fourier series and sturm-liouville eigenvalue...
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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
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
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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).
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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.
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Examples of Even and Odd Functions
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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
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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 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.
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Examples of Piecewise Functions
Figure: A piecewise smoothfunction.
Figure: Another piecewisesmooth function.
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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
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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.
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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.
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Kronecker’s Delta
Definition (Kronecker’s Delta)
δmn =
{1, m = n0, m 6= n.
(6)
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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.
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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
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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
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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.
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Graphs for Digital Pulse Train and its Fourier Series
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Examples of Fourier Series
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.
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Graphs for f (x) = |x|, f (x + 2) = f (x)
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Examples of Fourier Series
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.
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Graphs for f (x) = x3 + π, f (x + 2) = f (x)
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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.
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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.
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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.
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Examples Periodic Extensions
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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.
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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.
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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.
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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).
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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) =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.
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Figure: sN (x) converges to f(x),except at the jumps.
Figure: sN (x) convergesuniformly on the interval [-1, 1]
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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.
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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).
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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 WeierstraßM -test.
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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.
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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)
Y. K. Goh
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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
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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.
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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.
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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.
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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.
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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
Fourier Series and Sturm-Liouville Eigenvalue Problems
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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 .
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Example
Find the complex Fourier series for the 2π-periodic functionf(x) = ex defined in (−π, π).
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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 .
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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
Fourier Series and Sturm-Liouville Eigenvalue Problems
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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
.
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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
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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.
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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 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.
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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.
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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
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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.
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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.
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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.
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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.
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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
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.
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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.
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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).
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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).
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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.
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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.
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Example SL Problem: Harmonic Equation
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.
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Example SL Problem: Harmonic Equation
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
Fourier Series and Sturm-Liouville Eigenvalue Problems
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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
Fourier Series and Sturm-Liouville Eigenvalue Problems
![Page 66: Fourier Series and Sturm-Liouville Eigenvalue Problemsstaff.utar.edu.my/gohyk/UCCM3003/01_Fourier.pdfFourier Series and Sturm-Liouville Eigenvalue Problems Convergence of Fourier Series](https://reader034.vdocuments.site/reader034/viewer/2022042513/5fa3e8d17962ad28d4531469/html5/thumbnails/66.jpg)
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
Fourier Series and Sturm-Liouville Eigenvalue Problems
![Page 67: Fourier Series and Sturm-Liouville Eigenvalue Problemsstaff.utar.edu.my/gohyk/UCCM3003/01_Fourier.pdfFourier Series and Sturm-Liouville Eigenvalue Problems Convergence of Fourier Series](https://reader034.vdocuments.site/reader034/viewer/2022042513/5fa3e8d17962ad28d4531469/html5/thumbnails/67.jpg)
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 Problems