MA-Differential Equations for Engineers Lecture / : Fourier Series, Integral and Sturm-Liouville Problem Panchatcharam Mariappan Assistant Professor Department of Mathematics and Statistics IIT Tirupati, Tirupati December
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MA5023-Differential Equations for EngineersLecture 14/20 : Fourier Series, Integral and Sturm-Liouville Problem
Panchatcharam Mariappan1
1Assistant ProfessorDepartment of Mathematics and Statistics
IIT Tirupati, Tirupati
2 December 2020
Periodic Functions
1
Periodic Functions
• We have seen power series already• Let us look at another series called with trigonometric functions
Definition 1 (Fourier Series)These are infinite series that represent periodic functions in terms of cosinesand sines.
Definition 2 (Periodic Function)A function f(x) is called a periodic function if1. f(x) is defined for all real x except possibly at some points2. there is some positive number p such that
f(x+ p) = f(x), for allx2
Periodic Functions
Figure 1: Periodic Functions 3
Periodic Functions
Remarks• The positive number p is called a period of f• The smallest period is often called the fundamental period or Base
period• The interval containing full period is called window
Example 1• sinx, cosx, tanx, cotx, periodic, period 2π, 4π, 6π, · · · , 2nπ,• Fundamental period of sinx, cosx : 2π, window: [0, 2π), [2π, 4π),
[−π
2 ,π2
)• sinLx, cosLx, tanLx, cotLx, periodic, period 2π
L
• coshx, lnx, x, x3, not periodic4
Periodic Functions
Exercise 1 (Questions)• What is the period of sin4 x+ cos4 x?• What is the period of | sinx|+ | cosx|?• What is the period of sin
(πx2
)+ cos
(πx2
)?
• Have you come across any other periodic functions?• What will happen if we relax the condition p > 0 as p ≥ 0?
Exercise 2 (True or False)• Polynomials are periodic.• All hyperbolic functions are periodic
5
Trigonometric System
6
QuestionsWe have seen many periodic functions, mainly trigonometric functions. Letus consider only the sine and cosine functions. Let us consider the followingsystem
{1, cosx, sinx, cos 2x, sin 2x, · · · , cosnx, sinnx, · · · , } (1)
All the elements of these functions have the period 2π (1 is periodic with anyperiod). This system is called trigonometric system.
QuestionsFind the interval for which the following sequence converges, if it does so.1. {sinnx}∞n=1
2. {cosnx}∞n=1
7
Questions
Questions
1. If an converges, can you guarantee that∞∑n=1
an converges?
2. If∞∑n=1
an converges, can you guarantee that an converges?
3. Does∞∑n=0
sinnx converge? If so, for what values of x?
4. Does∞∑n=0
cosnx converge? If so, for what values of x?
5. Does∞∑n=0
(sinnx+ cosnx) converge? If so, for what values of x?
8
Big Question
Big Question
1. Does∞∑n=0
an(sinnx+ cosnx) converge? If so, for what values of x?
2. Does∞∑n=0
(an cosnx+ bn sinnx) converge? If so, for what values of x?
3. Suppose∞∑n=0
(an cosnx+ bn sinnx) converges, what will be the period of
this series for each x?
4. Suppose∞∑n=0
(an cosnx+ bn sinnx) converges, can we find an’s and bn’s?
9
Orthogonality(2D): Two vectors a and b are said to be orthogonal if
a.b = 0
(n D): Two vectors x, y ∈ Rn orthogonal if
xT y = yTx = 0
(Linear Algebra): Let V be an inner product space. Then u, v ∈ V are said tobe orthogonal if
〈u, v〉 = 0
where inner product is a mapping from V × V → R satisfying a set ofproperties.
10
OrthogonalityLet V be the set of all continuous function defined [a, b]. That is
V = C[a, b] = {f : [a, b]→ R : f is continuous}
Define
〈f, g〉 =b�
a
f(x)g(x)dx
Then f and g are orthogonal if
〈f, g〉 =b�
a
f(x)g(x)dx = 0
11
Orthogonality and OrthonormalityConsider the following {f1, f2, · · · } ⊂ C[a, b]. fi’s are said to be orthogonal if
〈fi, fj〉 = 0, for all i 6= j
fi’s are said to be orthonormal if
〈fi, fj〉 =
{0 i 6= j
1 i = j
12
Trigonometric System
Theorem 2 (Orthogonality of the Trigonometric System)The trigonometric system is orthogonal in the interval−π ≤ x ≤ π. That is, forn 6= m
π�
−π
sinnx sinmxdx =
π�
−π
cosnx cosmxdx =
π�
−π
cosnx sinmxdx = 0
π�
−π
cosnx sinnxdx = 0
13
Trigonometric SystemProof: We know that for n 6= m
π�
−π
cos(n±m)xdx =
[− sin(n±m)x
n±m
]π−π
= 0
Since sinx is an odd function, irrespective of n = m or n 6= m,
π�
−π
sin(n±m)xdx = 0
sin(A±B) = sinA cosB ± cosA sinB
cos(A±B) = cosA cosB ∓ sinA sinB14
Trigonometric SystemProof:
π�
−π
sinnx sinmxdx =1
2
π�
−π
cos(n−m)xdx− 1
2
π�
−π
cos(n+m)xdx = 0
π�
−π
cosnx cosmxdx =1
2
π�
−π
cos(n−m)xdx+1
2
π�
−π
cos(n+m)xdx = 0
π�
−π
sinnx cosmxdx =1
2
π�
−π
sin(n+m)xdx+1
2
π�
−π
sin(n−m)xdx = 0
Last equality is true irrespective of n = m or n 6= m.
15
Trigonometric SystemNow for n = m 6= 0
π�
−π
sin2 nxdx =1
2
π�
−π
dx− 1
2
π�
−π
cos 2nxdx = π
π�
−π
cos2 nxdx =1
2
π�
−π
dx+1
2
π�
−π
cos 2nxdx = π
When n = m = 0
π�
−π
sin2 nxdx = 0,
π�
−π
cos2 nxdx =
π�
−π
dx = 2π
16
Orthogonality and OrthonormalityBy the above theorem, the trigonometric system
{1, cosx, sinx, cos 2x, sin 2x, · · · , cosnx, sinnx, · · · , }
is orthogonal. By above discussion, the trigonometric system{1√2π,
1√πcosx,
1√πcos 2x, · · · , 1√
πcosnx, · · · ,
}is orthonormal. {
1√πsinx,
1√πsin 2x, · · · , 1√
πsinnx, · · · ,
}is also orthonormal.
17
Trigonometric System
• Let us exploit this trigonometric system and its orthogonality andorthonormality properties to obtain a few interesting series.• This series enjoys nice properties and has plenty of applications in
science and engineering.
Now let us revisit our Big Question. Suppose∞∑n=0
(an cosnx+ bn sinnx)
converges, what will be the period of this series for each x?Suppose this series converges, then you can find a function f such that
f(x) =
∞∑n=0
(an cosnx+ bn sinnx)
You can easily observe that the period of f is 2π.18
Fourier Series
19
Fourier SeriesSuppose the series converges then we can find the coefficient of the seriesan’s and bn’s by exploiting the orthogonality property of the trigonometricsystem.Multiply the above expression by sinmx,m 6= 0 and then integrate w.r.t. xfrom −π to π, then we get b0 = 0 and
π�
−π
f(x) sinmxdx =
π�
−π
∞∑n=0
(an cosnx sinmx+ bn sinnx sinmx)dx
=
∞∑n=0
an
π�
−π
cosnx sinmxdx+ bn
π�
−π
∞∑n=0
sinnx sinmxdx
= bmπ
20
Fourier SeriesMultiply the above expression by cosmx and then integrate w.r.t. x from −π toπ, then we get
π�
−π
f(x) cosmxdx =
π�
−π
∞∑n=0
(an cosnx cosmx+ bn sinnx cosmx)dx
=
∞∑n=0
an
π�
−π
cosnx cosmxdx+ bn
π�
−π
∞∑n=0
sinnx cosmxdx
π�
−π
f(x) cosmxdx =
{amπ m 6= 0
am2π m = 0
21
Fourier Series
Definition 3 (Fourier Series)Suppose that f is a given function of period 2π and is such that it can be rep-resented by
f(x) =∞∑n=0
(an cosnx+ bn sinnx) (2)
That is, the above series converges and its sum is f(x).Then, for all n = 1, 2, · · · we have
a0 =1
2π
π�
−π
f(x)dx, an =1
π
π�
−π
f(x) cosnxdx
b0 = 0, bn =1
π
π�
−π
f(x) sinnxdx
22
Fourier Series: ExampleFind the Fourier coefficients for the following periodic function
f(x) =
{−k −π < x < 0
k 0 < x < π
f(x+ 2π) = f(x) Note that f is an odd function.
a0 =1
2π
π�
−π
f(x)dx = 0, an =1
π
π�
−π
f(x) cosnxdx = 0
b0 = 0, bn =2k
π
π�
0
sinnxdx =2k
nπ(1− cosnπ)
23
Fourier Series
f(x) =
∞∑n=1
(an cosnx+ bn sinnx)
=2k
π
∞∑n=1
1− (−1)n
nsinnx =
4k
π
∞∑n=1
1
2n− 1sin(2n− 1)x
Note that
f(π2
)= k =
4k
π
∞∑n=1
1
2n− 1sin
((2n− 1)π
2
)=⇒ π
4=
∞∑n=1
(−1)n+1
2n− 1
24
Fourier SeriesUsually the Fourier Series is written as
f(x) = a0 +
∞∑n=1
(an cosnx+ bn sinnx) (3)
Theorem 3 (Representation by Fourier Series)Let f be periodic with period 2π and piecewise continuous in the interval[−π, π]. Furthermore, let f have a left-hand derivative and a right-hand deriva-tive at each point of that interval. Then the Fourier series converges. Its sumis f(x), except at points x0 where f(x) is discontinuous. There the sum of theseries is the average of the left- and right-hand limits of f at x0.
The proof of this theorem is an advanced topic, you can refer the text booklike Zygmund and Fefferman, Trigonometric Series 25
Fourier Series: Period 2LSuppose the period of f is p = 2L, then use the transformation,
x =p
2πv =⇒ v =
2π
x=π
Lx =⇒ dv =
π
L
Then v = ±π whenever x = ±L.
a0 =1
2π
π�
−π
f
(L
πv
)dv, an =
1
π
π�
−π
f
(L
πv
)cosnvdv
b0 = 0, bn =1
π
π�
0
f
(L
πv
)sinnvdv
26
Fourier Series: Period 2LHence,
f(x) = f
(L
πv
)=
∞∑n=0
(an cos
(nπL
)x+ bn sin
(nπL
)x)
(4)
a0 =1
2L
L�
−L
f(x)dx, an =1
L
L�
−L
f(x) cos(nπL
)xdx
b0 = 0, bn =1
L
L�
−L
f(x) sin(nπL
)xdx
27
Fourier Series: ExampleFind the Fourier coefficients for the following periodic function
f(x) =
{−k −2 < x < 0
k 0 < x < 2
f(x+ 4) = f(x) Here period 2L = 4 =⇒ L = 2. Note that f is an oddfunction. Again an = 0, a0 = 0, b0 = 0
bn =2k
π
2�
0
sin(nπx
2
)dx =
2k
nπ(1− cosnπ)
Hence
f(x) =4k
π
∞∑n=1
1
2n− 1sin
(2n− 1)π
2x
28
Fourier Cosine SeriesSuppose f is an even function, then bn vanishes and we obtain the FourierCosine Series
f(x) =
∞∑n=0
an cos(nπL
)x (5)
a0 =1
L
L�
0
f(x)dx, an =2
L
L�
0
f(x) cos(nπL
)xdx
29
Fourier Sine SeriesSuppose f is an odd function, then an vanishes and we obtain the FourierSine Series
f(x) =
∞∑n=0
bn sin(nπL
)x (6)
b0 = 0, bn =2
L
L�
0
f(x) sin(nπL
)xdx
30
Periodic Functions
Exercise 3 (Fourier Series)Find the appropriate Fourier Series• f(x) = x2,−1 < x < 1
• f(x) = 1− x2/4,−2 < x < 2
• f(x) = x|x|,−1 < x < 1
• Find what is half-range expansion and then find the Fourier series forf(x) = sinx, 0 < x < π
31
MA5023-Differential Equations for EngineersLecture 14/20 : Fourier Series, Integral and Sturm-Liouville Problem
Panchatcharam Mariappan1
1Assistant ProfessorDepartment of Mathematics and Statistics
IIT Tirupati, Tirupati
2 December 2020
32
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