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
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.