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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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0 < ‚ < 2n and have period 2n. As a consequence, the full Fourier series is required, rather than sine or cosine terms alone.
PROBLEMS
> 1. (a) Find the solution u(x, y) of Laplaceís equation in the rectangle 0 < x < a,0 < y < b,
also satisfying the boundary conditions
u(0, y) = 0, u(a, y) = 0, 0 < y < b,
u(x, 0) = 0, u(x, b) = g(x), 0 < x < a.
(b) Find the solution if
, , fx, 0 < x < a/2,
g(x) =1 nZZ
ya ó x, a/2 < x < a.
(c) For a = 3 and b = 1 plot u versus x for several values of y and also plot u versus y for several values of x .
(d) Plot u versus both x and y in three dimensions. Also draw a contour plot showing
several level curves of u(x, y) in the xy-plane.
2. Find the solution u(x, y) of Laplaceís equation in the rectangle 0 < x < a, 0 < y < b, also satisfying the boundary conditions
u(0, y) = 0, u(a, y) = 0, 0 < y < b,
u(x, 0) = h(x), u(x, b) = 0, 0 < x < a.
> 3. (a) Find the solution u(x, y) of Laplaceís equation in the rectangle 0 < x < a,0 < y < b,
also satisfying the boundary conditions
u(0, y) = 0, u(a, y) = f (y), 0 < y < b,
u(x, 0) = h(x), u(x, b) = 0, 0 < x < a.
Hint: Consider the possibility of adding the solutions of two problems, one with homogeneous boundary conditions except for u(a, y) = f (y), and the other with homogeneous boundary conditions except for u(x, 0) = h(x).
(b) Find the solution if h(x) = (x/a)2 and f (y) = 1 ó (y/b).
(c) Let a = 2 and b = 2. Plot the solution in several ways: u versus x, u versus y, u versus both x and y, and a contour plot.
612
Chapter 10. Partial Differential Equations and Fourier Series
4. Show how to find the solution u(x, y) of Laplaceís equation in the rectangle 0 < x < a, 0 < y < b, also satisfying the boundary conditions
u(0, y) = k(y), u(a, y) = f (y), 0 < y < b,
u(x, 0) = h(x), u(x, b) = g(x), 0 < x < a.
Hint: See Problem 3.
5. Find the solution u(r, ‚) of Laplaceís equation
urr + (1/r )ur + (1/r 2)u ‚‚ = 0 outside the circle r = a, also satisfying the boundary condition
u(a^) = f (‚), 0 < ‚ < 2Ô,
on the circle. Assume that u(r,ff) is single-valued and bounded for r > a.
> 6. (a) Find the solution u(r, ‚) of Laplaceís equation in the semicircular region r < a,
0 < ‚ < Ô, also satisfying the boundary conditions
u(r, 0) = 0, u(r,n) = 0, 0 < r < a,
u(a^) = f (‚), 0 < ‚ < Ô.
Assume that u is single-valued and bounded in the given region.
(b) Find the solution if f (‚) = ‚(Ô ó ‚).
(c) Let a = 2 and plot the solution in several ways: u versus r, u versus ‚, u versus both r and ‚ , and a contour plot.
7. Find the solution u(r, ‚) ofLaplaceís equation in the circular sector0 < r < a,0 < ‚ < a, also satisfying the boundary conditions
u(r, 0) = 0, u(r,a) = 0, 0 < r < a,
u(a, ‚) = f (‚), 0 < ‚ < a.
Assume that u is single-valued and bounded in the sector.
> 8. (a) Find the solution u(x, y) ofLaplaceís equation in the semi-infinite strip 0 < x < a,
y > 0, also satisfying the boundary conditions
u(0, y) = 0, u(a, y) = 0, y > 0,
u(x, 0) = f (x), 0 < x < a
and the additional condition that u(x, y) ^ 0as y ^ ś.
(b) Find the solution if f (x) = x(a ó x).
(c) Let a = 5. Find the smallest value of y0 for which u(x, y) < 0.1 for all y > y0.
9. Show that Eq. (23) has periodic solutions only if X is real.
Hint: Let X = ó/Ž2 where š = v + ia with v and a real.
10. Consider the problem of finding a solution u(x, y) ofLaplaceís equation in the rectangle 0 < x < a, 0 < y < b, also satisfying the boundary conditions
ux (0, ů) = 0, ux (a, y) = f (y), 0 < y < b,
uy (x, 0) = 0, uy (x, b) = 0, 0 < x < a.
This is an example of a Neumann problem.
(a) Show that Laplaceís equation and the homogeneous boundary conditions determine the fundamental set of solutions
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