편미분방정식(Partial Differential Equation, PDE)

Fourier Series

Pages 479-480b

Fourier Series Example(Sawtooth wave)

π+= xxf )( ππ <<− x)()2( xfxf =+ π

if

and

Find the Fourier series of the function

Solution of Fourier series

)3sin312sin

21(sin2)( ⋅⋅⋅+−+−+= xxxxf π

∫=L

n dxLxnxf

Lb 0sin)(2 π

ππ

πn

nnxdxxbn cos2sin2

0−== ∫

xL

nxfn

nb πsin)(1∑∞

=

=

π=)0(f

Euler formula

Graph of Function f(x) by Matlab>> x=[-12:0.1:12];>> y=pi;>> for(n=1:1000)y=y+(-2/n)*cos(n*pi)*sin(n*x);end>> plot(x,y)

Matlab Program for Partial Sums S1,S2,S3,S20

>> x=-pi:0.01:pi;>> a=pi;>> for n=1:1:1;a=a+(-2/n)*cos(n*pi)*sin(n*x);end>> subplot(2,2,1);>> plot(x,a)>> b=pi;>> for n=1:1:2;b=b+(-2/n)*cos(n*pi)*sin(n*x);end>> subplot(2,2,2);>> plot(x,b)

>> c=pi;>> for n=1:1:3;c=c+(-2/n)*cos(n*pi)*sin(n*x);end>> subplot(2,2,3);>> plot(x,c)>> d=pi;>> for n=1:1:20;d=d+(-2/n)*cos(n*pi)*sin(n*x);end>> subplot(2,2,4);>> plot(x,d)

Partial sums S1,S2,S3,S20

Partial sums S1,S2,S3,S20• >> x=-pi:0.01:pi;

• >> a=pi;

• >> for n=1:1:1;

• a=a+(-2/n)*cos(n*pi)*sin(n*x);

• end

• >> b=pi;

• >> for n=1:1:2;

• b=b+(-2/n)*cos(n*pi)*sin(n*x);

• end

• >> c=pi;

• >> for n=1:1:3;

• c=c+(-2/n)*cos(n*pi)*sin(n*x);

• end

• >> d=pi;

• >> for n=1:1:20;

• d=d+(-2/n)*cos(n*pi)*sin(n*x);

• end

• >> plot(x,[a; b; c; d])

Separation of Variables and Fourier Series

⎪⎪⎩

⎪⎪⎨

⎧

<<−

<<=

)2

()(2

)2

0(2

)(LxLxL

Lk

LxxLk

xf

Wave Equation with the following initial condition for the string fixed at both ends at x = 0 and L

PDE Wave Equation Analytical Solution

Matlab Program for Wave Equation Solution

)(*21&)(*

21

)(*21)(*

21),(

21 ctxfuctxfu

ctxfctxftxu

+=−=

++−=

∑∑

Graphs from the Matlab Program

Heat or Diffusion Equation