3.II.15D

Methods
Part IB, 2008

Let λ1<λ2<λn\lambda_{1}<\lambda_{2}<\ldots \lambda_{n} \ldots and y1(x),y2(x),yn(x)y_{1}(x), y_{2}(x), \ldots y_{n}(x) \ldots be the eigenvalues and corresponding eigenfunctions for the Sturm-Liouville system

Lyn=λnw(x)yn,\mathcal{L} y_{n}=\lambda_{n} w(x) y_{n},

where

Lyddx(p(x)dydx)+q(x)y,\mathcal{L} y \equiv \frac{d}{d x}\left(-p(x) \frac{d y}{d x}\right)+q(x) y,

with p(x)>0p(x)>0 and w(x)>0w(x)>0. The boundary conditions on yy are that y(0)=y(1)=0y(0)=y(1)=0.

Show that two distinct eigenfunctions are orthogonal in the sense that

01wynymdx=δnm01wyn2dx.\int_{0}^{1} w y_{n} y_{m} d x=\delta_{n m} \int_{0}^{1} w y_{n}^{2} d x .

Show also that if yy has the form

y=n=1anyn,y=\sum_{n=1}^{\infty} a_{n} y_{n},

with ana_{n} being independent of xx, then

01yLydx01wy2dxλ1\frac{\int_{0}^{1} y \mathcal{L} y d x}{\int_{0}^{1} w y^{2} d x} \geq \lambda_{1}

Assuming that the eigenfunctions are complete, deduce that a solution of the diffusion equation,

yt=1wLy\frac{\partial y}{\partial t}=-\frac{1}{w} \mathcal{L} y

that satisfies the boundary conditions given above is such that

12ddt(01wy2dx)λ101wy2dx.\frac{1}{2} \frac{d}{d t}\left(\int_{0}^{1} w y^{2} d x\right) \leq-\lambda_{1} \int_{0}^{1} w y^{2} d x .