Math 6410 • ODE

Consider the following inhomogeneous initial-Neumann problem:
\[
\begin{aligned}
  u_t &= Du_{xx}+\alpha tx, && 0<x<\pi,\quad 0<t,\\
  u(x,0) &= 1, && 0<x<\pi,\\
  u_x(0,t)&=u_x(\pi,t)=0, && 0<t;
\end{aligned}
\]
where \(\alpha\) is a constant.
\begin{enumerate}
\item[(a)] Determine the eigenfunctions of the homogeneous equation
\[
  u_{xx}=\lambda u,\qquad 0\le x\le\pi,\qquad u_x(0)=u_x(\pi)=0.
\]
\item[(b)] Solve the inhomogeneous initial-Neumann problem by carrying out an
eigenfunction expansion of \(u(x,t)\) in terms of the eigenfunctions obtained in
part (a). That is, denoting the eigenfunctions by \(v_k\), integer \(k\), set
\[
  u(x,t)=\sum_{k\ge0} c_k(t)v_k(x),
\]
and determine the time-dependent coefficients \(c_k(t)\).
\item[(c)] Give a physical interpretation of the solution in the limit \(\alpha\to0\).
\end{enumerate}