A pulse and its Fourier transform
Partial inverse Fourier transform defined by integrating only from -L to L. Plotted for L = 1, L = 10 and L = 100
ft := (a,L) -> int(fhat(a)*exp(I*omega*x),omega=-L..L)/(2*Pi):
p0 := plot(ft(2,1),x=-10..10,color=red):p1 := plot(ft(2,10),x=-10..10,color=red): p2 := plot(ft(2,100),x=-10..10,color=blue):
display([p0,p1,p2],insequence=true);
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Solution of heat equation with pulse as initial condition, plotted for several values of time.
with(plots):
u := (x,tt) -> (erf((x+a)/(2*alpha*sqrt(tt)))-erf((x-a)/(2*alpha*sqrt(tt))))/2:
f := x -> Heaviside(x+a)-Heaviside(x-a):
a := 2: p0:=display(plot(f(x),x=-3..3,color=red),textplot([8,0.8,t=0],font=[Helvetica,18]),view=[-15..15,0..1]):
a := 2: alpha := 1:clrs:=[blue,green,magenta,violet,brown,red]:
for k from 1 to 6 do
Digits:=1;tt := evalf(10^(k-4));Digits:=10;
cat(p,k) := display(plot(u(x,tt),x=-2*a-5*sqrt(tt)..2*a+5*sqrt(tt),u=0..1.05,color=clrs[k],view=[-15..15,0..1]),
textplot([8,0.8,t=tt],font=[Helvetica,18]));
end do:
display([p0,p1,p2,p3,p4,p5,p6],title="u(x,t) for t = 0.0, 0.001, 0.01, 0.1, 1.0, 10.0, and 100.0",titlefont=[Helvetica,14],insequence=true);
display([p6],title="u(x,t) for t = 100",titlefont=[Helvetica,14],view=[-45..45,0..0.15]);
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