Chapter 8
Separation of Variables
Lecture 10
8.1
Types of Boundary Value Problems:
Dirichlet Boundary Conditions
1. Heat Equation: α2 = Thermal Conductivity.
• Heat Flow in a Bar
•Heat Flow on a Disk
2. Wave Equation: c = Wave Speed.
• Vibration of a String
3. Laplace’s Equation:
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Separation of Variables
Neuman Boundary Conditions: What do you expect the solution to
look like as t → ∞?
Mixed Boundary Conditions:
Ice
Heat Bath u(0, t) = A
Ice
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u(L, t) = B Heat Bath 2.
8.2. SEPARATION OF VARIABLES:
8.2
Separation of Variables:
Consider the heat conduction in an insulated rod whose endpoints are held
at zero degree for all time and within which the initial temperature is given
by f (x).
Fourier’s Guess:
u(x, t) = X(x)T (t)
(8.1)
2
2
ut = X(x)Ṫ (t) = α uxx = α X (x)T (t)
÷α2 XT :
Ṫ (t)
X (x)
= 2
= Constant = −α2 .
X(x)
α T (t)
(8.2)
−>
dT
= −α2 λ2 dt
T
ln |T | = −α2 λ2 t + c
2 2
T (t) = De−α λ t .
Ṫ (t) = −α2 λ2 T (t)
(8.3)
x>
X (x) + λ2 X(x) = 0
Guess X(x) = erx ⇒ (r2 + λ2 )erx = 0 r = ±λi
(8.4)
X = c1 eiλx + c2? e−iλx
= A sin λx + B cos λx.
(8.5)
Impose the boundary conditions:
0 = u(0, t) = X(0)T (t) = BT (t) ⇒ B = 0
0 = u(L, t) = X(L)T (t) = (A sin λL)T (t).
(8.6)
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Separation of Variables
Now we do not want the trivial solution so A = 0. Thus we look for values
of λ such that
sin λL = 0 ⇒ λ =
nπ L
n = 1, 2, . . . .
nπx 2 nπ 2
Thus un (x, t) = e−α ( L ) t sin
L
are all solutions of ut = α2 uxx .
(8.7)
n = 1, 2, . . .
(8.8)
Since (8.8) (above eq. number) is linear, a linear combination of solutions
is again a solution. Thus the most general solution is
u(x, t) =
∞
bn sin
n=1
nπx L
2
2 nπ
e−α ( L ) t.
(8.9)
What about the initial condition u(x, 0) = f (x).
u(x, 0) = f (x) =
∞
n=1
bn sin
nπx L
.
(8.10)
Givenf (x) we
need to find the bn such that the infinite series of functions
nπx agrees with f on [0, L].
bn sin
L
Question: f (x) may
+ 2L) = f (x) but the series is
be periodic f(x
nπnot
nπx (x + 2L) = sin
.
periodic since sin
L
L
Answer: In fact they do agree on [0, L] and are different elsewhere.
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