fourier series and sturm-liouville eigenvalue...

Fourier Series and Sturm-Liouville

Eigenvalue Problems

Y. K. Goh

2009

Y. K. Goh

Fourier Series and Sturm-Liouville Eigenvalue Problems

Outline

I Functions

I Fourier Series Representation

I Half-range Expansion

I Convergence of Fourier Series

I Parseval’s Theorem and Mean Square Error

I Complex Form of Fourier Series

I Inner Products

I Orthogonal Functions

I Self-adjoint Operators

I Sturm-Liouville Eigenvalue Probelms

Y. K. Goh


Periodic Functions

Definition (Periodic Function)A 2L-periodic function f : R→ R ∪ {±∞} is a functionsuch 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 2π-periodic funtion withperiod 2π, since sin(x+ 2π) = sin(x).

Y. K. Goh


Even and Odd Functions

Definition (Even and Odd Functions)

I A function f is even if and only if f(−x) = f(x), ∀x.I A function f is odd if and only if f(−x) = −f(x),∀x.

For example

I sinx is an odd function since sin(−x) = − sinx.

I cosx 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.

Y. K. Goh


Examples of Even and Odd Functions

Y. K. Goh


Piecewise Continuous Functions

Definition (Piecewise Continuous Functions)A function f is said to be piecewise continuous on the interval[a, b] if

1. f(a+) and f(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 limitsexits

Y. K. Goh


Piecewise Smooth Functions

Definition (Piecewise Smooth Functions)A function f , defined on the interval [a, b], is said to bepiecewise smooth if f and f ′ are piecewise continous on [a, b].Thus f is 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 c where the one-sided limitslimx→c− f

′(x) and limx→c+ f′(x) exist. Furthermore,

limx→a+ f ′(x) and limx→b− f′(x) exist.

Y. K. Goh


Examples of Piecewise Functions

Figure: A piecewise smoothfunction.

Figure: Another piecewisesmooth function.

Y. K. Goh


Some useful integrations

I If f is even. Then, for any a ∈ R∫ a

−af(x) dx = 2

∫ a

0

f(x) dx.

I If f is odd. Then, for any a ∈ R∫ a

−af(x) dx = 0.

I If f is piecewise continuous and 2L-periodic. Then, forany a ∈ R ∫ 2L

0

f(x) dx =

∫ a+2L

a

f(x) dx.

Y. K. Goh


Orthogonal Functions

Definition (Orthogonal Functions)Two functions f and g 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.

Y. K. Goh


Orthogonal Properties of Trigonometric Functions

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

−πcosmx sinnx dx = 0, (3)∫ π

−πcosmx cosnx dx = πδmn, (4)∫ π

−πsinmx sinnx dx = πδmn (5)

where δmn is the Kronecker’s delta.

Y. K. Goh


Kronecker’s Delta

Definition (Kronecker’s Delta)

δmn =

{1, m = n0, m 6= n.

(6)

Y. K. Goh


Fourier Series

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

f(x) =a0

2+∞∑n=1

(an cosnωx+ bn sinnωx) (7)

and the Fourier series converges to f(x) if f is continuous atx and to 1

2[f(x+) + f(x−)] otherwise.

Here ω = 2π2L

is called the fundamental frequency, while theamplitudes a0, an, and bn are called Fourier coefficients of fand they are given by the Euler formula.

Y. K. Goh


Euler Formula

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

a0 =1

L

∫ L

−Lf(x) dx,

an =1

L

∫ L

−Lf(x) cosnωx dx =

1

L

∫ L

−Lf(x) cos

nπx

Ldx,

bn =1

L

∫ L

−Lf(x) sinnωx dx =

1

L

∫ L

−Lf(x) sin

nπx

Ldx.

for n = 1, 2, . . . .Y. K. Goh


Collorary

I If f is even and 2L-periodic, then the Fourier seriesrepresentation is

f(x) =a0

2+∞∑n=1

an cosnωx.

I If f is odd and 2L-periodic, then the Fourier seriesrepresentation is

f(x) =∞∑n=1

bn sinnωx.

Here, a0, an, and bn are given by the Euler formula.Y. K. Goh


Examples of Fourier Series

I (Odd function, digital impulses) Find the Fourierrepresentation of the periodic function f(x) with period2π, where

f(x) =

{−1, −π < x < 0,1, 0 < x < π.

[Answer:]

f(x) =∞∑

n odd

4 sinnx

nπ=∞∑k=1

4 sin(2k − 1)x

(2k − 1)π.

Y. K. Goh


Graphs for Digital Pulse Train and its Fourier Series

Y. K. Goh



I (Even function) Find the Fourier series for f(x) = |x| if−1 < x < 1 and f(x+ 2) = f(x).[Answer:]

f(x) = 12−∞∑n=1

4 cos(2n− 1)x

(2n− 1)2π2.

Y. K. Goh


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

Y. K. Goh



I Find the Fourier series of the 2-periodic functionf(x) = x3 + π if −1 < x < 1.[Answer:]

f(x) = π +∞∑n=1

(−1)n12− 2n2π2

n3π3sinnπx.

Y. K. Goh


Graphs for f (x) = x3 + π, f (x + 2) = f (x)

Y. K. Goh


Full-range Extensions

Consider a function f that 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 f defined in[0, p) is a p-periodic function given by

g(x) = f(x) if 0 ≤ x < p,

g(x) = g(x+ p) if x < 0,

g(x) = g(x− p) if x ≥ p.

Y. K. Goh


Half-range Periodic Extensions

Definition (Half-range Even Periodic Extension)The half-range even periodic extension fe of a function fdefined in [0, p) is a 2p-periodic even function given by

fe(x) =

{f(x) 0 ≤ x < p,

f(−x) −p ≤ x < 0.

Y. K. Goh


Half-range Periodic Extensions

Definition (Half-range Odd Periodic Extension)The half-range odd periodic extension fo of a function fdefined in [0, p) is a 2p-periodic odd function given by

fe(x) =

{f(x) 0 ≤ x < p,

−f(−x) −p ≤ x < 0.

Y. K. Goh


Examples Periodic Extensions

Y. K. Goh


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

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

f(x) =a0

2+∞∑n=1

(an cosnωx+ bn sinnωx) , 0 ≤ x < p, (8)

where ω = 2πp

and the Fourier coefficients

a0 =1

p/2

∫ p

0

f(x) dx, an =1

p/2

∫ p

0

f(x) cosnωx dx, and

bn =1

p/2

∫ p

0

f(x) sinnωx dx.

Y. K. Goh


Fourier Cosine Series for f defined on [0, p)

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

f(x) =a0

2+∞∑n=1

an cosnωx, 0 ≤ x < p, (9)

where ω = πp

and the Fourier coefficients a0 =2

p

∫ p

0

f(x) dx,

and an =2

p

∫ p

0

f(x) cosnωx dx.

Y. K. Goh


Fourier Since Series for f defined on [0, p)

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

f(x) =∞∑n=1

bn sinnωx, 0 ≤ x < p, (10)

where ω = πp

and the Fourier coefficients

bn =2

p

∫ p

0

f(x) sinnωx dx.

Y. K. Goh


Examples

Consider a signal f(t) = t measured from an experiment overthe duration given by 0 <= t < 4.

I Sketch the full-range periodic extension of f(t). Find thecorresponding Fourier expansion of f(t).

I Sketch the half-range even extension of f(t) and find thecorresponding Fourier cosine expansion of f(t).

I Sketch the half-range odd extension of f(t) and find thecorresponding Fourier sine expansion of f(t).

Y. K. Goh


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) =a

2+

N∑n=1

(an cosnωx+ bn sinnωx) .

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

I f(x), if f(x) is continuous for all x;

I1

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

Y. K. Goh


Figure: sN (x) converges to f(x),except at the jumps.

Figure: sN (x) convergesuniformly on the interval [-1, 1]

Y. K. Goh


Pointwise Convergence

Definition (Pointwise Convergence)A sequence of functions {sn} is said to converge pointwise tothe function f on the set E, if the sequence of numbers{sn(x)} converges to the number f(x), for each x in E.

Or,

if ∀x ∈ E, sn(x)→ f(x), then {sn} converge pointwise to f.

Y. K. Goh


Uniform Convergence

Definition (Uniform Convergence)We say that sn converges to f uniformly on a set E, and wewrite sn → f uniformly on E if, given ε > 0, we can find apositive integer N such that for all n ≥ N

|sn(x)− f(x)| < ε,∀x ∈ E.

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

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

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

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

Y. K. Goh


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 WeierstraßM -test.

Y. K. Goh


WeierstraßM -Test

Theorem (WeierstraßM -Test)Let {uk}∞k=0 be a sequence of real- or complex-valued functionson E. If there exists a sequence {Mk}∞k=0 of nonnegative realnumbers such that the following two conditions hold:

I |uk(x)| ≤Mk,∀x ∈ E, and

I∑∞

k=0Mk <∞.

Then∞∑k=0

uk(x) converges uniformly on E.

Y. K. Goh


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)

Y. K. Goh


Mean Square Error

Since sN converges to f only when N →∞, for mostpractical purposes, we need N to 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

∫ L

−L[f(x)]2 dx− 1

4a2

0 −1

2

N∑n=1

(a2n + b2n

).

Y. K. Goh


Mean Square Approximation

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

∫ L−L |f(x)|2 dx on

[−L,L]. Then sN , the Nth partial sum of the Fourier series off , approximates f in the mean square sense with an error ENthat decreases to zero as N →∞.

limN→∞

EN =1

2L

∫ L

−L[f(x)]2 dx− 1

4a2

0 −1

2

N∑n=1

(a2n + b2n

)= 0.

Y. K. Goh


Bessel’s Inequality and Parseval’s Identity

Since EN > 0, from the definition of EN , we get

Definition (Bessel’s Inequality)

1

4a2

0 +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 (Parseval’s Identity)

1

4a2

0 +1

2

∞∑n=1

(a2n + b2n) =

1

2L

∫ L

−L[f(x)]2 dx.

Y. K. Goh


Multiplication Theorem

A generalisation of the Parseval’s identity is the multiplicationtheorem.

Theorem (Multiplication (Inner Product) Theorem)If f and g are two 2L-periodic piecewise smooth functions

1

2L

∫ L

−Lf(x)g(x) dx =

∞∑n=−∞

cnd∗n

where cn and dn are the Fourier coefficients for the complexFourier series of f and g respectively.

Y. K. Goh


Complex Form of Fourier Series

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

∞∑n=−∞

cneinωx,

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

cn = 12L

∫ L−L f(x)e−inωx dx, n = 0,±1,±2, . . . .

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

continuous at x, and to1

2[f(x+) + f(x−)] otherwise.

Y. K. Goh


Relations of Complex and Real Fourier Coefficients

c−n = c∗n;

c0 =1

2a0

cn =1

2(an − ibn);

c−n =1

2(an + ibn).

a0 = 2c0;

an = cn + c−n;

bn = i(cn − c−n).

Y. K. Goh


Complex Form of Parseval’s Identity

Theorem (Complex Form of Parseval’s Identity)Suppose f is a square integrable 2L-periodic piecewise smoothfunction on [−L,L]. Then

1

2L

∫ L

−L[f(x)]2 dx =

∞∑n=−∞

cnc∗n =

∞∑n=−∞

|cn|2

where cn is the complex Fourier coefficients of f .

Y. K. Goh


Example

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

Y. K. Goh


Frequency Spectra

The distribution of the magnitude of complex Fouriercoefficients |cn| in frequency domain is called the amplitudespectrum of f .

The distribution of p0 = |c0|2 and pn = |cn|2 in frequencydomain is called the power spectrum of f .

Y. K. Goh


Inner Products

Definition (Inner Products)Let ψ and φ be (possibly complex) functions of x on theinterval (a, b). Then the inner product of ψ and φ is

〈ψ|φ〉 =

∫ b

a

ψ∗(x)φ(x) dx.

Note that the notation of inner product in some books is

(φ, ψ) =

∫ b

a

ψ∗(x)φ(x) dx.

Please take note on the order of φ and ψ in the brackets.Y. K. Goh


Norm

Definition (Norm)Let f be (possibly complex) function of x on the interval(a, b). Then the norm of f is

||ψ|| =√〈f |f〉 =

(∫ b

a

|f(x)|2 dx)1/2

.

Y. K. Goh


Orthogonal Functions

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

〈f |g〉 =

∫ b

a

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

I ||Fn|| 6= 0 for all n; and

I 〈Fm|Fn〉 = 0, for m 6= n.Y. K. Goh


Normalisation and Orthonormal Set

Definition (Normalisation)A normalised function fn for a function Fn, with ||Fn|| 6= 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

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

I 〈fm|fn〉 = 0, for m 6= n;

or simply, 〈fm|fn〉 = δmn, where δmn is the Kronecker’s delta.

Y. K. Goh


Generalized Fourier Series

Theorem (Generalized Fourier Series)If {f1, f2, f3, . . . } is a complete set of orthogonal functions on(a, b) and if f can be represented as a linear combination offn, then the generalised Fourier series of f is given by

f(x) =∞∑n=1

anfn(x) =∞∑n=1

〈fn|f〉||fn||2

fn(x),

where an = 〈fn|f〉||fn||2 is the generalised Fourier coefficient.

Y. K. Goh


Parseval’s Identity

Theorem (Generalized Parseval’s Identity)If {f1, f2, f3, . . . } is a complete set of orthogonal functions on(a, b) and let f be such that ||f || is finite. Then∫ b

a

|f(x)|2 dx =∞∑n=1

|〈fn|f〉|2

||fn||2.

Y. K. Goh


Orthogonality with respect to a Weight, w(x)

I (Inner product)

〈f |g〉 =

∫ b

a

f ∗(x)g(x)w(x) dx

I (Orthogonality)

〈fm|fn〉 =

∫ b

a

f ∗m(x)fn(x)w(x) dx = ||fm||2δmnI (Generalised Fourier Series)

f(x) =∞∑n=1

〈fn|f〉||fn||2

fn(x)

I (Generalised Parseval’s Identity)∫ b

a

|f(x)|2w(x) dx =∞∑n=1

|〈fn|f〉|2

||fn||2

Y. K. Goh


Adjoint and Self-adjoint Operators

Definition (Adjoints of Differential Operators)Suppose u and v are (possible complex) functions of x in (a, b)and let L be a linear differential operator. Then, the formaladjoint M of L is another operator such that for all u and v

〈f |L[g]〉 = 〈M [f ]|g〉.

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

Y. K. Goh


Example: Adjoint for L[φ] ≡ p(x)dφ/dx

Suppose L[φ] ≡ p(x)dφ/dx, then

〈u|L[v]〉 =

∫ b

a

u∗[pd

dx

]v dx = [u∗pv]ba −

∫ b

a

v

[d

dx(p∗u)

]∗dx

= 〈M [u]|v〉

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

I Formal adjoint M [φ] ≡ − ddx

(p∗φ); and

I Boundary conditions [u∗pv]ba = 0.

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

Y. K. Goh


Adjoint for 2nd Order Linear Differential Operators

Suppose L[φ] ≡ a0(x)φ′′ + a1(x)φ

′ + a2(x)φ. Then,

〈u|L[v]〉 = [u∗a0v′ + u∗a1v − v(a0u

∗)′]ba +∫ b

a

v [(a0u∗)′′ − (a1u

∗)′ + a2u∗] dx = 〈M [u]|v〉 with

appropriate choice of boundary conditions. Note that

I M [φ] ≡ a0φ′′ + (2a′0 − a1)φ

′ + (a2 − a′1 + a′′0)φ.

I M can be made self-adjoint if a1 = a′0.

I The self-adjoint operator S is

S[φ] =d

dx

(a0(x)

dφ

dx

)+ a2(x)φ.

I The neccessary boundary condition is[a0u

∗v′ − a0v(u∗)′]

ba = 0.

Y. K. Goh


Eigenvalue Problems & Sturm-Liouville Equation

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

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

Definition (Sturm-Liouville Equation)

[S + λw(x)] y =d

dx

(a0(x)

dy

dx

)+ a2(x)y + λw(x)y = 0 for

a < x < b.

Y. K. Goh


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

d

dx

(a0(x)

dy

dx

)+ a2(x)y + λw(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 of c1 and c2 and at least one of d1 and d2

are non-zero.

Y. K. Goh


Singular Sturm-Liouville Problems

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

I a0(x), a′0(x), a2(x) and w(x) are continuous in [a, b];

I a0(x) > 0 and w(x) > 0.

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

I fails the regularity conditions; or

I infinite boundary conditions; or

I one or more of the coefficients become singular.

Y. K. Goh


Solutions of SL Problems

I A trivial (not useful) solution to the SL problem is y = 0.I Other non-trivial solutions would be the eigenfunctionsym, and for each of these eigenfunctions there is acorresponding eigenvalue λm.

I 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.

I Any function f in the Hilbert space can be expressed as alinear combination of the eigenfunctions,

f(x) =∞∑n=1

anyn(x).

Y. K. Goh


Eigenvalues of Sturm-Liouville Problems

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

I All eigenvalues are real and compose a countably infinitecollections satisfying λ1 < λ2 < λ3 < . . . where λj →∞as j →∞.

I To each eigenvalue λj there corresponds only to oneindependent eigenfunction yj(x).

I The eigenfunctions yj(x), j = 1, 2, . . . , compose acomplete orthogonal set with appropriate to the weightfunctions w(x) in doubly-integrable functions spaceL2(a, b).

Y. K. Goh


Eigenfunction Expansions

Theorem (Eigenfunction Expansions)If f ∈ L2(a, b) then eigenfunction expansion of f on{y1, y2, . . . } is

f(x) =∞∑n=1

Anyn, a < x < b,

where

An =〈yn|f〉||yn||2

=

∫ b

a

y∗n(x)f(x)w(x) dx/

∫ b

a

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

Y. K. Goh


Example SL Problem: Harmonic Equation

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

I ODE : y′′ + λy = 0, 0 < x < L.

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

I Eigenvalues: λn = k2n =

n2π2

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

I Eigenfunctions: yn(x) = sin(nπLx)

for n = 1, 2, . . . ;

I Eigenfunction expansion: f(x) =∞∑n=1

bn sinnπ

Lx.

Y. K. Goh



Again, the Harmonic Equation with another boundarycondition.

I ODE : y′′ + λy = 0, 0 < x < L.

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

I Eigenvalues: λn = k2n =

n2π2

L2;

I Eigenfunctions: yn(x) = cos(nπLx)

for n = 0, 1, 2, . . . ;

I Eigenfunction expansion: f(x) =a0

2+∞∑n=1

an cosnπ

Lx.

Y. K. Goh



The Harmonic Equation with periodic boundary condition.

I ODE: y′′ + λy = 0, 0 < x < 2π.

I Periodic boundary condition: y(0) = y(2π).

I Eigenvalues: λn = k2n = n2;

I Eigenfunctions: yn(x) = einx for n = 0,±1,±2, . . . ;

I Eigenfunction expansion: f(x) =∞∑

n=−∞

cneinx.

Y. K. Goh


Example SL Problem: (Parametric) Bessel

Equation

Bessel Equation. x2y′′ + xy′ + (λ2x2 − µ2)y = 0 or

[x2y′]′ + (λ2x− µ2

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

I Since a0(x) = x2 is zero at x = 0, =⇒ singular SLproblem.

I ODE: x2y′′ + xy′ + (λ2x2 − µ2)y = 0;I Boundary conditions: y(x) is bounded, and y(L) = 0;I Eigenvalues: λ2 = λ2

n = αn

L, n = 1, 2, . . . , where αn is

the nth-root of Jµ, i.e. Jµ(αn) = 0;I Eigenfunctions: yn(x) = Jµ(λnx), n = 1, 2, . . . ;I Weight: w(x) = x;I Eigenfunction expansion: f(x) =

∑∞n=1 anJµ(λnx).

Y. K. Goh


Example SL Problem: Spherical Bessel Equation

Bessel Equation. x2y′′ + xy′ + (λ2x2 − n(n+ 1))y = 0 in theinterval 0 < x <∞.

I Since a0(x) = x2 is zero at x = 0, =⇒ singular SLproblem.

I ODE: x2y′′ + xy′ + (λ2x2 − µ(µ+ 1))y = 0;

I Boundary conditions: y(x) is bounded, and y(L) = 0;

I Eigenvalues: λ2 = λ2n = αn

L, n = 1, 2, . . . , where αn is

the nth-root of jµ, i.e. jµ(αn) = 0;

I Eigenfunctions: yn(x) = jµ(λnx), n = 1, 2, . . . ;

I Weight: w(x) = x;

I Eigenfunction expansion: f(x) =∑∞

n=1 anjµ(λnx).

Y. K. Goh


Example SL Problem: Legendre Equation

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

I a0(x) = (1− x2) and a0(±1) = 0 =⇒ singular SLproblem.

I ODE: (1− x2)y′′ − 2xy′ + λy = 0;

I Boundary conditions: y(x) is bounded at x = ±1;

I Eigenvalues: λ = λn = n(n+ 1), n = 0, 1, 2, . . . ;

I Eigenfunctions: yn(x) = Pn(x), n = 0, 1, 2, . . . ;

I Eigenfunction expansion: f(x) =∑∞

n=0 anPn(x).

Y. K. Goh


fourier series and sturm-liouville eigenvalue...

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