Applications of Fourier Applications of Fourier

Transforms in Solving Transforms in Solving

Differential EquationsDifferential Equations

MIDN 1/C Allison TylerMIDN 1/C Allison Tyler

United States Naval AcademyUnited States Naval Academy

18 April 200718 April 2007

HistoryHistory

�� Daniel BernoulliDaniel Bernoulli

�� Known for the Known for the

Bernoulli PrincipleBernoulli Principle

�� Solved the wave Solved the wave

equation in 1720equation in 1720’’s s

using techniques using techniques

Fourier later would Fourier later would

employemploy

�� Worked with Euler in Worked with Euler in

St. PetersburgSt. Petersburg

HistoryHistory

�� Jean le Rond d'Alembert Jean le Rond d'Alembert

�� French mathematicianFrench mathematician

�� Worked with Euler and Worked with Euler and

Bernoulli on the wave Bernoulli on the wave

equationequation

�� ““FoundFound”” the Cauchythe Cauchy--

Riemann equations Riemann equations

decades before Cauchy decades before Cauchy

or Riemann didor Riemann did

HistoryHistory

�� Joseph FourierJoseph Fourier

�� French mathematicianFrench mathematician

�� Served in Egypt while Served in Egypt while

in Napoleonin Napoleon’’s Armys Army

�� Solved the heat Solved the heat

equation and won equation and won

award for it in 1811, award for it in 1811,

published work in published work in

1822 in 1822 in ThThééorie orie

analytique de la analytique de la

chaleurchaleur

HistoryHistory

�� Erwin SchrErwin Schröödingerdinger

�� ViennaVienna--born physicistborn physicist

�� Artilleryman in WWIArtilleryman in WWI

�� Best known for Best known for ““SchrSchröödingerdinger’’s s

CatCat”” and the Schrand the Schröödinger dinger

EquationEquation

�� Won the Nobel Prize in 1933Won the Nobel Prize in 1933

�� Worked with EinsteinWorked with Einstein

�� Fled Germany and Austria Fled Germany and Austria

several times after several times after ““insultinginsulting””

the Nazisthe Nazis

ConceptsConcepts

�� The following concepts were usedThe following concepts were used

Fourier Coefficients

dxexfP

C P

inxP

n

π2

0)(

1−

∫=

Fourier Series

∑∞

−∞=

=n

P

inx

neCxfFS

π2

))((

Superposition Principle

The Superposition Principle states that for a linear system,

the sum of solutions for that system is also a solution.

ConceptsConcepts

�� OrthogonalityOrthogonality of of SinesSines

1 ;sin)( ≥

= n

L

xnxf

π

∫ =

L

dxL

xn

L

xm

L 00sinsin

2 ππ

nm ≠

Given

for

,

ConceptsConcepts

�� Proof of Proof of OrthogonalityOrthogonality

( )

( )

( )

( )

( )( )( )

( )( )( )

( ) ( )( )

( ) ( )

0

002

002

0sin0sin2

2

sin

2

sin2

2

sin

2

sin2

sinsin2

0

0

=

−−−=

−−

+

+−

−

−=

+

+

−−

−

=

∫

LL

Lnm

nm

nm

nm

L

L

xnm

L

xnm

L

xnm

L

xnm

Ldx

L

xn

L

xm

L

L

L

π

π

π

π

π

π

π

π

ππ

Wave EquationWave Equation

�� People studied the People studied the

wave equation wave equation

because they were because they were

intrigued by string intrigued by string

instruments. instruments.

�� Bernoulli defined Bernoulli defined

nodes and nodes and

frequencies of frequencies of

oscillation oscillation

Derivation of Wave EquationDerivation of Wave Equation

�� To solve the wave equation, we assume that it is a To solve the wave equation, we assume that it is a

separable equation:separable equation:

u(x,tu(x,t)=)=X(x)T(tX(x)T(t))

)()0,( )()0,( :Conditions Initial

0),( 0),0( :conditionsBoundary 2

22

2

2

xgxdt

uxfxu

tLutux

u

t

u

=∂

=

==∂

∂=

∂

∂ω

The Wave Equation

DerivationDerivation

'' 2

2

XTt

uXT'

t

u=

∂

∂=

∂

∂

constant a equal they if is equal be will

sides twoheonly way t thebecause ''''

:get weVariables of SeparationBy

''''

'' '

2

2

2

22

2

2

2

2

λω

ωω

−==

=⇒∂

∂=

∂

∂⇒

=∂

∂=

∂

∂

X

X

T

T

TXXTx

u

t

u

TXx

uTX

x

u

Solve for X as in the heat equation and you get

sin1

∑∞

=

=

n

n xL

nCX

π

Solve for T

)sin()cos( :Therefore

0)(

0'' ''

21

22

22

tDtDT

niL

nDTD

TTTT

nn

n

γγ

γπω

λω

λωλω

+=

Ζ∈±=

±=⇒=+

=+⇒−=

Thus we get

( )

( ))sin()cos( sin),(

Principleion Superpositby )sin()cos( sin),(

)()(),(

1

1

tEtDxL

ntxu

tEtDxL

nCtxu

tTxXtxu

nnnn

n

nnnn

n

n

γγπ

γγπ

+

=

+

=

=

∑

∑∞

=

∞

=

( )

( )

( ) )( sin)0cos()0sin( sin)0,(

)cos()sin( sin),(

)()0,(

)sin()(2

where

)( sin)0sin()0cos( sin)0,(

)()0,( :Conditions Initial theUsing

11

1

0

11

xgxL

nEEDx

L

nx

t

u

tEtDxL

ntx

t

u

xgxt

u

dxxxfL

D

xfxL

nDEDx

L

nxu

xfxu

n

nnnnn

n

n

nnnn

n

n

L

nn

n

nnnnn

n

=

=+−

=

∂

∂

+−

=

∂

∂

=∂

∂

=

=

=+

=

=

∑∑

∑

∫

∑∑

∞

=

∞

=

∞

=

∞

=

∞

=

πγγ

πγ

γγπ

γ

λ

πγγ

π

∫=L

n

n

n dxxxgL

E0

)sin()(2

where λγ

Final Solution

( )

∫∫

∑

==

+

=

∞

=

L

n

n

n

L

nn

nnnn

n

dxxxgL

EdxxxfL

D

tEtDxL

ntxu

00

1

)sin()(2

and )sin()(2

where

)sin()cos( sin),(

λγ

λ

γγπ

Unfiltered Wave EquationUnfiltered Wave Equation

Filtered Wave EquationFiltered Wave Equation

Heat EquationHeat Equation

�� The reason we solve the heat equation is The reason we solve the heat equation is

to find out what happens to a length of to find out what happens to a length of

metal/plastic when heat is applied (the metal/plastic when heat is applied (the

temperature distribution over time)temperature distribution over time)

Derivation of Heat EquationDerivation of Heat Equation

�� The assumption made by Bernoulli and adopted The assumption made by Bernoulli and adopted

by Fourier was that the equation by Fourier was that the equation u(x,tu(x,t) was ) was

separable:separable:

u(x,tu(x,t)=)=X(x)T(tX(x)T(t))

)()0,( :Conditions Initial

0),( 0),0( :ConditionsBoundary x

k2

2

xfxu

tLutuu

t

u

=

==∂

∂=

∂

∂

The Heat Equation

DerivationDerivation

TXx

uTX

x

u

XTt

u

'' '

'

2

2

=∂

∂=

∂

∂

=∂

∂

TkXXTx

uk '''

t

u2

2

=⇒∂

∂=

∂

∂

The only way this can occur is if both sides equal a constant

λ−==X

X

kT

''T'

Ζ∈

=⇒==⇒==

====

+=⇒

±=⇒=+

=+⇒−=

n L

n when 0sin 0sin)(

0)0sin( and 1)0cos( because 0)0(

:get weconditionsboundary theUsing

)sin()cos(

0)(

0'' ''

2

2

1

21

2

πλπλλλ

λλ

λλ

λλ

nnLLLCLX

CX

xCxC X

iDXD

XXXX

∑∞

=

=

1

Principleion Superposit by the sin :Thereforen

n xL

nAX

π

Now Solve for X

Now Solve for T

( )∑ ∑∞

=

∞

=

−−

−

==

=

=⇒

−=⇒=+

=+⇒−=

1 1

sin)sin(),(

)()(),( :get weThus,

0)(

0' '

n n

kt

nnnn

kt

kt

nexAxABetxu

tTxXtxu

BeT

kDTkD

kTTkTT

λλ

λ

λλ

λλ

λλ

dxxxfL

A

xfxAxu

xfxu

L

nn

n

nn

∫

∑

=⇒

==

=∞

=

0

1

)sin()(1

)()sin()0,(

)()0,( :condition initial theUsing

λ

λ

Final Solution

∑ ∫∞

=

− ==1

0)sin()(

1 where)sin(),(

n

L

nn

kt

nn dxxxfL

AexAtxu n λλ λ

Unfiltered Heat EquationUnfiltered Heat Equation

Filtered Heat EquationFiltered Heat Equation

Euler and Euler and dd’’AlembertAlembert

�� They too solved the wave equation, but by They too solved the wave equation, but by using 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 AlongUntil 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:Modern Application:SchrSchröödingerdinger

�� The SchrThe Schröödinger Equation was published dinger Equation was published

in a series of papers on wave mechanics in a series of papers on wave mechanics

in 1926in 1926

�� The solutions to the SchrThe solutions to the Schröödinger Equation dinger Equation

are wave functions that express the are wave functions that express the

probability that a particle (specifically the probability that a particle (specifically the

electron of a hydrogen atom) will be in a electron of a hydrogen atom) will be in a

certain location at any observationcertain location at any observation

Solution to SchrSolution to Schröödingerdinger

�� As with the heat and wave equations, we assume that the As with the heat and wave equations, we assume that the

SchrSchröödinger Equation is separable:dinger Equation is separable:

u(x,tu(x,t)=)=X(x)T(tX(x)T(t))

The Schrödinger Equation

serg

xfx

tLttm

i

t

−=

=

==∂

∂=

∂

∂

27-

2

2

1.054x10

)()0,( :Conditions Initial

0),( 0),0( : ConditionsBoundary 2

h

h

ψ

ψψψψ

SolutionSolution

Equations WaveandHeat theSimilar to '''

:get we variablesof separationBy

'''

2Let ''

2

'

)()(),(

2

2

2

2

cX

X

isT

T

TisXXT

xis

t

msX

m

i

x

XTt

tTxXtx

−==

=

∂

∂=

∂

∂

==∂

∂

=∂

∂

=

ψψ

ψ

ψ

ψ

hh

Assume Ψ(x,t) is separable

Solve for T:

isct

disctdisct

beT

eeeT

disctT

iscT

T

−

−+−

=

==

+−=

=

)ln(

' :Tfor solve To

Solve for X:

)sin()cos(

0)(

0'' ''

'' :Xfor solve To

21

2

xcaxcaX

icDXcD

cXXcXX

cX

X

+=

±=⇒=+

=+⇒−=

−=

( )

∑∞

=

−

−

=

=

=

∈

===

1

2

2

n12

2

sin),(

principleion superposit By the

sin),(

)()(),(get weTherefore

and 0 Where sin

n

tL

nis

n

isct

exL

nbtx

beLcatx

tTxXtx

ZnL

ncaLcaX

ππ

ψ

ψ

ψ

π

Using the boundary conditions we get the same results as before:

Using the initial conditions:

∫

∑

=⇒

=

=

∞

=

L

n

n

n

dxxL

nxf

Lb

xfxL

nbx

0

1

sin)(2

)(sin)0,(

π

πψ

Final Solution

∫∑

=

=

∞

=

− L

n

n

tL

nis

n dxxL

nxf

Lbex

L

nbtx

01

sin)(2

Wheresin),(

2

ππψ

π

SAGESAGE

�� The graphs were produced using the open The graphs were produced using the open

source program SAGE by Prof. W. David source program SAGE by Prof. W. David

Joyner, USNAJoyner, USNA

httphttp://://modular.math.washington.edumodular.math.washington.edu/sage/sage

ReferencesReferences

�� DebnathDebnath, , LokenathLokenath. . Integral Transforms and Their Applications.Integral Transforms and Their Applications. New York: CRC New York: CRC Press, 1995.Press, 1995.

�� FarlowFarlow, Stanley J. , Stanley J. Partial Differential Equations for Scientists and EngineersPartial Differential Equations for Scientists and Engineers. New . New York: Dover Publications, Inc., 1993.York: Dover Publications, Inc., 1993.

�� OO’’Connor, J.J. and E.F. Robertson. Connor, J.J. and E.F. Robertson. ““Daniel Bernoulli.Daniel Bernoulli.”” University of St. Andrews University of St. Andrews School of Mathematics and StatisticsSchool of Mathematics and Statistics (accessed 10 February 2007).(accessed 10 February 2007).

�� ------. . ““Erwin Rudolf Josef Alexander SchrErwin Rudolf Josef Alexander Schröödinger.dinger.”” University of St. Andrews School of University of St. Andrews School of Mathematics and StatisticsMathematics and Statistics (accessed 12 March 2007).(accessed 12 March 2007).

�� ------. . ““Jean le Jean le RondRond dd’’AlembertAlembert..”” University of St. Andrews School of Mathematics and University of St. Andrews School of Mathematics and StatisticsStatistics (accessed 11 March 2007).(accessed 11 March 2007).

�� ------. . ““Joseph Fourier.Joseph Fourier.”” University of St. Andrews School of Mathematics and StatisticsUniversity of St. Andrews School of Mathematics and Statistics(accessed 12 March 2007).(accessed 12 March 2007).

�� Taylor, John, Chris Taylor, John, Chris ZafiratosZafiratos and Michael A. and Michael A. DubsonDubson. . Modern Physics for Scientists Modern Physics for Scientists and Engineersand Engineers. San Francisco: Benjamin. San Francisco: Benjamin--Cummings Pub. Co., 2003.Cummings Pub. Co., 2003.

�� Van Van VleckVleck, Edward B. , Edward B. ““The Influence of FourierThe Influence of Fourier’’s Series upon the Development of s Series upon the Development of

Mathematics.Mathematics.”” ScienceScience New Series, Vol. 39, No. 995. (Jan. 23, 1914) 113New Series, Vol. 39, No. 995. (Jan. 23, 1914) 113--124. 124.

�� Walker, James S. Walker, James S. Fast Fourier Transforms 2nd Ed.Fast Fourier Transforms 2nd Ed. New York: CRC Press, 1996.New York: CRC Press, 1996.