By Smirnov V.I.
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Extra resources for A course of higher mathematics, vol. 2
C) Does the addition of air resistance increase or decrease the flight time? 4. p(x,c)y = 0, y" for 0 < x < 1, where y(O) = y(l) = O. For small c assume p ~ 1 + c/-L(x), where /-L(x) is positive and continuous. 3). (a) Find Yo and Ao· (b) Find Y1 and AI. 5. An interesting problem that can trace its origins back to Bernoulli and Euler is that of a whirling elastic string that is held fixed at each end (something like a jump-rope). The problem comes down to solving for a function y( x) and a frequency of rotation w that satisfy (Caughey, 1970) y "+ 2(1 ,(I + c 2y2)l/2 + a 2) y = 0 ,lor 0 < x < 1, 1" W where y'(O) = y'(l) = O.
B) Suppose Yo = 1, Y1 = 0, and 0: = 2. Calculate the exact solution, and the two-term expansion you found in part (a), for n = 2,3, ... ,10. Comment on the accuracy of the approximation. 12. The transport of calcium in a cell is described by the following nonlinear diffusion problem for the concentration e(x, t) of calcium in a bounded domain 0 (Holmes and Bell, 1991): \7 2 c = c[8t e + f(c)], for x where E 0 and t > 0, for x E 80 0 for x E 80 1 and e(x, 0) = O. Here 8n is the normal derivative and 80 = 80 0 u80 1 is the boundary of O.
We will not pursue this here but point out that the approximations we have calculated so far are x '" ±1 - E. To illustrate the accuracy of this result, one of the exact solutions is plotted in Fig. 4 along with its asymptotic approximation. It is clear that they are in very good agreement for small values of E. 19). 9990005···. 20 --- 1. 4. 20) and the two-term asymptotic expansion x rv 1 + c. It is seen that for small c the asymptotic approximation is very close to the exact value. It is appropriate to comment on the notation used in the previous paragraph.