Recall that the Laplace operator takes the following form in polar coordinates: 1 0²u 10 rər (rou). ² 20² √²u = Uxx + Uyy where, the coordinates r, are given as follows: + x = r cos 0, = r sin 0, 0 = tan-¹(²), | r = √√√x² + y². We found by the method of separation of variables that any product of the form
Recall that the Laplace operator takes the following form in polar coordinates: 1 0²u 10 rər (rou). ² 20² √²u = Uxx + Uyy where, the coordinates r, are given as follows: + x = r cos 0, = r sin 0, 0 = tan-¹(²), | r = √√√x² + y². We found by the method of separation of variables that any product of the form
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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![1. Recall that the Laplace operator takes the following form in polar coordinates:
1 ə
rər ər
² u = Uxx + Uyy
where, the coordinates r, are given as follows:
V²u =
x = r cos 0,
y = r sin 0,
=
=
n=1
r =
We found by the method of separation of variables that any product of the form
Ao(Co+Do lnr)
(An cos(n) + Bn sin(nº))(Cnrª + Dnr¯n), n = 1, 2, 3, ...
satisfies the equation (not necessarily the boundary conditions however), as well as the periodicity in the
angle 0., ie u(r, 0 + 2π) = u(r, 0).
u(1,0)
tan-¹(2/2),
√x² + y².
Recall, further, that in order to have a solution that does not go to infinity near the origin, we set Dk = 0
for k = 0, 1, 2, ... in the expressions above. Finally, we obtained the following infinite sum as the solution,
whose coefficients ao,
an,
andbn will be determined by the boundary condition.
1 0²u
r² 20²
u(r, 0): = ao + [an cos(nº) + bn sin(nº)]rn
=
a. Find the temperature distribution in an insulated metals disc (say, a coin) of radius 1, whose circular
edge is kept at T = 1 at the lower semi-circle, at Tu 0 at the upper semi-circle. Namely, in polar
coordinates,
=
{
1 θε[π,2π].
0 0 = [0, π]
b. Find the temperature ar the following point on the disc: (x,y) = (0,0), and (0, ½1).
c. (*) Consider an annulus of inner radius r₁ =
to u(2,0) = 1 and u(1, 0) be as in part (a.).
1 and outer radius r₂ 2. Let the temperature be set
Find the solution to the Laplace equation in this case.
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91f76606-a4d9-42f0-be0d-7b1366a5593f%2F512f60be-578e-46bd-bcd2-888e93148880%2Fjm3xa8l_processed.png&w=3840&q=75)
Transcribed Image Text:1. Recall that the Laplace operator takes the following form in polar coordinates:
1 ə
rər ər
² u = Uxx + Uyy
where, the coordinates r, are given as follows:
V²u =
x = r cos 0,
y = r sin 0,
=
=
n=1
r =
We found by the method of separation of variables that any product of the form
Ao(Co+Do lnr)
(An cos(n) + Bn sin(nº))(Cnrª + Dnr¯n), n = 1, 2, 3, ...
satisfies the equation (not necessarily the boundary conditions however), as well as the periodicity in the
angle 0., ie u(r, 0 + 2π) = u(r, 0).
u(1,0)
tan-¹(2/2),
√x² + y².
Recall, further, that in order to have a solution that does not go to infinity near the origin, we set Dk = 0
for k = 0, 1, 2, ... in the expressions above. Finally, we obtained the following infinite sum as the solution,
whose coefficients ao,
an,
andbn will be determined by the boundary condition.
1 0²u
r² 20²
u(r, 0): = ao + [an cos(nº) + bn sin(nº)]rn
=
a. Find the temperature distribution in an insulated metals disc (say, a coin) of radius 1, whose circular
edge is kept at T = 1 at the lower semi-circle, at Tu 0 at the upper semi-circle. Namely, in polar
coordinates,
=
{
1 θε[π,2π].
0 0 = [0, π]
b. Find the temperature ar the following point on the disc: (x,y) = (0,0), and (0, ½1).
c. (*) Consider an annulus of inner radius r₁ =
to u(2,0) = 1 and u(1, 0) be as in part (a.).
1 and outer radius r₂ 2. Let the temperature be set
Find the solution to the Laplace equation in this case.
-
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