4. Consider diffusion inside an enclosed circular tube. Let its length (circum- ference) be 21. Let x denote the arc length parameter where −1≤ x ≤l. Then the concentration of the diffusing substance satisfies ut = kuxx for − 1 ≤ x ≤ 1 u(−l, t) = u(l, t) and ux(-l, t) = ux(l, t). These are called periodic boundary conditions. (a) Show that the eigenvalues are λ = (nл/1)² for n = 0, 1, 2, 3, . . . . > (b) Show that the concentration is u(x, t) = 1 2 +Σ(An n=1 Ao + COS nπX 1 + Bn sin nπX 17/ ²x² ) e-n²x ²k1/1² =
4. Consider diffusion inside an enclosed circular tube. Let its length (circum- ference) be 21. Let x denote the arc length parameter where −1≤ x ≤l. Then the concentration of the diffusing substance satisfies ut = kuxx for − 1 ≤ x ≤ 1 u(−l, t) = u(l, t) and ux(-l, t) = ux(l, t). These are called periodic boundary conditions. (a) Show that the eigenvalues are λ = (nл/1)² for n = 0, 1, 2, 3, . . . . > (b) Show that the concentration is u(x, t) = 1 2 +Σ(An n=1 Ao + COS nπX 1 + Bn sin nπX 17/ ²x² ) e-n²x ²k1/1² =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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[Second Order Equations] How do you solve 4?
![1. Solve the diffusion problem ut = kuxx in 0 < x < 1, with the mixed
boundary conditions u(0, t) = ux(l, t) = 0.
2. Consider the equation utt = c²uxx for 0 < x < 1, with the boundary con-
ditions ux(0, t) = 0, u(l, t) = 0 (Neumann at the left, Dirichlet at the
right).
(a) Show that the eigenfunctions are cos[(n + 1)Ãx/l].
(b) Write the series expansion for a solution u(x, t).
4.
3. Solve the Schrödinger equation u₁ = ikuxx for real k in the interval
0 < x < / with the boundary conditions ux(0, t) = 0, u(l, t) = 0.
Consider diffusion inside an enclosed circular tube. Let its length (circum-
ference) be 21. Let x denote the arc length parameter where -1 ≤ x ≤l.
Then the concentration of the diffusing substance satisfies
ut = kuxx for − 1 ≤ x ≤ 1
ux(−l,
u(-l, t) = u(l, t) and
These are called periodic boundary conditions.
(a) Show that the eigenvalues are λ = (në/1)² for
Show that the concentration is
(b)
ux(-1, t) = ux(l, t).
u(x, t) = 2²
= =4o + Σ (Ancos
1
n=1
n =
: (ní /1)² for n = 0, 1, 2, 3, . . . .
nπX
1
+ B₁ sin ”7-²) e-n²x²k¹/1²¸](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08fe1bb2-2ec5-4dff-a549-31bfc208ed8e%2Ff7d847b6-0c9f-437e-9334-b1f25cc9231e%2Fluqu13i_processed.png&w=3840&q=75)
Transcribed Image Text:1. Solve the diffusion problem ut = kuxx in 0 < x < 1, with the mixed
boundary conditions u(0, t) = ux(l, t) = 0.
2. Consider the equation utt = c²uxx for 0 < x < 1, with the boundary con-
ditions ux(0, t) = 0, u(l, t) = 0 (Neumann at the left, Dirichlet at the
right).
(a) Show that the eigenfunctions are cos[(n + 1)Ãx/l].
(b) Write the series expansion for a solution u(x, t).
4.
3. Solve the Schrödinger equation u₁ = ikuxx for real k in the interval
0 < x < / with the boundary conditions ux(0, t) = 0, u(l, t) = 0.
Consider diffusion inside an enclosed circular tube. Let its length (circum-
ference) be 21. Let x denote the arc length parameter where -1 ≤ x ≤l.
Then the concentration of the diffusing substance satisfies
ut = kuxx for − 1 ≤ x ≤ 1
ux(−l,
u(-l, t) = u(l, t) and
These are called periodic boundary conditions.
(a) Show that the eigenvalues are λ = (në/1)² for
Show that the concentration is
(b)
ux(-1, t) = ux(l, t).
u(x, t) = 2²
= =4o + Σ (Ancos
1
n=1
n =
: (ní /1)² for n = 0, 1, 2, 3, . . . .
nπX
1
+ B₁ sin ”7-²) e-n²x²k¹/1²¸
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