Math 6410 • ODE

Consider the integral
\[
u(x,t)=\frac{1}{2\sqrt{\pi t}}\int_{-\infty}^{\infty}
f(\xi)e^{-(x-\xi)^2/4t}\,d\xi.
\]

\begin{enumerate}
\item[(a)] Prove that if \(f(x)\) is continuous and bounded for all \(x\) then \(u\) is a
solution of the initial value problem
\[
u_t=u_{xx}\quad\text{for all }-\infty<x<\infty,\ t>0.
\]
\[
u(x,0)=f(x)\quad\text{for all }-\infty<x<\infty.
\]
Formulate clearly what you mean by a solution.

\item[(b)] Show how the results of part a) can be used to solve the initial-boundary
value problem
\[
u_t=u_{xx}\quad\text{for all }0<x<1,\ t>0,
\]
\[
u(x,0)=g(x)\quad\text{for }0\le x\le1,
\]
\[
u(0,t)=u(1,t)=0\quad\text{for }t\ge0.
\]
Define what you mean by a solution and state what conditions on \(g\) guarantee that
such a solution exists.
\end{enumerate}

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\textit{DE Qualifying Examination 1992}