Consider the partial differential equation for heat in a one-dimensional rod with temperature u(x, du g²u Ət 8x² Assume initial condition: u(x,0) = 15 -1.25x = D.
Consider the partial differential equation for heat in a one-dimensional rod with temperature u(x, du g²u Ət 8x² Assume initial condition: u(x,0) = 15 -1.25x = D.
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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![Consider the partial differential equation for heat in a one-dimensional rod with temperature u(x, t):
du
8²u
Ət
8x²
Assume initial condition:
u(x,0) = 15 -1.25.x
and boundary conditions:
(a) Determine the steady state temperature distribution:
u(x) =
bo
(b) Find the full time dependent solution using Fourier series with trigonometric functions chosen appropriately for the boundary conditions. Simplify your
answers.
u(x, t) = A + box + Σ1 ane
cos(T) + 1 b₂e16 sin(²)
'n=]
Ao
||
an =
bn
-
= D
D₂²²
16
u(0,t) = 15 u(4, t) = 6](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3306ea0d-0f79-489f-b4ee-38af79ef2243%2F3f49d911-c011-43b1-a419-ee4879df0253%2F65dsu1c_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the partial differential equation for heat in a one-dimensional rod with temperature u(x, t):
du
8²u
Ət
8x²
Assume initial condition:
u(x,0) = 15 -1.25.x
and boundary conditions:
(a) Determine the steady state temperature distribution:
u(x) =
bo
(b) Find the full time dependent solution using Fourier series with trigonometric functions chosen appropriately for the boundary conditions. Simplify your
answers.
u(x, t) = A + box + Σ1 ane
cos(T) + 1 b₂e16 sin(²)
'n=]
Ao
||
an =
bn
-
= D
D₂²²
16
u(0,t) = 15 u(4, t) = 6
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