a? dx² (2) ду? To separate variables in (1) and (2), we assume a product solution of the form u(x, y, t) = X(x)Y(y)T(t). We note that a²u = X"YT, dx² a²u = XY"T, ду? ди ΧΥΤ'. at and As we see next, with appropriate boundary conditions, boundary-value problems involving (1) and (2) lead to the concept of Fourier series in two variables. EXAMPLE 1 Temperatures in a Plate Find the temperature u(x, y, t) in the plate shown in Figure 12.8.1(a) if the initial temperature is f(x, y) throughout and if the boundaries are held at temperature zero for time t> 0. late SOLUTION We must solve (a²u a²u ди k + 0 0 ду? at' subject to и(0, у, 1) %3D 0, и(b, у, t) %3D 0, 0 <у<с, 1>0 и (х, 0, г) %3D 0, и(х, с, 1) — 0, 0 0 u(x, y, 0) = f(x, y), 0
a? dx² (2) ду? To separate variables in (1) and (2), we assume a product solution of the form u(x, y, t) = X(x)Y(y)T(t). We note that a²u = X"YT, dx² a²u = XY"T, ду? ди ΧΥΤ'. at and As we see next, with appropriate boundary conditions, boundary-value problems involving (1) and (2) lead to the concept of Fourier series in two variables. EXAMPLE 1 Temperatures in a Plate Find the temperature u(x, y, t) in the plate shown in Figure 12.8.1(a) if the initial temperature is f(x, y) throughout and if the boundaries are held at temperature zero for time t> 0. late SOLUTION We must solve (a²u a²u ди k + 0 0 ду? at' subject to и(0, у, 1) %3D 0, и(b, у, t) %3D 0, 0 <у<с, 1>0 и (х, 0, г) %3D 0, и(х, с, 1) — 0, 0 0 u(x, y, 0) = f(x, y), 0
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
I don't understand when you use separation of variables you get X"/X=-Y"/Y+T'/KT. Can you please explain it to me. Thank you
![11:47 PM Sun May 16
* 83%
Done
DIFFERENTIAL EQUATIONS with Boundary-Value Problemsa Zill Cullen
(b, с)
are small, free, and undamped, u(x, y, t) satisfies the two-dimensional wave
equation
2²u
3²u
(2)
dx²
ду?
To separate variables in (1) and (2), we assume a product solution of the form
u(x, y, t) = X(x)Y(y)T(t). We note that
(a)
ди
- ΧΥΤ'.
dt
X"YT,
:XY"T,
ду?
and
dx?
As we see next, with appropriate boundary conditions, boundary-value problems
involving (1) and (2) lead to the concept of Fourier series in two variables.
EXAMPLE 1
Temperatures in a Plate
Find the temperature u(x, y, t) in the plate shown in Figure 12.8.1(a) if the initial
temperature is f (x, y) throughout and if the boundaries are held at temperature zero
for time t> 0.
(b)
12.8.1 (a) Rectangular plate
tangular membrane
SOLUTION We must solve
ди
0 < x < b, 0 < y< c, t>0
at'
k
dy²,
subject to
и(0, у, 1) %3D 0, и(b, у, ) %3D 0, 0 <у<с, 1>0
u(x, 0, t) = 0, u(x, c, t) = 0, 0 < x < b, t> 0
u(x, y, 0) = f(x, y), 0<x<b, 0<y< c.
Substituting u(x, y, t) = X(x)Y(y)T(t), we get
T'
Y"
+
X"
k (X'YT + ΧY'T) XYT'
(3)
or
X
Y
kT
Since the left-hand side of the last equation in (3) depends only on x and the right
side depends only on y and t, we must have both sides equal to a constant –A:
X"
Y"
T'
Y
kT
and so
X" + λX-
(4)
Y"
T'
+ A.
kT
(5)
Y
By the same reasoning, if we introduce another separation constant -u in (5), then
Y"
T'
and
||
Y
kT
yield
Y" + µY = 0
and
T' + k(^ + µ)T = 0.
(6)
Now the homogeneous boundary conditions
и(0, у, 1) %3D 0, и(b, у, ) %3D 01
и(х, 0, г) %3D 0, и (х, с, 1) %3D 0]
[X(0) = 0, X(b) = 0
imply that
Y(0) = 0, Y(c) = 0.
%3D
Thus we have two Sturm-Liouville problems:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca708747-373f-4dbe-b127-10492ca0e68f%2F2382a840-febd-45b1-84d0-ffa86f345d76%2Fw9lczb_processed.png&w=3840&q=75)
Transcribed Image Text:11:47 PM Sun May 16
* 83%
Done
DIFFERENTIAL EQUATIONS with Boundary-Value Problemsa Zill Cullen
(b, с)
are small, free, and undamped, u(x, y, t) satisfies the two-dimensional wave
equation
2²u
3²u
(2)
dx²
ду?
To separate variables in (1) and (2), we assume a product solution of the form
u(x, y, t) = X(x)Y(y)T(t). We note that
(a)
ди
- ΧΥΤ'.
dt
X"YT,
:XY"T,
ду?
and
dx?
As we see next, with appropriate boundary conditions, boundary-value problems
involving (1) and (2) lead to the concept of Fourier series in two variables.
EXAMPLE 1
Temperatures in a Plate
Find the temperature u(x, y, t) in the plate shown in Figure 12.8.1(a) if the initial
temperature is f (x, y) throughout and if the boundaries are held at temperature zero
for time t> 0.
(b)
12.8.1 (a) Rectangular plate
tangular membrane
SOLUTION We must solve
ди
0 < x < b, 0 < y< c, t>0
at'
k
dy²,
subject to
и(0, у, 1) %3D 0, и(b, у, ) %3D 0, 0 <у<с, 1>0
u(x, 0, t) = 0, u(x, c, t) = 0, 0 < x < b, t> 0
u(x, y, 0) = f(x, y), 0<x<b, 0<y< c.
Substituting u(x, y, t) = X(x)Y(y)T(t), we get
T'
Y"
+
X"
k (X'YT + ΧY'T) XYT'
(3)
or
X
Y
kT
Since the left-hand side of the last equation in (3) depends only on x and the right
side depends only on y and t, we must have both sides equal to a constant –A:
X"
Y"
T'
Y
kT
and so
X" + λX-
(4)
Y"
T'
+ A.
kT
(5)
Y
By the same reasoning, if we introduce another separation constant -u in (5), then
Y"
T'
and
||
Y
kT
yield
Y" + µY = 0
and
T' + k(^ + µ)T = 0.
(6)
Now the homogeneous boundary conditions
и(0, у, 1) %3D 0, и(b, у, ) %3D 01
и(х, 0, г) %3D 0, и (х, с, 1) %3D 0]
[X(0) = 0, X(b) = 0
imply that
Y(0) = 0, Y(c) = 0.
%3D
Thus we have two Sturm-Liouville problems:
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