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**Unformatted text preview: ** CHAPTER NINE
BOUNDARY-LAYER THEORY His career has been an extraordinary one. He is a man of good
birth and excallent education, endowed by nature with a phenom- enal mathematical faculty. At the age of twenty-one he wrote a
treatise upon the binomial theorem, which has had a European
vogue. On the strength of it he won the mathematical chair at one of our smaller universities, and had, to all appearances, a
most brilliant career before him, But the man had hereditary
tendencies of the most diabolical kind. A criminal strain ran in his blood, which. instead of being modiﬁed, was increased and rendered inﬁnitely more dangerous by his extraordinary mental powers.
(Sherlock Holmes, The Final Problem Sir Arthur Conan Doyle 9.1 INTRODUCTION TO BOUNDARY-LAYER THEORY In this and the next chapter we discuss perturbative methods for solving a differ-
ential equation whose highest derivative is multiplied by the perturbing parameter
a. The most elementary of these methods is called boundary-layer theory. A boundary layer is a narrow region where the solution of a differential
equation changes rapidly. By deﬁnition, the thickness of a boundary layer must
approach 0 as e —> 0. In this chapter we will be concerned with differential equa-
tions whose solutions exhibit only isolated (well—separated) narrow regions of
rapid variation. It is possible for a solution to a perturbation problem to undergo
rapid variation over a thick region (one whose thickness does not vanish with 3).
However, such a region is not a boundary layer. We will consider such problems in Chap. 10.
Here is a simple boundary—value problem whose solution exhibits boundary- layer structure. Example 1 Exactly soluble boundary-layer problem. Consider the differential equation 5y" + (1 + £]y’ + y = 0, y(0}=0, y(l) = 1. (9.1.1) The exact solution of this equation is e: __ ﬁzzle
Lia (9.1.2) y{x}$ 24; r 8:}[5' 419 ...

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