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Fourier Series and Sturm-Liouville

Eigenvalue Problems

Y. K. Goh

2009

Y. K. Goh

Fourier Series and Sturm-Liouville Eigenvalue Problems
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Outline

Functions

Fourier Series Representation

Half-range Expansion

Convergence of Fourier Series Parsevals Theorem and Mean Square Error

Complex Form of Fourier Series

Inner Products

Orthogonal Functions

Self-adjoint Operators

Sturm-Liouville Eigenvalue Probelms

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Periodic Functions

Definition (Periodic Function)A 2L-periodicfunction f : R R {} is a function

such that there exists a constant L >0 such that

f(x) =f(x+ 2L), x R. (1)

Here 2L is called the fundamental period or just period.

For example,f(x) = sin(x) is a 2-periodic funtion withperiod2, since sin(x+ 2) = sin(x).

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Even and Odd Functions

Definition (Even and Odd Functions)

A function f is even if and only iff(x) =f(x), x.

A function f is odd if and only iff(x) =f(x), x.

For example

sin x is an odd function since sin(x) = sin x.

cos x is an even function since cos(x) = cos x.Note that even function is symmetric about the y-axis. On theother hand, odd function is symmetric about the origin.

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Examples of Even and Odd Functions

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Piecewise Continuous Functions

Definition (Piecewise Continuous Functions)A function fis said to be piecewise continuous on the interval

[a, b] if

1. f(a+) andf(b) exist, and

2. f is defined and continous on (a, b) except at a finitenumber of points in (a, b) where the left and right limits

exits

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Piecewise Smooth Functions

Definition (Piecewise Smooth Functions)A function f, defined on the interval[a, b], is said to bepiecewise smooth iff andf are piecewise continous on [a, b].Thus fis piecewise smooth if

1. f is piecewise continous on [a, b],

2. f exists and is continous in (a, b) except possibly atfinitely many points cwhere the one-sided limitslimxcf(x) and limxc+f(x) exist. Furthermore,limxa+f

(x) andlimxbf(x) exist.

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Examples of Piecewise Functions

Figure: A piecewise smoothfunction.

Figure: Another piecewisesmooth function.

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Some useful integrations

Iff is even. Then, for any a R aa

f(x) dx= 2

a0

f(x) dx.

Iff is odd. Then, for any a R aa

f(x) dx= 0.

Iff is piecewise continuous and 2L-periodic. Then, for

anya R 2L

0

f(x) dx=

a+2La

f(x) dx.

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Definition (Orthogonal Functions)Two functions f andg are said to be orthogonal in theinterval[a, b] if b

a

f(x)g(x) dx= 0. (2)

We will come back to orthogonal functions again later.

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Orthogonal Properties of Trigonometric Functions

The orthogonal properties of sine and cosine functions aresummarised as follow:

cos mx sin nxdx = 0, (3)

cos mx cos nxdx = mn, (4)

sin mx sin nxdx = mn (5)

where mn is the Kroneckers delta.

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Kroneckers Delta

Definition (Kroneckers Delta)

mn=

1, m= n0, m=n.

(6)

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Fourier Series

Theorem (Fourier Series Representation)Supposef is a 2L-periodic piecewise smooth function, thenFourier series off is given by

f(x) = a0

2 +

n=1

(ancos nx+ bnsin nx) (7)

and the Fourier series converges tof(x) iff is continuous at

x and to 1

2 [f(x+) + f(x)] otherwise.

Here = 22L

is called the fundamental frequency, while theamplitudesa0, an, andbn are called Fourier coefficients offand they are given by the Euler formula.

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Euler Formula

Definition (Euler Formula)The Fourier coefficients for a 2L-periodic function fare givenby

a0 = 1L

LL

f(x) dx,

an = 1

L

LL

f(x)cos nxdx= 1

L

LL

f(x)cosnx

L dx,

bn = 1

L

L

L

f(x)sin nxdx= 1L

L

L

f(x)sinnxL

dx.

forn= 1, 2, . . . .Y. K. Goh

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Collorary

Iff is even and 2L-periodic, then the Fourier seriesrepresentation is

f(x) =a0

2 +

n=1

ancos nx.

Iff is odd and 2L-periodic, then the Fourier seriesrepresentation is

f(x) =n=1

bnsin nx.

Here,a0, an, and bn are given by the Eulerformula.

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Examples of Fourier Series

(Odd function, digital impulses) Find the Fourierrepresentation of the periodic function f(x)with period2, where

f(x) =

1, < x

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Graphs for Digital Pulse Train and its Fourier Series

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(Even function) Find the Fourier series for f(x) =|x| if1< x

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Graphs forf(x) =|x|, f(x+ 2) =f(x)

Y. K. GohFourier Series and Sturm-Liouville Eigenvalue Problems
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Find the Fourier series of the 2-periodic functionf(x) =x3 + if1< x

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Graphs forf(x) =x3 +, f(x+ 2) =f(x)

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Full-range Extensions

Consider a function fthat is only defined in the interval [0, p).We could always extension the function outside the range toproduce a new function. Of course, we have infinite manyways to extend the function, but here we will focus only onthree specific extensions.

Definition (Full-range Periodic Extension)The full-range periodic extension g of a function fdefined in[0, p) is a p-periodic function given by

g(x) = f(x) if0x < p,

g(x) = g(x+p) ifx

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Half-range Periodic Extensions

Definition (Half-range Even Periodic Extension)The half-range even periodic extension fe of a function f

defined in [0, p) is a 2p-periodic even function given by

fe(x) =

f(x) 0 x < p,

f(x) px

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Half-range Periodic Extensions

Definition (Half-range Odd Periodic Extension)The half-range odd periodic extension fo of a function f

defined in [0, p) is a 2p-periodic odd function given by

fe(x) =

f(x) 0x < p,

f(x) px

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Examples Periodic Extensions

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Full-range Fourier Series for fdefined on[0, p)

TheoremIff(x) is a piecewise smooth function defined on an interval[0, p), then fhas a full-range Fourier series expansion

f(x) = a02

+n=1

(ancos nx+ bnsin nx) , 0x < p, (8)

where= 2p

and the Fourier coefficients

a0= 1p/2

p0

f(x) dx, an= 1p/2

p0

f(x)cos nx dx, and

bn= 1

p/2

p0

f(x)sin nx dx.

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Fourier Cosine Series for fdefined on[0, p)

TheoremIff(x) is a piecewise smooth function defined on an interval[0, p), then fhas a half-range Fourier cosine series expansion

f(x) =a0

2 +

n=1

ancos nx,0x < p, (9)

where=

p and the Fourier coefficients

a0=

2

p p0 f(x) dx

,

andan=2

p

p0

f(x)cos nx dx.

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Fourier Since Series forfdefined on[0, p)

TheoremIff(x) is a piecewise smooth function defined on an interval[0, p), then fhas a half-range Fourier sine series expansion

f(x) =n=1

bnsin nx, 0 x < p, (10)

where=

p

and the Fourier coefficients

bn=2

p

p

0

f(x)sin nxdx.

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Examples

Consider a signal f(t) =t measured from an experiment overthe duration given by 0

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Convergence of Fourier Series

In the Fourier Series Representation Theorem, we were sayingthat for every 2L-periodic piecewise smooth function f, wecould construct a partial sum

sN(x) = a2

+Nn=1

(ancos nx+ bnsin nx) .

And, when N , the partial sum sN(x) converges to

f(x), iff(x) is continuous for all x;

1

2[f(x+) + f(x)], at the discontinuous points, or jumps.

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Figure: sN(x)

converges tof(x)

,except at the jumps.

Figure: sN(x) converges

uniformly on the interval [-1, 1]

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Pointwise Convergence

Definition (Pointwise Convergence)A sequence of functions {sn} is said to converge pointwise to

the function fon the set E, if the sequence of numbers{sn(x)}converges to the number f(x), for each x in E.

Or,

ifxE, sn(x)f(x), then{sn}converge pointwise to f.

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Uniform Convergence

Definition (Uniform Convergence)We say that sn converges to funiformly on a set E, and wewrite snfuniformly on E if, given >0, we can find apositive integer Nsuch that for allnN

|sn(x) f(x)|< , xE.

Definition (Uniform Convergence Series)A series s(x) =

k=0 uk(x) is said to converge uniformly to

f(x) on a set Eif the sequence of partial sumssn(x) =

nk=0 uk(x)converges uniformly to f(x).

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Note that if a sequence of partial sums sn converges uniformlyto f, then sn is also pointwise convergence. However, theconverse is not always true.

In order to determine is a sn is uniformly convergence, we usethe WeierstraM-test.

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WeierstraM-Test

Theorem (WeierstraM-Test)Let{uk}

k=0 be a sequence of real- or complex-valued functions

on E. If there exists a sequence{Mk}k=0 of nonnegative real

numbers such that the following two conditions hold:

|uk(x)| Mk, xE, and

k=0 Mk

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Gibbs Phenomena

Here is an example of non-uniform convergence. The peaksremain same height but the width of the peaks changes.

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1

N=10

sqr(x)S10(x)

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1

N=30

sqr(x)S30(x)

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1

N=50

sqr(x)S50(x)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 -0.5 0 0.5 1

N=10

err(x)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 -0.5 0 0.5 1

N=30

err(x)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 -0.5 0 0.5 1

N=50

err(x)

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Mean Square Error

SincesNconverges to fonly when N , for mostpractical purposes, we need Nto be large but finite. Thus, weare approximating f with sN, and it is important for us tokeep track of the error of the approximation.

Definition (Mean Square Error)The mean square error of the partial sum sN relative to f is

EN = 1

2L L

L

[f(x) sN(x)]2 dx

= 1

2L

LL

[f(x)]2 dx 1

4a20

1

2

Nn=1

a2n+ b

2n

.

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Mean Square Approximation

Theorem (Mean Square Approximation)Suppose thatf is square integrable, i.e.

L

L|f(x)|2 dx on

[L, L]. Then sN, theNth partial sum of the Fourier series of

f, approximatesf in the mean square sense with an errorENthat decreases to zero asN .

limN

EN= 1

2L L

L

[f(x)]2 dx 1

4

a20 1

2

N

n=1

a2n+ b2n= 0.

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Bessels Inequality and Parsevals Identity

SinceEN>0, from the definition ofEN, we getDefinition (Bessels Inequality)

1

4a20+

1

2

N

n=1

(a2n+ b2n)

1

2L L

L

[f(x)]2 dx.

A stronger result is when taking the limit N

Definition (Parsevals Identity)

1

4a20+

1

2

n=1

(a2n+ b2n) =

1

2L

LL

[f(x)]2 dx.

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Multiplication Theorem

A generalisation of the Parsevals identity is the multiplicationtheorem.

Theorem (Multiplication (Inner Product) Theorem)

Iff andg are two2L-periodic piecewise smooth functions

1

2L

LL

f(x)g(x) dx=

n=

cndn

wherecn anddn are the Fourier coefficients for the complexFourier series off andg respectively.

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Complex Form of Fourier Series

Theorem (Complex Form of Fourier Series)Letf be a 2L-periodic piecewise smooth function. Thecomplex form of the Fourier series off is

n=

cneinx,

where the frequency = 2/2L and the Fourier coefficients

cn= 12LLL f(x)einx dx, n= 0, 1, 2, . . . .

For all x,the complex Fourier series converges to f(x) iff is

continuous at x, and to 1

2[f(x+) + f(x)] otherwise.

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Relations of Complex and Real Fourier Coefficients

cn = cn;

c0 = 1

2a0

cn = 1

2(an ibn);

cn = 1

2(an+ ibn).

a0 = 2c0;an = cn+ cn;

bn = i(cn cn).

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Complex Form of Parsevals Identity

Theorem (Complex Form of Parsevals Identity)Supposef is a square integrable2L-periodic piecewise smooth

function on [L, L]. Then

1

2L

LL

[f(x)]2 dx=

n=

cncn=

n=

|cn|2

wherecn is the complex Fourier coefficients off.

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Example

Find the complex Fourier series for the 2-periodic functionf(x) =ex defined in (, ).

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Frequency Spectra

The distribution of the magnitude of complex Fouriercoefficients |cn| in frequency domain is called the amplitude

spectrum off.

The distribution ofp0=|c0|2 and pn=|cn|

2 in frequencydomain is called the power spectrum off.

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Inner Products

Definition (Inner Products)Let and be (possibly complex) functions ofx on theinterval(a, b). Then the inner product of andis

|= ba

(x)(x) dx.

Note that the notation of inner product in some books is

(, ) =

b

a

(x)(x) dx.

Please take note on the order of and inthebrackets.Y. K. Goh

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Norm

Definition (Norm)Letfbe (possibly complex) function ofx on the interval

(a, b). Then the norm off is

||||=

f|f=

ba

|f(x)|2 dx

1/2.

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Definition (Orthogonal Functions)The functions f andg are called orthogonal on the interval(a, b) if their inner product is zero,

f|g= ba

f(x)g(x) dx= 0.

Definition (Orthogonal Set of Functions)

A set of functions {F1

, F2

, F3

, . . . } defined on the interval(a, b) is called an orthogonal set if

||Fn|| = 0for all n; and

Fm|Fn= 0, form=n.Y. K. Goh


N l d O h l S
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Normalisation and Orthonormal Set

Definition (Normalisation)A normalised function fn for a function Fn, with||Fn|| = 0, isdefined as

fn(x) =Fn(x)

||Fn||

.

Definition (Orthonormal Set of Functions)A set of functions {f1, f2, f3, . . . }defined on the interval(a, b) is called an orthonormal set if

||fn||= 1for all n; and

fm|fn= 0, form=n;

or simply, fm|fn=mn, where mn is theKroneckersdelta.

Y. K. Goh


G li d F i S i
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Generalized Fourier Series

Theorem (Generalized Fourier Series)If{f1, f2, f3, . . . } is a complete set of orthogonal functions on(a, b) and if fcan be represented as a linear combination of

fn, then the generalised Fourier series of f is given by

f(x) =n=1

anfn(x) =n=1

fn|f

||fn||2fn(x),

wherean= fn|f||fn||2

is the generalised Fourier coefficient.

Y. K. Goh


P l Id i
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Parsevals Identity

Theorem (Generalized Parsevals Identity)If{f1, f2, f3, . . . } is a complete set of orthogonal functions on

(a, b) and letf be such that||f|| is finite. Then ba

|f(x)|2 dx=n=1

|fn|f|2

||fn||2 .

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O h li i h W i h ( )
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Orthogonality with respect to a Weight,w(x)

(Inner product)f|g=

ba

f(x)g(x)w(x) dx

(Orthogonality)

fm|fn= ba

fm(x)fn(x)w(x) dx=||fm||

2

mn

(Generalised Fourier Series)

f(x) =

n=1fn|f

||fn||2fn(x)

(Generalised Parsevals Identity) ba

|f(x)|2w(x) dx=n=1

|fn|f|2

||fn||2

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Adj i t d S lf dj i t O t
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Adjoint and Self-adjoint Operators

Definition (Adjoints of Differential Operators)Suppose uandv are (possible complex) functions ofxin(a, b)and let L be a linear differential operator. Then, the formaladjointM ofL is another operator such that for all u andv

f|L[g]= M[f]|g.

Definition (Self-adjoint Operators)Suppose M is the formal adjoint operator for a linear operatorL in space S. IfM=L, then the operator L is said to beformally self-adjoint or formally Hermitian.

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E l Adj i t f L[] ( )d/d
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Example: Adjoint for L[]p(x)d/dx

Suppose L[]p(x)d/dx,then

u|L[v] =

ba

up

d

dx

v dx= [upv]ba

ba

v

d

dx(pu)

dx

= M[u]|v

In the last step, we set the boundary term to zero. The theadjoint for the operator L consists of

Formal adjoint M[] ddx

(p);and

Boundary conditions [upv]ba= 0.

Furhermore, ifp(x) is a pure imaginary constant, thenM=L. i.e. L is self-adjoint.

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Adjoi t fo 2nd O de Li ea Diffe e tial O e ato s
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Adjoint for 2nd Order Linear Differential Operators

Suppose L[]a0(x)

+ a1(x)

+ a2(x). Then,u|L[v]= [ua0v + ua1v v(a0u

)]ba+ b

a

v [(a0u) (a1u

) + a2u] dx=M[u]|vwith

appropriate choice of boundary conditions. Note that

M[] a0 + (2a0 a1)

+ (a2 a1+ a

0).

Mcan be made self-adjoint ifa1=a0.

The self-adjoint operator S is

S[] = d

dx

a0(x)d

dx

+ a2(x).

The neccessary boundary condition is[a0u

v a0v(u)]

ba= 0.

Y. K. Goh


Eigenvalue Problems & Sturm Liouville Equation
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Eigenvalue Problems & Sturm-Liouville Equation

Definition (Eigenvalue Problem)The eigenvalue problem associated to a differential operator Lis the equation Ly+ y = 0, where is called the eigenvalue,andy is called the eigenfunction.

It is possible to find a weight factor w(x)> 0 forL such thatS[y] w(x)L[y] is self-adjoint. The resulting eigenvalueequation is called the Sturm-Liouville Equation

Definition (Sturm-Liouville Equation)

[S+ w(x)] y= d

dx

a0(x)

dy

dx

+ a2(x)y+ w(x)y= 0for

a < x < b.

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Regular Sturm Liouville Problems
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Regular Sturm-Liouville Problems

Definition (Regular Sturm-Liouville Problem)A regular SL problem is a boundary value problem on a closedfinite interval[a, b] of the form

ddx

a0(x) dydx

+ a2(x)y+ w(x)y= 0, a < x < b,

satisfying regularity conditions and boundary conditions

c1y(a) + c2y

(a) = 0, d1y(b) + d2y

(b) = 0,

where at least one ofc1 andc2 and at least one ofd1 andd2are non-zero.

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Singular Sturm Liouville Problems
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Singular Sturm-Liouville Problems

Definition (Regularity Conditions)The regularity conditions of a regular SL problem are

a0(x), a0(x), a2(x) andw(x) are continuous in [a, b];

a0(x)> 0andw(x)> 0.

Definition (Singular Sturm-Liouville Problem)A singular SL problem is a boundary value problem consists ofSturm-Liouville equation, but either

fails the regularity conditions; or infinite boundary conditions; or

one or more of the coefficients become singular.

Y. K. Goh


Solutions of SL Problems
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Solutions of SL Problems

A trivial (not useful) solution to the SL problem is y= 0. Other non-trivial solutions would be the eigenfunctions

ym, and for each of these eigenfunctions there is acorresponding eigenvalue m.

There are infinite many of these eigenfunctions, and theset of eigenfunctions {y1, y2, . . . , ym, . . . }forms acomplete orthogonal set of functions that span theinfinite dimensional Hilbert space.

Any function f in the Hilbert space can be expressed as a

linear combination of the eigenfunctions,

f(x) =n=1

anyn(x).

Y. K. Goh


Eigenvalues of Sturm-Liouville Problems
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Eigenvalues of Sturm-Liouville Problems

Theorem (Sturm-Liouville Problem)The eigenvalues and eigenfunctions of a SL problem has the

properties of

All eigenvalues are real and compose a countably infinite

collections satisfying1< 2< 3< . . . wherej asj .

To each eigenvaluej there corresponds only to oneindependent eigenfunction yj(x).

The eigenfunctionsyj(x), j = 1, 2, . . . , compose acomplete orthogonal set with appropriate to the weight

functionsw(x) in doubly-integrable functions spaceL2(a, b).

Y. K. Goh


Eigenfunction Expansions
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Eigenfunction Expansions

Theorem (Eigenfunction Expansions)Iff L2(a, b) then eigenfunction expansion off on{y1, y2, . . . } is

f(x) =n=1

Anyn, a < x < b,

whereAn=

yn|f

||yn||2 =

b

a

yn(x)f(x)w(x) dx/

b

a

|yn(x)|2w(x) dx.

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Example SL Problem: Harmonic Equation
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An example of SL equation is the Harmonic Equation withboundary condition.

ODE : y + y= 0, 0< x < L.

Dirichlet Boundary condition: y(0) =y(L) = 0.

Eigenvalues: n=k2n=

n22

L2 , n= 1, 2, . . . ;

Eigenfunctions: yn(x) = sin

n

Lx

forn= 1, 2, . . . ;

Eigenfunction expansion: f(x) =n=1

bnsinnL

x.

Y. K. Goh


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Again, the Harmonic Equation with another boundarycondition.

ODE : y + y= 0, 0< x < L.

Neumann Boundary condition: y(0) =y(L) = 0.

Eigenvalues: n=k2n=

n22

L2 ;

Eigenfunctions: yn(x) = cos

n

Lx

forn= 0, 1, 2, . . . ;

Eigenfunction expansion: f(x) = a02

+n=1

ancosnL

x.

Y. K. Goh


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The Harmonic Equation with periodic boundary condition.

ODE:y + y= 0, 0< x

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Example SL Problem: (Parametric) Bessel

Equation

Bessel Equation. x2y + xy + (2x2 2)y = 0or

[x2y] + (2x 2

x)y = 0 in the interval 0< x < L.

Since a0(x) =x2 is zero at x= 0, = singular SL

problem. ODE:x2y + xy + (2x2 2)y= 0; Boundary conditions: y(x) is bounded, and y(L) = 0; Eigenvalues: 2 =2n=

nL

, n= 1, 2, . . . , where n is

thenth-root ofJ, i.e. J(n) = 0; Eigenfunctions: yn(x) =J(nx), n= 1, 2, . . . ; Weight: w(x) =x; Eigenfunction expansion: f(x) =

n=1 anJ(nx).

Y. K. Goh


Example SL Problem: Spherical Bessel Equation
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Example SL Problem: Spherical Bessel Equation

Bessel Equation. x

2

y

+ xy

+ (

2

x

2

n(n+ 1))y= 0in theinterval0< x

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p g q

Legendre Equation. (1 x2)y 2xy + n(n+ 1)y= 0or[(1 x2)y] + n(n+ 1)y= 0, in the interval 1< x

01 Fourier

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