Multiple Fourier Series

8
Multiple Fourier Series William O. Bray Herein we will explore a few examples of multiple Fourier sine series on the square D = @0, ΠD · @0, ΠD. These have the form: f Hx, yL ~ n,m f Hn, mL sinHnxL sinHnyL, f Hn, mL = 4 Π 2 D f Hx, yL sinHnxL sinHmyL dA Hx,yL The following command forms the square partial sums of the Fourier sine series with Fourier coefficients given as a table. In[23]:= parsum@x_, y_, N_, a_D := Sum@a@@n, mDD Sin@nxD Sin@myD, 8n, 1, N<, 8m, 1, N<D Example 1 In[24]:= f1@x_, y_D := x HPi - xL y HPi - yL In[25]:= Plot3D@f1@x, yD, 8x, 0, Pi<, 8y, 0, Pi<, AxesLabel 8x, y, z<D Out[25]= 0 1 2 3 x 0 1 2 3 y 0 2 4 6 z This function is actually a product of a function of x with a function of y, hence the double Fourier sine coeffi- cients are the products of the one dimensional Fourier coefficients. We can implement this as follows.

Transcript of Multiple Fourier Series

Page 1: Multiple Fourier Series

Multiple Fourier SeriesWilliam O. Bray

Herein we will explore a few examples of multiple Fourier sine series on the square D = @0, ΠD ´ @0, ΠD. Thesehave the form:

f Hx, yL ~ Ún,m f`Hn, mL sinHn xL sinHn yL, f

`Hn, mL =

4

Π2 ÙDf Hx, yL sinHn xL sinHm yL dAHx,yL

The following command forms the square partial sums of the Fourier sine series with Fourier coefficientsgiven as a table.

In[23]:=parsum@x_, y_, N_, a_D :=

Sum@a@@n, mDD Sin@n xD Sin@m yD, 8n, 1, N<, 8m, 1, N<D

Example 1In[24]:=

f1@x_, y_D := x HPi - xL y HPi - yL

In[25]:=Plot3D@f1@x, yD, 8x, 0, Pi<, 8y, 0, Pi<, AxesLabel ® 8x, y, z<D

Out[25]=

0

1

2

3

x

0

1

2

3

y

0

2

4

6

z

This function is actually a product of a function of x with a function of y, hence the double Fourier sine coeffi-cients are the products of the one dimensional Fourier coefficients. We can implement this as follows.

Page 2: Multiple Fourier Series

In[26]:=g@k_D := FourierSinCoefficient@x HPi - xL, x, kD

In[27]:=f1hat@n_, m_D := g@nD g@mD

In[28]:=f1hat@n, mD

Out[28]= 16 H-1 + H-1LmL H-1 + H-1LnL

m3 n3Π

2

In[29]:=a1 = Table@f1hat@n, mD, 8n, 1, 10<, 8m, 1, 10<D;

In[30]:=Plot3D@parsum@x, y, 10, a1D, 8x, 0, Pi<, 8y, 0, Pi<D

Out[30]=

0

1

2

3 0

1

2

3

0

2

4

6

Visually, we cannot see much difference between the partial sum plot and that of the actual function. Thefollowing is a depiction of the error.

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Page 3: Multiple Fourier Series

In[31]:=Plot3D@Abs@f1@x, yD - parsum@x, y, 10, a1DD,

8x, 0, Pi<, 8y, 0, Pi<, PlotPoints ® 40D

Out[31]=

0

1

2

3 0

1

2

3

0.000

0.005

0.010

Notice the growth in error along the boundaries; undoubtably this is due to the fact that the partial derivativesof the function are discontinuous across the boundary (think in terms of the periodic extension of this function).

Example 2In[32]:=

f2@x_, y_D := Piecewise@88x y, 0 £ x £ Pi � 2 && 0 £ y £ Pi � 2<,8x HPi - yL, 0 £ x < Pi � 2 && Pi � 2 £ y £ Pi<,8HPi - xL y, Pi � 2 £ x £ Pi && 0 £ y £ Pi � 2<,8HPi - xL HPi - yL, Pi � 2 £ x £ Pi && Pi � 2 £ y £ Pi<<D

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Page 4: Multiple Fourier Series

In[33]:=Plot3D@f2@x, yD, 8x, 0, Pi<, 8y, 0, Pi<D

Out[33]=

0

1

2

3 0

1

2

3

0

1

2

In[34]:=Integrate@f2@x, yD Sin@n xD Sin@m yD, 8x, 0, Pi<, 8y, 0, Pi<D

Out[34]= 1

m2 n2K4 SinB

m Π

2F SinB

n Π

2F - 2 Sin@m ΠD SinB

n Π

2F -

2 SinBm Π

2F Sin@n ΠD + Sin@m ΠD Sin@n ΠDO

In[35]:=a2 = H4 � Pi^2L Table@%, 8n, 1, 10<, 8m, 1, 10<D;

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Page 5: Multiple Fourier Series

In[36]:=Plot3D@parsum@x, y, 10, a2D, 8x, 0, Pi<, 8y, 0, Pi<D

Out[36]=

0

1

2

3 0

1

2

3

0.0

0.5

1.0

1.5

2.0

For a look at the error, consider the following.

In[37]:=Plot3D@Abs@f2@x, yD - parsum@x, y, 10, a2DD,

8x, 0, Pi<, 8y, 0, Pi<, PlotRange ® Full, PlotPoints ® 25D

Out[37]=

0

1

2

3 0

1

2

3

0.00

0.05

0.10

0.15

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Page 6: Multiple Fourier Series

Example 3In[38]:=

f3@x_, y_D :=

Piecewise@881, 0 £ x £ Pi && 0 £ y £ x<, 80, 0 £ x £ Pi && x < y £ Pi<<D

In[39]:=Plot3D@f3@x, yD, 8x, 0, Pi<, 8y, 0, Pi<D

Out[39]=

0

1

2

3 0

1

2

3

0.0

0.5

1.0

In[40]:=Integrate@Sin@n xD Sin@m yD, 8x, 0, Pi<, 8y, 0, x<D

Out[40]= 1 - Cos@n ΠD

m n+

n - n Cos@m ΠD Cos@n ΠD - m Sin@m ΠD Sin@n ΠD

m3- m n2

In[41]:=a3 =

Table@H4 � Pi^2L Integrate@Sin@n xD Sin@m yD, 8x, 0, Pi<, 8y, 0, x<D,8n, 1, 15<, 8m, 1, 15<D;

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Page 7: Multiple Fourier Series

In[42]:=Plot3D@parsum@x, y, 5, a3D, 8x, 0, Pi<, 8y, 0, Pi<D

Out[42]=

0

1

2

3 0

1

2

3

0.0

0.5

1.0

In[43]:=Plot3D@parsum@x, y, 10, a3D, 8x, 0, Pi<, 8y, 0, Pi<D

Out[43]=

0

1

2

3 0

1

2

3

0.0

0.5

1.0

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In[44]:=Plot3D@parsum@x, y, 15, a3D, 8x, 0, Pi<, 8y, 0, Pi<, PlotPoints ® 30D

Out[44]=

0

1

2

3 0

1

2

3

0.0

0.5

1.0

A view of the partial sums relative to the periodic extension of this function is seen as follows.

In[45]:=Plot3D@parsum@x, y, 15, a3D, 8x, 0, 2 Pi<, 8y, 0, 2 Pi<D

Out[45]=

0

2

4

6 0

2

4

6

-1

0

1

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