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The first three approximate solutions are: u[ll (t) u[2l(t) u[3l (t) + 0(t)3 3 u(t) + 360t + 0(t)4 u(t) - 90t 2 = u( t) - 1080t4 + O( t)5. From the above we see that exactly one additional correct term is picked up in each iteration. The reason is evident from a consideration of the error term, f[kl(t) = u[kl(t)-u(t), which satisfies (5). IT the partial derivatives in that equation are evaluated at suitable points near the solution, the higher order terms can be ignored in that equation, so we find that f[k+ll = l G(r)f[kl(r)dr where G is the Greens function for the left hand side of (5).

Math. 6(1988),88-97. Feng Kang, Wu Hua-mo, Qin Meng-zhao, Wang Dao-liu, Jour. Camp. Math. 7(1989),72-96. Li Chun-wang, Qin Meng-zhao, Jour. Compo Math. 6:2 (1989). Wu Yu-hua, Computer Math. & Appl. 15:12(1989), 1041-1050. Qin Mengzhao, Math. Meth. in Appl. Sci. 11(1989), 543-557. R. Hamilton, Mathematical Papers, v. 2, Cambridge, 1940. L. Synge, Scripta Math. 10(1944), 13-24. Klein, Entwickelung der Mathematik im 19 Jahrhundert, Teubner, 1928. E. Schroedinger, Scripta Math. 10(1944), 92-94. I. Arnold, Mathematical Methods of Classical Mechanics, Nauka, 1974.

3) must be omitted. 1) has a single globally attracting fixed point U1. 1) will tend to U1 as T ...... 00. As a consequence it is to be expected that, for f sufficiently small, the solution u(x, t) will tend to the constant U1 as T ...... 00. 1. o(x,r) -> U1 as r -> 00. 3) in general. 3). (x,r,e) holds for x near aw. We call it a boundary layer expansion. It can be found by first introducing the stretched variable v = nje, where n(x) is the distance from x to aw, and n - 1 coordinates y orthogonal to n(x).