A First Look at Perturbation Theory (Dover Books on Physics) by James G., Mann Jr., James E., Physics Simmonds

By James G., Mann Jr., James E., Physics Simmonds

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Extra info for A First Look at Perturbation Theory (Dover Books on Physics) 2nd edition by Simmonds, James G., Mann Jr., James E., Physics (1997) Paperback

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In the chapters to follow, we shall develop systematic methods for re­ ducing singular perturbation problems to regular ones. These will then be solved by constructing regular perturbation expansions. 4. Explain how one or more of the points dis­ cussed at the beginning of the Chapter is illustrated by each of the following problems. Also explain why each is a s i ngul ar perturbation problem. (Note th at all may be solved exactly. ) (a) ii + ( 1 + f)Y = 0, 0 < t < 7r y e O ) = 0 , y ( 7r ) = fP , p > O.

C) ii + y = f cos t , O < t y eO) = 0, yeO) = 1 .

4. Most importantly, the solution of the DE will be a function of both f and the independent variable, say t, which immediately raises the question of whether certain approximate solution are uniformly accu­ rate for all values of t in the domain of interest . Fortunately, when f is small , characterizing the f-dependence of the solution of a DE with Ie 's or Be's simplifies. The rest of this book is 39 44 Singular Perturbations in Ordin ary Differential Equations The term e-t / f is called a boundary layer ( or the b oundary layer con­ tribution to yet , f» because it is significant only in a narrow layer of width O ( f) near the b oun d ary t = o.

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