18MAB201T-U2-part2.pdf

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18MAB201T-Transforms and Boundary Value

Problems

Prepared by

Dr. S. ATHITHAN

Assistant Professor

Department of of Mathematics

Faculty of Engineering and Technology

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FOURIER SERIES

TOPICS:

? Introduction of Fourier series -Dirichlet’s conditions for existence of Fourier Series

? Fourier series -related problems

? Half Range sine series-related problems

? Half Range cosine series-related problems

? Parseval’s Identity (without proof)-related problems

? Harmonic Analysis for finding fundamental harmonic

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1

Fourier Series

1.1

Harmonic Analysis

Example : 1

Compute the first three harmonics of the Fourier series for the following data: x 0 π/3 2π/3 π 4π/3 5π/3 2π

y 1.0 1.4 1.9 1.7 1.5 1.2 1.0

Hints/Solution:

Here length of the interval2l = 6.

∴The Fourier series is given by

f(x) = a0

2 + a1cos _πx

3

+ a2cos

_2πx 3

+a3cos

_3πx 3

+ b1sin _πx

3

+

b2sin

_2πx 3

+b3sin

_3πx 3

x y ycosx ysinx ycos 2x ysin 2x ycos 3x ysin 3x

0 1.0 1 0 1 0 1 0

π

3 1.4 0.7 1.212 -0.7 1.212 -1.4 0 2π

3 1.9 -0.95 1.65 -0.95 -1.645 1.9 0 3π

3 1.7 -1.7 0 1.7 0 -1.7 0

4π

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a0 = 2 _Σy

n

= 2

_8.7 6

= 2.9,

a1 = 2

_Σycosx

n

= −0.37,

a2 = 2

_{Σycos 2x}

n

= −0.1,

a3 = 2

_{Σycos 3x}

n

= 0.03,

b1 = 2

_Σysinx

n

= 0.17,

b2 = 2

_{Σysin 2x}

n

= −0.06, b3 = 0

∴ f(x) = 2.9

2 −0.37 cosx−0.1 cos 2x+ 0.03 cos 3x+ 0.17 sinx−0.06 sin 2x

Example : 2

(a). Find the Fourier series upto 2 harmonics for the following data: x 0 1 2 3 4 5

y 9 18 24 28 26 20

Hints/Solution:

(a). Here length of the interval2l = 6.

∴The Fourier series is given by

f(x) = a0

2 +a1cos

πx

3

+a2cos

2πx

3

+b1sin

πx

3

+b2sin

2πx

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x _u πx

3 y ycos _πx 3 ysin _πx 3 ycos _2πx 3 ysin _2πx 3

0 9 9 0 9 0

1=π

3 18 9 15.7 -9 15.6

2=2π

3 24 -12 20.9 -12 -20.9

3=3π

3 28 -28 0 28 0

4=4π

3 26 -13 -22.6 -13 22.6

5=5π

3 20 10 -17.4 -10 -17.4

125 -25 -3.4 -7 0.01 Now

a0 = 2 _Σy n = 2 ₁₂₅ 6

= 41.66,

a1 = 2

Σycos πx₃

n

!

=−8.33,

a2 = 2

Σycos 2πx₃

n

!

= −6.33,

b1 = 2

Σysin πx₃

n

!

=−1.13,

b2 = 2

Σysin 2πx₃

n

!

= 6.9

∴ f(x) = 41.66 −8.33 cos

πx

− 6.33 cos

2πx

− 1.13 sin

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2

Exercise/Practice/Assignment Problems

1. Compute the first three harmonics of the Fourier series for the following data:

x◦ 0 30 60 90 120 150 180 210 240 270 300 330 360

y 2.34 3.01 3.69 4.15 3.69 2.20 0.83 0.51 0.88 1.09 1.19 1.64 2.34

Ans:f(x) = 2.102 + 0.559 cosx+ 1.535 sinx−0.519 cos 2x−0.091 sin 2x+ 0.20 cos 3x+ 0 sin 3x

2. Compute the first three harmonics of the Fourier series for the following data:

x 0 1 2 3 4 5 6 7 8 9 10 11

y 6.824 7.976 8.026 7.204 5.676 3.674 1.764 0.552 0.262 0.904 2.492 4.736

Ans:f(x) = 4.174+2.450 cosπx

6 +3.160 sin

πx

6 +0.120 cos 2πx

6 +0.034 sin 2πx

6 +

0.080 cos3πx

6 + 0.010 sin 3πx

6

3. Compute the first three harmonics of the Fourier series for the following data:

x◦ 0 45 90 135 180 225 270 315 360

y 6.824 8.001 7.204 4.675 1.764 0.407 0.904 3.614 6.824

Ans: f(x) = 4.174 + 2.420 cosx+ 3.105 sinx+ 0.12 cos 2x+ 0.03 sin 2x+ 0.110 cos 3x+ 0.045 sin 3x

4. Compute the first three harmonics of the Fourier series for the following data:

(a) x 0 π/6 2π/6 3π/6 4π/6 5π/6 π y 2.34 2.2 1.6 0.83 0.51 0.88 1.19

Ans:f(x) = 1.4 sinx+ 1.21 sin 2x+ 0.75 sin 3x

(b) x 0 T /6 2T /6 3T /6 4T /6 5T /6 T y 1.98 1.3 1.05 1.3 -0.88 -0.25 1.98 (c) x 0 30

◦

60◦ 90◦ 120◦ 150◦ 180◦

y 0 5224 8097 7850 5499 2626 0