[39] Consider the heat conduction in a circular cylinder: x² + y² < a²,0 < z
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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39
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![[39] Consider the heat conduction in a circular cylinder: x² + y² < a²,0 < z < b. Assume
that the following parts of the boundary are insulated for all times:
the bottom face: x² + y² <a², z =
the lateral boundary surface: x² + y² = a²,0 <z<b.
0; and
Also assume that the boundary temperature on the top face is a prescribed radially
symmetric function f(r), where r = (x² + y²) ¹/2.
8
(a) Write down the boundary value problem in cylindrical coordinates, for the steady-
state temperature distribution.
(b) Find all product solutions u = R(r) Z(z) of the PDE and the homogeneous boundary
conditions you gave in (a).
(c) Find the solution formula for the boundary value problem you gave in (a).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a4dcd82-baf4-45bc-b40b-693a3e683492%2F3cf1e56d-4feb-49f1-913e-4b372fc6cf43%2F0hh1hlv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[39] Consider the heat conduction in a circular cylinder: x² + y² < a²,0 < z < b. Assume
that the following parts of the boundary are insulated for all times:
the bottom face: x² + y² <a², z =
the lateral boundary surface: x² + y² = a²,0 <z<b.
0; and
Also assume that the boundary temperature on the top face is a prescribed radially
symmetric function f(r), where r = (x² + y²) ¹/2.
8
(a) Write down the boundary value problem in cylindrical coordinates, for the steady-
state temperature distribution.
(b) Find all product solutions u = R(r) Z(z) of the PDE and the homogeneous boundary
conditions you gave in (a).
(c) Find the solution formula for the boundary value problem you gave in (a).
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