Chapter 1 - Explorations
1.1.3 - Approximating Fourier's Solution
reset();
var('x,w');
def z(x,w,a):
var('k');
s = 0;
for k in [1..len(a)]:
s += a[k-1]*exp((1/2*(2*k-1)*(-pi)*w))*(cos(1/2*(2*k-1)*pi*x));
return s;
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a is the list of coefficients. The length of a determines the number of terms in the series.
a = [1, -1/2, 1/3, -1/4, 1/5];
z(x,w,a)
1/5*e^(-9/2*pi*w)*cos(9/2*pi*x) - 1/4*e^(-7/2*pi*w)*cos(7/2*pi*x) + 1/3*e^(-5/2*pi*w)*cos(5/2*pi*x) - 1/2*e^(-3/2*pi*w)*cos(3/2*pi*x) + e^(-1/2*pi*w)*cos(1/2*pi*x) 1/5*e^(-9/2*pi*w)*cos(9/2*pi*x) - 1/4*e^(-7/2*pi*w)*cos(7/2*pi*x) + 1/3*e^(-5/2*pi*w)*cos(5/2*pi*x) - 1/2*e^(-3/2*pi*w)*cos(3/2*pi*x) + e^(-1/2*pi*w)*cos(1/2*pi*x) |
The first command produces a 3-dimensional plot of z(x,w), the second produces the cross-section parallel to the w-axis at x=0, and the third produces the cross-section above the x-axis.
plot3d(lambda x,w: z(x,w,a), (x,-1,1), (w,0,2)).show();
plot(lambda w: z(0,w,a), (w,0,2)).show();
plot(lambda x: z(x,0,a), (x,-1,1)).show();
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This function is the first n terms of Fourier's approximation to the constant function 1.
def FS(n,x):
return 4/pi*sum((-1)^(k-1)/(2*k-1)*cos(1/2*(2*k-1)*pi*x), k, 1, n);
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Plot Fourier's approximation with 6 terms.
plot(lambda x: FS(6,x), (x,-1,1)).show();
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This function is the first n terms approximation of the solution to the heat equation with the boundary function as the constant function 1.
def FSS(n,x,w):
return 4/pi*sum((-1)^(k-1)/(2*k-1)*exp(1/2*(-2*k+1)*pi*w)*cos(1/2*(2*k-1)*pi*x), k, 1, n);
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Plot the 6-term solution to the heat equation.
plot3d(lambda x,w: FSS(6,x,w), (x,-1,1), (w,0,2)).show();
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