tailieunhanh - numerical mathematics and scientific computation volume 1 Episode 6

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 172 Chapter 3. Series Operators and Continued Fractions We first consider an analytic equation f z e 0 and seek the roots Zi e . It is practical to move the origin to Zilfi . See Problem or . If Zị fi is a simple root . if dộ dz Ạ 0 e j o 0 then a theorem of complex analysis tells us that Zi e is an analytic function in a neighborhood of the origin hence the expansion Zi e Co C1 c2e2 . exists. We call this a regular perturbation problem. If Zi 0 is a -fold root then under certain conditions there exists an expansion of the form Zi e Co Cie1 c2f lk . for each of the k k th roots of . If one tries to determine the coefficients in an expansion of the wrong form one usually runs into contradictions. A simple perturbation example for a differential equation is given in Problem 12. More interesting examples are presented in Sec. . Example . We shall expand the roots of f z e ez2 z 1 0 into powers of . The reduced problem z 1 0 has only one finite root z-1 0 1. Set z 1 xe X Cl c2e c3 2 . Then xe e e 1 xe 2 X 0 . 1 C1 c2e2 . . 2 ci c2 c3 2 . 0. Matching the coefficients of o1 e2 we obtain the system 1 Cl 0 Cl 1 2ci c2 0 c2 2 2c2 c2 c3 0 C3 5 hence z-1 e 1 2e2 5e3 . Now the easiest way to obtain the expansion for the second root z2 e is to use the fact that the sum of the roots of the quadratic equation equals -1 hence z2 e 1 1 2 2 . Note the appearance of the term -1. This is due to a characteristic feature of this example. The degree of the polynomial is lower for the reduced problem than it is for Ạ 0 one of the roots escapes to 00 as t 0. This is an example of a singular perturbation problem an important type of problem for differential equations see Sec. . Although we have already determined z2 e we shall use this problem to illustrate a general balancing procedure recommended in Lin-Segel 25 Sec. where it is applied to singular perturbation problems for differential equations too. The basic idea is very simple each term in an .