Applications of Fourier
Transforms in Solving
Differential Equations
MIDN 1/C Allison Tyler
United States Naval Academy
18 April 2007
History
Daniel Bernoulli
Known for the
Bernoulli Principle
Solved the wave
equation in 1720’s
using techniques
Fourier later would
employ
Worked with Euler in
St. Petersburg
History
Jean le Rond d'Alembert
French mathematician
Worked with Euler and
Bernoulli on the wave
equation
“Found” the CauchyRiemann equations
decades before Cauchy
or Riemann did
History
Joseph Fourier
French mathematician
Served in Egypt while
in Napoleon’s Army
Solved the heat
equation and won
award for it in 1811,
published work in
1822 in Théorie
analytique de la
chaleur
History
Erwin Schrödinger
Vienna-born physicist
Artilleryman in WWI
Best known for “Schrödinger’s
Cat” and the Schrödinger
Equation
Won the Nobel Prize in 1933
Worked with Einstein
Fled Germany and Austria
several times after “insulting”
the Nazis
Concepts
The following concepts were used
Fourier Coefficients
−2πinx
1 P
Cn = ∫ f ( x)e P dx
P 0
Fourier Series
∞
FS ( f )( x) =
∑C e
2πinx
P
n
n = −∞
Superposition Principle
The Superposition Principle states that for a linear system,
the sum of solutions for that system is also a solution.
Concepts
Orthogonality of Sines
Given
 nπx 
f ( x) = sin 
; n ≥ 1 ,
 L 
2 L  mπx   nπx 
sin 
 sin 
dx = 0
∫
0
L
 L   L 
for
m≠n
Concepts
Proof of Orthogonality
L
  (m − n )πx 
 (m + n )πx  
 sin 
 sin 

L
2
2 
L
L
 mπx   nπx 
− 

dx
=
sin
sin




2(m + n )πx 
L ∫0
L  2(m − n )πx
 L   L 


L
L

0
=
2  sin ((m − n )π ) sin ((m + n )π )  2

 − (sin (0 ) − sin (0))
−
L  2(m − n )π
2(m + n )π  L
2
2
(
)
= 0 − 0 − (0 − 0 )
L
L
=0
Wave Equation
People studied the
wave equation
because they were
intrigued by string
instruments.
Bernoulli defined
nodes and
frequencies of
oscillation
Derivation of Wave Equation
To solve the wave equation, we assume that it is a
separable equation:
u(x,t)=X(x)T(t)
The Wave Equation
2
∂ 2u
2 ∂ u
=ω
2
∂t
∂x 2
Boundary conditions : u (0, t ) = 0
Initial Conditions :
u ( x,0) = f ( x)
u ( L, t ) = 0
∂u
( x,0) = g ( x)
dt
Derivation
∂u
= XT'
∂t
∂ 2u
= XT ' '
2
∂t
∂u
∂ 2u
= X 'T
= X ' 'T
2
∂x
∂x
2
∂
u
∂ 2u
2
⇒
=
⇒ T ' ' X = ω 2 X ' 'T
ω
2
2
∂t
∂x
By Separation of Variables we get :
T ''
X ''
=
= −λ
because the only way the two sides
2
X
ωT
will be equal is if they equal a constant
Solve for X as in the heat equation and you get
 nπ 
X = ∑ C n sin 
x
 L 
n =1
∞
Solve for T
T ' ' = −ω 2 λT ⇒ T ' '+ω 2 λT = 0
 ωn π 
( D 2 + ω 2 λ )T = 0 ⇒ D = ±
i = ±γ n n ∈ Ζ
 L 
Therefore : T = D1 cos(γ nt ) + D2 sin(γ nt )
Thus we get
u ( x, t ) = X ( x)T (t )
 nπ 
u ( x, t ) = ∑ Cn sin 
x  (Dn cos(γ nt ) + En sin(γ nt ) ) by Superposition Principle
L


n =1
∞
 nπ 
u ( x, t ) = ∑ sin 
x  (Dn cos(γ nt ) + En sin(γ nt ) )
 L 
n =1
∞
Using the Initial Conditions : u ( x,0) = f ( x )
∞
 nπ 
 nπ 
u ( x,0) = ∑ sin
x  (Dn cos(γ n 0) + En sin(γ n 0) ) = ∑ Dn sin
x  = f ( x)
 L 
 L 
n =1
n =1
2 L
where Dn = ∫ f ( x) sin( λn x)dx
L 0
∂u
( x,0) = g ( x)
∂t
∞
∂u
 nπ 
( x, t ) = ∑ γ n sin 
x  (− Dn sin(γ n t ) + En cos(γ nt ) )
∂t
 L 
n =1
∞
∞
∂u
 nπ 
 nπ 
( x,0) = ∑ γ n sin 
x  (− Dn sin(γ n 0) + En cos(γ n 0) ) = ∑ En sin 
x  = g ( x)
∂t
 L 
 L 
n =1
n =1
∞
where En =
2 L
g ( x) sin( λn x)dx
γ n L ∫0
Final Solution
 nπ 
u ( x, t ) = ∑ sin 
x  (Dn cos(γ nt ) + En sin(γ nt ) )
 L 
n =1
2 L
2 L
where Dn = ∫ f ( x) sin( λn x )dx and En =
g ( x) sin( λn x)dx
∫
0
0
L
γ nL
∞
Unfiltered Wave Equation
Filtered Wave Equation
Heat Equation
The reason we solve the heat equation is
to find out what happens to a length of
metal/plastic when heat is applied (the
temperature distribution over time)
Derivation of Heat Equation
The assumption made by Bernoulli and adopted
by Fourier was that the equation u(x,t) was
separable:
u(x,t)=X(x)T(t)
The Heat Equation
∂u
∂ 2u
=k 2
∂t
∂x
Boundary Conditions : u (0, t ) = 0
Initial Conditions :
u ( x ,0 ) = f ( x )
u ( L, t ) = 0
Derivation
∂u
= XT '
∂t
∂u
= X 'T
∂x
∂u
∂ 2u
=k 2
∂t
∂x
∂ 2u
= X ' 'T
2
∂x
⇒ XT ' = kX ' ' T
The only way this can occur is if both sides equal a constant
T' X ' '
=
= −λ
kT
X
Now Solve for X
X ' ' = −λX ⇒ X ' '+λX = 0
(D2 + λ ) X = 0 ⇒ D = ± λi
⇒ X = C1 cos( λ x) + C2 sin( λ x)
Using the boundary conditions we get :
X (0) = C1 = 0 because cos(0) = 1 and sin(0) = 0
2
 nπ 
X ( L) = C2 sin λ L = 0 ⇒ sin λ L = 0 when λ L = nπ ⇒ λn = 
 n∈Ζ
L


∞
 nπ 
Therefore : X = ∑ An sin 
x  by the Superposition Principle
L


n =1
Now Solve for T
T ' = −λkT ⇒ T '+λkT = 0
( D + λk )T = 0 ⇒ D = −λk
⇒ T = Be −λkt
Thus, we get : u ( x, t ) = X ( x)T (t )
u ( x, t ) = Be
− λkt
∞
∞
n =1
n =1
(
)
− λn kt
sin(
λ
)
sin
λ
A
x
=
A
x
e
∑ n
∑ n
n
n
Using the initial condition : u ( x,0) = f ( x)
∞
u ( x,0) = ∑ An sin( λn x) = f ( x)
n =1
1 L
⇒ An = ∫ f ( x) sin( λn x)dx
L 0
Final Solution
∞
u ( x, t ) = ∑ An sin( λn x)e
n =1
− λn kt
where
1 L
An = ∫ f ( x) sin( λn x)dx
L 0
Unfiltered Heat Equation
Filtered Heat Equation
Euler and d’Alembert
They too solved the wave equation, but by
using “arbitrary functions representing two
waves, one moving along the string to the
right, the other to the left, with a velocity
equal to the constant c.”
Bernoulli’s solution was a summation of
sine waves
No one could tell if the two solutions could
be reconciled
Until Fourier Came Along
“Fourier showed that almost any function,
when regarded as a periodic function over
a given interval, can be represented by a
trigonometric series of the form
f(x)=ao+a1cosx+a2cos2x+a3cos3x… +
b1sinx+b2sin2x+b3sin3x…
where the coefficients ai and bi can be
found from f(x) by computing certain
integrals.”
Modern Application:
Schrödinger
The Schrödinger Equation was published
in a series of papers on wave mechanics
in 1926
The solutions to the Schrödinger Equation
are wave functions that express the
probability that a particle (specifically the
electron of a hydrogen atom) will be in a
certain location at any observation
Solution to Schrödinger
As with the heat and wave equations, we assume that the
Schrödinger Equation is separable:
u(x,t)=X(x)T(t)
The Schrödinger Equation
∂ψ
ih ∂ 2ψ
=
∂t
2m ∂t 2
Boundary Conditions : ψ (0, t ) = 0
Initial Conditions :
h = 1.054x10 -27 erg − s
ψ ( x,0) = f ( x)
ψ ( L, t ) = 0
Solution
Assume Ψ(x,t) is separable
ψ ( x, t ) = X ( x)T (t )
∂ψ
= XT '
∂t
∂ 2ψ
ih
=
X ''
2
∂x
2m
∂ψ
∂ 2ψ
= is 2
∂t
∂x
XT ' = isX ' ' T
Let s =
h
2m
By separation of variables we get :
T ' X ''
=
= −c
isT
X
Similar to the Heat and Wave Equations
Solve for T:
T'
To solve for T :
= isc
T
ln(T ) = −isct + d
T = e −isct + d = e −isct e d
T = be −isct
Solve for X:
X ''
= −c
X
X ' ' = −cX ⇒ X ' '+cX = 0
To solve for X :
( D 2 + c ) X = 0 ⇒ D = ± ci
X = a1 cos( c x) + a2 sin( c x)
Using the boundary conditions we get the same results as before:
 nπ 
Where a1 = 0 and cn = 
X = a2 sin c L

L


Therefore we get ψ ( x, t ) = X ( x)T (t )
( )
ψ ( x, t ) = a2 sin c L be −isct
By the superposition principle
∞
 nπ 
x e
 L 
ψ ( x, t ) = ∑ bn sin 
n =1
2
 nπ 
−is 
 t
 L 
Using the initial conditions:
∞
 nπ 
x  = f ( x)
L


 nπ 
f ( x) sin
x dx
 L 
ψ ( x,0) = ∑ bn sin
n =1
⇒ bn =
2 L
L ∫0
2
n∈Z
Final Solution
 nπ
ψ ( x, t ) = ∑ bn sin 
 L
n =1
∞

x e

2
 nπ 
−is 
 t
 L 
2 L
 nπ
Where bn = ∫ f ( x) sin 
L 0
 L

x dx

SAGE
The graphs were produced using the open
source program SAGE by Prof. W. David
Joyner, USNA
http://modular.math.washington.edu/sage
References
Debnath, Lokenath. Integral Transforms and Their Applications. New York: CRC
Press, 1995.
Farlow, Stanley J. Partial Differential Equations for Scientists and Engineers. New
York: Dover Publications, Inc., 1993.
O’Connor, J.J. and E.F. Robertson. “Daniel Bernoulli.” University of St. Andrews
School of Mathematics and Statistics (accessed 10 February 2007).
---. “Erwin Rudolf Josef Alexander Schrödinger.” University of St. Andrews School of
Mathematics and Statistics (accessed 12 March 2007).
---. “Jean le Rond d’Alembert.” University of St. Andrews School of Mathematics and
Statistics (accessed 11 March 2007).
---. “Joseph Fourier.” University of St. Andrews School of Mathematics and Statistics
(accessed 12 March 2007).
Taylor, John, Chris Zafiratos and Michael A. Dubson. Modern Physics for Scientists
and Engineers. San Francisco: Benjamin-Cummings Pub. Co., 2003.
Van Vleck, Edward B. “The Influence of Fourier’s Series upon the Development of
Mathematics.” Science New Series, Vol. 39, No. 995. (Jan. 23, 1914) 113-124.
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