this situation one should consider the case where an arbitrary boundary layer is located on the right this case is reduced to the previous one with the change of variable x 1 - z . In Example 5 above the boundary layer is on the left if k 0 and on the right if k 0. There is a procedure for matching subsequent asymptotic terms of the expansion see the seventh row and last column in Table . In its general form this procedure can be represented as inner expansion of the outer expansion y-expansion for x 0 outer expansion of the inner expansion y-expansion .