2. Consider the equation ut = = 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 + ½)πx/l]. (b) Write the series expansion for a solution u(x, t).
2. Consider the equation ut = = 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 + ½)πx/l]. (b) Write the series expansion for a solution u(x, t).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
[Second Order Equations] How do you solve 2?
![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%2F166c3c05-8f7c-4144-a957-9187cb099dfe%2Fdcpme7_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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