Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions •...
Transcript of Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions •...
![Page 1: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/1.jpg)
Today
• Fourier Series examples - even and odd extensions, other symmetries
• Using Fourier Series to solve the Diffusion Equation
![Page 2: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/2.jpg)
Examples - calculate the Fourier Series
![Page 3: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/3.jpg)
Examples - calculate the Fourier Series
![Page 4: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/4.jpg)
Examples - calculate the Fourier Series
![Page 5: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/5.jpg)
Examples - calculate the Fourier Series
![Page 6: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/6.jpg)
Examples - calculate the Fourier Series
![Page 7: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/7.jpg)
Examples - calculate the Fourier Series
![Page 8: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/8.jpg)
Examples - calculate the Fourier Series
![Page 9: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/9.jpg)
Even and odd extensions
• For a function f(x) defined on [0,L], the even extension of f(x) is the function
fe(x) =�
f(x) for 0 ≤ x ≤ L,f(−x) for − L ≤ x < 0.
![Page 10: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/10.jpg)
Even and odd extensions
• For a function f(x) defined on [0,L], the even extension of f(x) is the function
fe(x) =�
f(x) for 0 ≤ x ≤ L,f(−x) for − L ≤ x < 0.
• For a function f(x) defined on [0,L], the odd extension of f(x) is the function
fo(x) =�
f(x) for 0 ≤ x ≤ L,−f(−x) for − L ≤ x < 0.
![Page 11: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/11.jpg)
Even and odd extensions
• For a function f(x) defined on [0,L], the even extension of f(x) is the function
fe(x) =�
f(x) for 0 ≤ x ≤ L,f(−x) for − L ≤ x < 0.
• For a function f(x) defined on [0,L], the odd extension of f(x) is the function
fo(x) =�
f(x) for 0 ≤ x ≤ L,−f(−x) for − L ≤ x < 0.
• Because these functions are even/odd, their Fourier Series have a couple simplifying features:
fo(x) =∞�
n=1
bn sinnπx
L
fe(x) =a0
2+∞�
n=1
an cosnπx
L
bn =2L
� L
0f(x) sin
nπx
Ldx
an =2L
� L
0f(x) cos
nπx
Ldx
![Page 12: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/12.jpg)
Even and odd extensions
• For a function f(x) defined on [0,L], the even extension of f(x) is the function
fe(x) =�
f(x) for 0 ≤ x ≤ L,f(−x) for − L ≤ x < 0.
• For a function f(x) defined on [0,L], the odd extension of f(x) is the function
fo(x) =�
f(x) for 0 ≤ x ≤ L,−f(−x) for − L ≤ x < 0.
• Because these functions are even/odd, their Fourier Series have a couple simplifying features:
fo(x) =∞�
n=1
bn sinnπx
L
fe(x) =a0
2+∞�
n=1
an cosnπx
L
bn =2L
� L
0f(x) sin
nπx
Ldx
an =2L
� L
0f(x) cos
nπx
Ldx
![Page 13: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/13.jpg)
Even and odd extensions
• For a function f(x) defined on [0,L], the even extension of f(x) is the function
fe(x) =�
f(x) for 0 ≤ x ≤ L,f(−x) for − L ≤ x < 0.
• For a function f(x) defined on [0,L], the odd extension of f(x) is the function
fo(x) =�
f(x) for 0 ≤ x ≤ L,−f(−x) for − L ≤ x < 0.
• Because these functions are even/odd, their Fourier Series have a couple simplifying features:
fo(x) =∞�
n=1
bn sinnπx
L
fe(x) =a0
2+∞�
n=1
an cosnπx
L
bn =2L
� L
0f(x) sin
nπx
Ldx
an =2L
� L
0f(x) cos
nπx
Ldx
![Page 14: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/14.jpg)
Even and odd extensions
• For a function f(x) defined on [0,L], the even extension of f(x) is the function
fe(x) =�
f(x) for 0 ≤ x ≤ L,f(−x) for − L ≤ x < 0.
• For a function f(x) defined on [0,L], the odd extension of f(x) is the function
fo(x) =�
f(x) for 0 ≤ x ≤ L,−f(−x) for − L ≤ x < 0.
• Because these functions are even/odd, their Fourier Series have a couple simplifying features:
fo(x) =∞�
n=1
bn sinnπx
L
fe(x) =a0
2+∞�
n=1
an cosnπx
L
bn =2L
� L
0f(x) sin
nπx
Ldx
an =2L
� L
0f(x) cos
nπx
Ldx
![Page 15: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/15.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
![Page 16: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/16.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
![Page 17: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/17.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
![Page 18: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/18.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
![Page 19: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/19.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
![Page 20: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/20.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
![Page 21: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/21.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
![Page 22: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/22.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
![Page 23: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/23.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
![Page 24: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/24.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
bn =2L
� L
0f(x) sin
nπx
Ldx
f(x) =∞�
n=1
bn sinnπx
L
![Page 25: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/25.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
bn =2L
� L
0f(x) sin
nπx
Ldx
f(x) =∞�
n=1
bn sinnπx
L
![Page 26: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/26.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
bn =2L
� L
0f(x) sin
nπx
Ldx
f(x) =∞�
n=1
bn sinnπx
Lb1 = 0
![Page 27: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/27.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
bn =2L
� L
0f(x) sin
nπx
Ldx
f(x) =∞�
n=1
bn sinnπx
Lb1 = 0
![Page 28: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/28.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
bn =2L
� L
0f(x) sin
nπx
Ldx
f(x) =∞�
n=1
bn sinnπx
Lb1 = 0b2 �= 0
![Page 29: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/29.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
bn =2L
� L
0f(x) sin
nπx
Ldx
f(x) =∞�
n=1
bn sinnπx
Lb1 = 0b2 �= 0
![Page 30: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/30.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
bn =2L
� L
0f(x) sin
nπx
Ldx
f(x) =∞�
n=1
bn sinnπx
Lb1 = 0b2 �= 0b3 = 0
![Page 31: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/31.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
bn =2L
� L
0f(x) sin
nπx
Ldx
f(x) =∞�
n=1
bn sinnπx
Lb1 = 0b2 �= 0b3 = 0
![Page 32: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/32.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
bn =2L
� L
0f(x) sin
nπx
Ldx
f(x) =∞�
n=1
bn sinnπx
Lb1 = 0b2 �= 0b3 = 0b4 = 0
![Page 33: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/33.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
bn =2L
� L
0f(x) sin
nπx
Ldx
f(x) =∞�
n=1
bn sinnπx
Lb1 = 0b2 �= 0b3 = 0b4 = 0
• bn=0 for n = odd or 4k
![Page 34: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/34.jpg)
Fourier Series for functions with other symmetries
• Find the Fourier Sine Series for f(x):
• Because we want the sine series, we use the odd extension.
• The Fourier Series for the odd extension has an=0 because of the symmetry about x=0.
• What other symmetries does f have?
bn =2L
� L
0f(x) sin
nπx
Ldx
f(x) =∞�
n=1
bn sinnπx
Lb1 = 0b2 �= 0b3 = 0b4 = 0
• bn=0 for n = odd or 4k• Calculate bn
![Page 35: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/35.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) = cos3πx
2
![Page 36: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/36.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) = cos3πx
2
The IC is an eigenvector! Note that it satisfies the BCs.
![Page 37: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/37.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) = cos3πx
2
v3(x) = cos3πx
2
The IC is an eigenvector! Note that it satisfies the BCs.
![Page 38: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/38.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) = cos3πx
2vn(x) = cos
nπx
2
v3(x) = cos3πx
2
The IC is an eigenvector! Note that it satisfies the BCs.
![Page 39: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/39.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) = cos3πx
2vn(x) = cos
nπx
2
v3(x) = cos3πx
2
un(x, t) = eλnt cosnπx
2
The IC is an eigenvector! Note that it satisfies the BCs.
![Page 40: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/40.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) = cos3πx
2vn(x) = cos
nπx
2
v3(x) = cos3πx
2
un(x, t) = eλnt cosnπx
2
The IC is an eigenvector! Note that it satisfies the BCs.
∂
∂tun(x, t) = λneλnt cos
nπx
2
![Page 41: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/41.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) = cos3πx
2vn(x) = cos
nπx
2
v3(x) = cos3πx
2
un(x, t) = eλnt cosnπx
2
The IC is an eigenvector! Note that it satisfies the BCs.
∂
∂tun(x, t) = λneλnt cos
nπx
2∂2
∂x2un(x, t) = −n2π2
4eλnt cos
nπx
2
![Page 42: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/42.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) = cos3πx
2vn(x) = cos
nπx
2
v3(x) = cos3πx
2
un(x, t) = eλnt cosnπx
2
The IC is an eigenvector! Note that it satisfies the BCs.
∂
∂tun(x, t) = λneλnt cos
nπx
2∂2
∂x2un(x, t) = −n2π2
4eλnt cos
nπx
24
4
![Page 43: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/43.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) = cos3πx
2vn(x) = cos
nπx
2
v3(x) = cos3πx
2
un(x, t) = eλnt cosnπx
2
The IC is an eigenvector! Note that it satisfies the BCs.
∂
∂tun(x, t) = λneλnt cos
nπx
2∂2
∂x2un(x, t) = −n2π2
4eλnt cos
nπx
24
4
![Page 44: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/44.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) = cos3πx
2vn(x) = cos
nπx
2
v3(x) = cos3πx
2
un(x, t) = eλnt cosnπx
2
The IC is an eigenvector! Note that it satisfies the BCs.
∂
∂tun(x, t) = λneλnt cos
nπx
2∂2
∂x2un(x, t) = −n2π2
4eλnt cos
nπx
24
4
![Page 45: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/45.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) = cos3πx
2vn(x) = cos
nπx
2
v3(x) = cos3πx
2
un(x, t) = eλnt cosnπx
2
The IC is an eigenvector! Note that it satisfies the BCs.
∂
∂tun(x, t) = λneλnt cos
nπx
2∂2
∂x2un(x, t) = −n2π2
4eλnt cos
nπx
2λn = −n2π2
44
![Page 46: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/46.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) = cos3πx
2vn(x) = cos
nπx
2
v3(x) = cos3πx
2
un(x, t) = eλnt cosnπx
2
The IC is an eigenvector! Note that it satisfies the BCs.
∂
∂tun(x, t) = λneλnt cos
nπx
2∂2
∂x2un(x, t) = −n2π2
4eλnt cos
nπx
2λn = −n2π2
44
u(x, t) = e−9π2t cos3πx
2
So the solution is
![Page 47: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/47.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) =a0
2+∞�
n=1
an cosnπx
L
![Page 48: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/48.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) =a0
2+∞�
n=1
an cosnπx
L
The IC is the sum of eigenvectors!
![Page 49: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/49.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) =a0
2+∞�
n=1
an cosnπx
L
The IC is the sum of eigenvectors!
un(x, t) = eλnt cosnπx
2
![Page 50: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/50.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) =a0
2+∞�
n=1
an cosnπx
L
The IC is the sum of eigenvectors!
un(x, t) = eλnt cosnπx
2λn = −n2π2
![Page 51: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/51.jpg)
Using Fourier Series to solve the Diffusion Equation
ut = 4uxx
du
dx
����x=0,2
= 0
u(x, 0) =a0
2+∞�
n=1
an cosnπx
L
The IC is the sum of eigenvectors!
un(x, t) = eλnt cosnπx
2λn = −n2π2
u(x, t) =a0
2+∞�
n=1
ane−n2π2t cosnπx
2
![Page 52: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/52.jpg)
Using Fourier Series to solve the Diffusion Equation
u(x, 0) = x
ut = 4uxx
du
dx
����x=0,2
= 0
![Page 53: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/53.jpg)
Using Fourier Series to solve the Diffusion Equation
u(x, 0) = x
ut = 4uxx
du
dx
����x=0,2
= 0
bn =(−1)n+14
nπ
u(x, t) =a0
2+∞�
n=1
ane−n2π2t cosnπx
2
u(x, t) =∞�
n=1
bne−n2π2t sinnπx
2
(0 for n odd)
a0 = 1, an = − 8n2π2
for n even(A)
(B)
![Page 54: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/54.jpg)
Using Fourier Series to solve the Diffusion Equation
u(x, 0) = x
ut = 4uxx
du
dx
����x=0,2
= 0
bn =(−1)n+14
nπ
u(x, t) =a0
2+∞�
n=1
ane−n2π2t cosnπx
2
u(x, t) =∞�
n=1
bne−n2π2t sinnπx
2
(0 for n odd)
a0 = 1, an = − 8n2π2
for n even(A)
(B)
![Page 55: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/55.jpg)
Using Fourier Series to solve the Diffusion Equation
u(x, 0) = x
ut = 4uxx
du
dx
����x=0,2
= 0
bn =(−1)n+14
nπ
u(x, t) =a0
2+∞�
n=1
ane−n2π2t cosnπx
2
u(x, t) =∞�
n=1
bne−n2π2t sinnπx
2
(0 for n odd)
a0 = 1, an = − 8n2π2
for n even(A)
(B)
![Page 56: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/56.jpg)
Using Fourier Series to solve the Diffusion Equation
u(x, 0) = x
ut = 4uxx
du
dx
����x=0,2
= 0
bn =(−1)n+14
nπ
u(x, t) =a0
2+∞�
n=1
ane−n2π2t cosnπx
2
u(x, t) =∞�
n=1
bne−n2π2t sinnπx
2
(0 for n odd)
a0 = 1, an = − 8n2π2
for n even(A)
(B)
![Page 57: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/57.jpg)
Using Fourier Series to solve the Diffusion Equation
u(0, t) = u(2, t) = 0
u(x, 0) = x
ut = 4uxx
https://www.desmos.com/calculator/yt7kztckeu
https://www.desmos.com/calculator/wcdvgrveez
![Page 58: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/58.jpg)
Using Fourier Series to solve the Diffusion Equation
u(0, t) = u(2, t) = 0
u(x, 0) = x
ut = 4uxx
bn =(−1)n+14
nπ
u(x, t) =a0
2+∞�
n=1
ane−n2π2t cosnπx
2
u(x, t) =∞�
n=1
bne−n2π2t sinnπx
2
(0 for n odd)
a0 = 1, an = − 8n2π2
for n even(A)
(B)
https://www.desmos.com/calculator/yt7kztckeu
https://www.desmos.com/calculator/wcdvgrveez
![Page 59: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/59.jpg)
Using Fourier Series to solve the Diffusion Equation
u(0, t) = u(2, t) = 0
u(x, 0) = x
ut = 4uxx
bn =(−1)n+14
nπ
u(x, t) =a0
2+∞�
n=1
ane−n2π2t cosnπx
2
u(x, t) =∞�
n=1
bne−n2π2t sinnπx
2
(0 for n odd)
a0 = 1, an = − 8n2π2
for n even(A)
(B)
https://www.desmos.com/calculator/yt7kztckeu
https://www.desmos.com/calculator/wcdvgrveez
![Page 60: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/60.jpg)
Using Fourier Series to solve the Diffusion Equation
u(0, t) = u(2, t) = 0
u(x, 0) = x
ut = 4uxx
bn =(−1)n+14
nπ
u(x, t) =a0
2+∞�
n=1
ane−n2π2t cosnπx
2
u(x, t) =∞�
n=1
bne−n2π2t sinnπx
2
(0 for n odd)
a0 = 1, an = − 8n2π2
for n even(A)
(B)
https://www.desmos.com/calculator/yt7kztckeu
https://www.desmos.com/calculator/wcdvgrveez
![Page 61: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/61.jpg)
Using Fourier Series to solve the Diffusion Equation
u(0, t) = u(2, t) = 0
u(x, 0) = x
ut = 4uxx
bn =(−1)n+14
nπ
u(x, t) =a0
2+∞�
n=1
ane−n2π2t cosnπx
2
u(x, t) =∞�
n=1
bne−n2π2t sinnπx
2
(0 for n odd)
a0 = 1, an = − 8n2π2
for n even(A)
(B)
https://www.desmos.com/calculator/yt7kztckeu
https://www.desmos.com/calculator/wcdvgrveez
![Page 62: Fourier Series examples - even and odd extensions, other …€¦ · Even and odd extensions • For a function f(x) defined on [0,L], the even extension of f(x) is the function](https://reader033.fdocuments.net/reader033/viewer/2022053023/6052c3d4d6a3361bcb4f1f51/html5/thumbnails/62.jpg)
Using Fourier Series to solve the Diffusion Equation
u(0, t) = u(2, t) = 0
u(x, 0) = x
ut = 4uxx
bn =(−1)n+14
nπ
u(x, t) =a0
2+∞�
n=1
ane−n2π2t cosnπx
2
u(x, t) =∞�
n=1
bne−n2π2t sinnπx
2
(0 for n odd)
a0 = 1, an = − 8n2π2
for n even(A)
(B)
• Show Desmos movies.https://www.desmos.com/calculator/yt7kztckeu
https://www.desmos.com/calculator/wcdvgrveez