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Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
Download (direct link): elementarydifferentialequations2001.pdf
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(a) Find the temperature u (x, t).
(b) Plot u versus x for several values of t. Also plot u versus t for several values of x.
(c) Plot u(4, t) and u(11, t) versus t. Observe that the points x = 4 and x = 11 are symmetrically located with respect to the initial temperature pulse, yet their temperature plots are significantly different. Explain physically why this is so.
15. Consider a uniform bar of length L having an initial temperature distribution given by f (x), Q < x < L. Assume that the temperature at the end x = Q is held at 0C, while the end x = L is insulated so that no heat passes through it.
(a) Show that the fundamental solutions of the partial differential equation and boundary conditions are
sin[(2n 1)nx/2L], n = 1, 2, 3,....
(b) Find a formal series expansion for the temperature u (x, t),

that also satisfies the initial condition u(x, Q) = f (x).
Hint: Even though the fundamental solutions involve only the odd sines, it is still possible
590
Chapter 10. Partial Differential Equations and Fourier Series
to represent f by a Fourier series involving only these functions. See Problem 39 of Section 10.4.
> 16. In the bar of Problem 15 suppose that L 30, a2 = 1, and the initial temperature distri-
bution is f (x) 30 x for 0 < x < 30.
(a) Find the temperature u(x, t).
(b) Plot u versus x for several values of t. Also plot u versus t for several values of x.
(c) How does the location xm of the warmest point in the bar change as t increases? Draw a graph of xm versus t.
(d) Plot the maximum temperature in the bar versus t.
> 17. Suppose that the conditions are as in Problems 15 and 16 except that the boundary condition
at x = 0 is u (0, t) 40.
(a) Find the temperature u(x, t).
(b) Plot u versus x for several values of t. Also plot u versus t for several values of x.
(c) Compare the plots you obtained in this problem with those from Problem 16. Explain how the change in the boundary condition at x = 0 causes the observed differences in the behavior of the temperature in the bar.
18. Consider the problem
X" + kX 0, X'(0) 0, X'(L) 0. (i)
Let k 2, where v + i a with v and a real. Show that if 0, then the only solution of Eqs. (i) is the trivial solution X(x) = 0.
Hint: Use an argument similar to that in Problem 17 of Section 10.1.
19. The right end of a bar of length a with thermal conductivity Kj and cross-sectional area Aj is joined to the left end of a bar of thermal conductivity k2 and cross-sectional area A2. The composite bar has a total length L. Suppose that the end x = 0 is held at temperature zero, while the end x = L is held at temperature T. Find the steady-state temperature in the composite bar, assuming that the temperature and rate of heat flow are continuous at x = a.
Hint: See Eq. (2) of Appendix A.
20. Consider the problem
a2uxx ut, 0 < x < L, t > 0;
u(0, t) = 0, ux(L, t) + u(L, t) = 0, t > 0; (i)
u(x, 0) = f (x), 0 < x < L.
(a) Let u(x, t) = X(x)T(t) and show that
X" + kX = 0, X (0) = 0, X'(L) + Y X (L) = 0, (ii)
and
T' + ka2 T = 0,
where k is the separation constant.
(b) Assume that k is real, and show that problem (ii) has no nontrivial solutions if k < 0.
(c) If k > 0, let k with > 0. Show that proble^m (ii) has nontrivial solutions only if is a solution of the equation
cos -L + Y sin L 0. (iii)
(d) Rewrite Eq. (iii) as tan L /Y. Then, by drawing the graphs of y tan L and y L/yL for > 0 on the same set of axes, show that Eq. (iii) is satisfied by infinitely many positive values of ; denote these by 1,2,...,, ..., ordered in increasing size.
(e) Determine the set of fundamental solutions un (x, t) corresponding to the values found in part (d).
10.7 The Wave Equation: Vibrations of an Elastic String
591
10.7 The Wave Equation: Vibrations of an Elastic String
A second partial differential equation occurring frequently in applied mathematics is the wave7 equation. Some form of this equation, or a generalization of it, almost inevitably arises in any mathematical analysis ofphenomena involving the propagation of waves in a continuous medium. For example, the studies of acoustic waves, water waves, electromagnetic waves, and seismic waves are all based on this equation.
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