Boundary value problem

Question

Consider the boundary value problem

\begin{align} \frac{du}{dt}&= \frac{d^2u}{dx^2} , \\ u(0,t)&=0 \\ u(L,t)&=0 \\ u(x,0)&=f(x) \end{align}

I know how to solve it using variation of parameters and the equilibrium occurs when $\frac{du}{dt}=0$.

I found that the equilibrium solution tends to zero as $t$ tends to infinity, but what is the physical interpretation of the problem? Also, assuming that $L> \pi$ and $L \neq n \pi$, how do you determine the equilibrium solution and show that as $t$ tends to infinity the analytical solution approaches equilibrium?

Answer 1

This equation models, for example, a diffusion phenomenon when at the initial moment you have some distribution $u=f(x)$ and for $t>0$ the initial inhomogeneity tends to some equilibrium (homogeneity of $u(x)$). Factually the zero boundary conditions mean that everything not only become homogeneous, but also zero due to non zero flux $j\propto\partial u/\partial x$ through the boundaries.

Whatever $L$ is, one can represent the solution as a spectral series: $$u(x,t)=\sum_n A_n \sin(\pi n x/L)\rm{e}^{-\lambda_n t}$$ with $\lambda_n\propto n^2/L^2$. Each exponential is fading with time, so when $t\to\infty$ the whole solution tends to zero.