Review of separation of variables
Recall the standard methods for a PDE such as utt = c2 uxx (from MATH2401):
• Look for a solution of the form u(x, t) = X(x)T (t)
• Substitute in and arrange so all the x dependence is on one side and all the t dependence on
the other: thus each side must be a constant
• Assuming the value of the constant, solve each resulting ODE
• Allow the boundary conditions to rule out some values of the constant
• Determine all possible values of the constant (the eigenvalues of the system) from the zero
boundary conditions
• The solution is a linear combination of the resulting eigenfunctions X(x)T (t)
• Determine the remaining constants from the nonzero boundary conditions or initial conditions.
Familiar cases here involve Fourier series.
Review of Fourier series
Orthogonality
Remember the trigonometric formulae:
1
cos a cos b = 2 (cos(a − b) + cos(a + b)) ,
1
sin a sin b = 2 (cos(a − b) − cos(a + b)) ,
1
sin a cos b = 2 (sin(a + b) − sin(a − b)) .
We can use them to show:
L L
cos(mπx/L) cos(nπx/L) dx = sin(mπx/L) sin(nπx/L) dx = Lδmn ,
−L −L
L
sin(mπx/L) cos(nπx/L) dx = 0.
−L
We say that the set of functions {sin(nπx/L), cos(nπx/L)} , n = 0, 1, 2, . . . are orthogonal with
L
respect to the inner product (f, g) = −L f (x)g(x) dx.
Fourier Coefficients
The results above imply that if we can write
∞
a0
f (x) = + an cos(nπx/L) + bn sin(nπx/L)
2 n=1
then on multiplying through, for example, by cos(mπx/L) and integrating over the interval [−L, L],
only one term survives on the right hand side and the result is Lam . Similarly bm can be found and
we have
1 L 1 L
an = f (x) cos(nπx/L) dx, bn = f (x) sin(nπx/L) dx. (1)
L −L L −L
It is in fact possible to show that you can in fact write f (x) in this way.
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Graphs
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0.5
-3 -2 -1 1 2 3
-0.5
-1
Graphs of sin nx, n = 1, 2, 3, 4, 5
1
0.5
-3 -2 -1 1 2 3
-0.5
-1
Graphs of cos nx, n = 0, 1, 2, 3, 4, 5
Note how one set is odd and the other even and how their rate of oscillation increases with n.
Large values of n represent the rapidly changing components of f (x) in its Fourier series. Such
components can be expected if f (x) has a discontinuity.
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