Page 645 - Advanced_Engineering_Mathematics o'neil

17.2 The Heat Equation on [0, L] 625

SECTION 17.2 PROBLEMS

In each of Problems 1 through 7, write a solution of the its left end maintained at zero temperature, while its
initial-boundary value problem. Graph the twentieth par- right end is perfectly insulated. The bar has an initial
2
tial sum of the series for u(x,t) on the same set of axes for temperature f (x) = x for 0 ≤ x ≤ 2. Determine the
different values of the time. temperature distribution u(x,t) and lim t→0 u(x,t).
11. Show that the partial differential equation
2
∂u ∂ u
1. = k for 0 < x < L,t > 0,
2
∂t ∂x 2 ∂u ∂ u ∂u
u(0,t)= u(L,t) = 0for t ≥ 0, ∂t = k ∂x 2 + A ∂x + Bu
u(x,0)= x(L − x) for 0 ≤ x ≤ L
can be transformed into a standard heat equation for v
2
∂u ∂ u by letting u(x,t) = e αx+βt v(x,t) and choosing α and
2. = 4 for 0 < x < L,t > 0,
∂t ∂x 2 β appropriately.
u(0,t)= u(L,t) = 0for t ≥ 0, 12. Use the idea of Problem 11 to solve
2
u(x,0)= x (L − x) for 0 ≤ x ≤ L
2
∂u ∂ u ∂u
2
∂u ∂ u = + 4 + 2u for 0 < x <π,t > 0,
3. = 3 for 0 < x < L,t > 0, ∂t ∂x 2 ∂x
∂t ∂x 2
u(0,t) = u(π,t) = 0for t ≥ 0,
u(0,t)= u(L,t) = 0for t ≥ 0,
u(x,0)= L(1 − cos(2πx/L)) for 0 ≤ x ≤ L u(x,0) = x(π − x) for 0 ≤ x ≤ π.
2
∂u ∂ u 13. Solve
4. = for 0 < x <π,t > 0,
∂t ∂x 2 2
∂u ∂u ∂u = ∂ u + 6 ∂u for 0 < x < 4,t > 0,
(0,t)= (π,t) = 0for t ≥ 0, 2
∂x ∂x ∂t ∂x ∂x
u(x,0)= sin(x) for 0 ≤ x ≤ π u(0,t) = u(4,t) = 0for t ≥ 0,
2
∂u ∂ u u(x,0) = 1for0 ≤ x ≤ 4.
5. = 4 for 0 < x < 2π,t > 0,
∂t ∂x 2
∂u ∂u Graph the twentieth partial sum of the solution for
(0,t)= (2π,t) = 0for t ≥ 0, selected times.
∂x ∂x
u(x,0)= x(2π − x) for 0 ≤ x ≤ 2π 14. Solve
2
2
∂u ∂ u ∂u ∂ u ∂u
6. = 4 for 0 < x < 3,t > 0, = − 6 for 0 < x <π,t > 0,
∂t ∂x 2 ∂t ∂x 2 ∂x
∂u ∂u u(0,t) = u(π,t) = 0for t ≥ 0,
(0,t)= (3,t) = 0for t ≥ 0,
∂x ∂x
2
u(x,0)= x for 0 ≤ x ≤ 3 u(x,0) = x(π − x) for 0 ≤ x ≤ π.
2
∂u ∂ u Graph the twentieth partial sum of the solution for
7. = 2 for 0 < x < 6,t > 0, selected times.
∂t ∂x 2
∂u ∂u 15. Solve
(0,t)= (6,t) = 0for t ≥ 0,
∂x ∂x ∂u ∂ u
2
u(x,0)= e −x for 0 ≤ x ≤ 6 = 16 for 0 < x < 1,t > 0,
∂t ∂x 2
8. A thin, homogeneous bar of length L has insulated u(0,t) = 2,u(1,t) = 5for t ≥ 0,
ends and initial temperature B, a positive constant. 2
u(x,0) = x for 0 ≤ x ≤ 1.
Find the temperature distribution in the bar.
9. A thin homogeneous bar of length L has initial tem- 16. Solve
perature f (x) = B where the right end x = L is insu- ∂u ∂ u
2
lated, while the left end is kept at zero temperature. = k for 0 < x < L,t > 0,
∂t ∂x 2
Find the temperature distribution in the bar.
u(0,t) = T,u(L,t) = 0for t ≥ 0,
10. A thin, homogeneous bar having thermal diffusivity
of 9 and a length of 2 cm has insulated sides and u(x,0) = x(L − x) for 0 ≤ x ≤ L.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
October 14, 2010 15:25 THM/NEIL Page-625 27410_17_ch17_p611-640