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Solved Problems

00
0
2
X = −λ X, X(0) = X (L) = 0.
The problem for X is a regular Sturm-Liouville eigenvalue problem. From the
Rayleigh quotient,

0 L
0 2
0 2
− [φφ ] + R L (φ ) dx R L (φ ) dx
2
λ = 0 0 0
R L φ dx = R L φ dx
2
2
0 0
2
we see that there are only positive eigenvalues. For λ > 0 the general solution of
the ordinary differential equation is
X = a cos(λx) + a sin(λx).
2
1
The solution that satisfies the left boundary condition is

X = a sin(λx).

For non-trivial solutions, the right boundary condition imposes the constraint,

cos (λL) = 0,

π 1
λ = n − , n ∈ N.
L 2
The eigenvalues and eigenfunctions are

(2n − 1)π (2n − 1)πx
λ = , X = sin , n ∈ N.
n
n
2L 2L
The differential equation for T becomes

(2n − 1)π 2
00
T = −c 2 T,
2L

which has the two linearly independent solutions,

(2n − 1)cπt (2n − 1)cπt
T n (1) = cos , T n (2) = sin .
2L 2L

The eigenvalues and eigen-solutions of the partial differential equation are,

(2n − 1)π
λ = , n ∈ N,
n
2L

(2n − 1)πx (2n − 1)cπt
u (1) = sin cos ,
n
2L 2L

(2n − 1)πx (2n − 1)cπt
u (2) = sin sin .
n
2L 2L

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