편미분방정식 (partial differential equation, pde) · 2 ( ) (2) 2 (0 2 ( ) x l l l x l k l x x...
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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