Let λ1<λ2<…λn… and y1(x),y2(x),…yn(x)… be the eigenvalues and corresponding eigenfunctions for the Sturm-Liouville system
Lyn=λnw(x)yn,
where
Ly≡dxd(−p(x)dxdy)+q(x)y,
with p(x)>0 and w(x)>0. The boundary conditions on y are that y(0)=y(1)=0.
Show that two distinct eigenfunctions are orthogonal in the sense that
∫01wynymdx=δnm∫01wyn2dx.
Show also that if y has the form
y=n=1∑∞anyn,
with an being independent of x, then
∫01wy2dx∫01yLydx≥λ1
Assuming that the eigenfunctions are complete, deduce that a solution of the diffusion equation,
∂t∂y=−w1Ly
that satisfies the boundary conditions given above is such that
21dtd(∫01wy2dx)≤−λ1∫01wy2dx.