$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$

Crucial in many problems is formula (3.2.14) rewritten as \begin{equation} u(x,t)=\int _{-\infty}^\infty G(x,y,t) g(y)\,dy. \label{a} \end{equation} with \begin{equation} G(x,y,t)=\frac{1}{2\sqrt{k\pi t}}e^{-\frac{(x-y)^2}{4kt}} \label{b} \end{equation} This formula solves IVP for a heat equation \begin{equation} u_t=ku_{xx} \label{c} \end{equation} with the initial function $g(x)$.

In many problems below for a modified standard problem you need to derive a similar formula albeit with modified $G(x,y,t)$. Consider \begin{equation*} \erf(z)= \frac{2}{\sqrt{\pi}}\int_0^ze^{-z^2}\,dz \tag{Erf}\label{eq-Erf} \end{equation*} as a standard function.

**Problem 1.**
Using method of continuation obtain formula similar to
(\ref{a})-(\ref{b}) for solution of IBVP for a heat equation on
${x>0,t>0}$ with the initial function $g(x)$ and with

- Dirichlet boundary condition $u|_{x=0}=0$;
- Neumann boundary condition $u_x|_{x=0}=0$;

**Problem 2.**
Using method of continuation obtain formula similar to
(\ref{a})-(\ref{b}) for solution of IBVP for a heat equation on
${x>0,t>0}$ with the initial function $g(x)$ and with

- Dirichlet boundary condition on both ends $u|_{x=0}=u|_{x=L}=0$;
- Neumann boundary condition on both ends $u_x|_{x=0}=u_x|_{x=L}=0$;
- Dirichlet boundary condition on one end and Neumann boundary condition on another $u|_{x=0}=u_x|_{x=L}=0$.

**Problem 3.**
Consider heat equation with a convection term
\begin{equation}
u_t+\underbracket{c u_x}_{\text{convection term}} =ku_{xx}.
\label{d}
\end{equation}

Prove that it is obtained from the ordinary heat equation with respect to $U$ by a change of variables $U(x,t)=u(x+ct, t)$. Interpret (\ref{d}) as equation describing heat propagation in the media moving to the right with the speed $c$.

Using change of variables $u(x,t)=U(x-vt,t)$ reduce it to ordinary heat equation and using (\ref{a})-(\ref{b}) for a latter write a formula for solution $u (x,t)$.

Can we use the method of continuation

*directly*to solve IBVP with Dirichlet or Neumann boundary condition at $x>0$ for (\ref{d}) on $\{x>0,t>0\}$? Justify your answer.Plugging $u(x,t)= v(x,t)e^{\alpha x +\beta t}$ with appropriate constants $\alpha,\beta$ reduce (\ref{d}) to ordinary heat equation.

Using (d) write formula for solution of such equation on the half-line or an interval in the case of Dirichlet boundary condition(s). Can we use this method in the case of Neumann boundary conditions? Justify your answer.

**Problem 4.**
Using either formula (\ref{a})-(\ref{b}) or its modification (if
needed)

- Solve IVP for a heat equation (\ref{c}) with $g(x)=e^{-\varepsilon |x|}$; what happens as $\varepsilon \to +0$?
- Solve IVP for a heat equation with convection (\ref{d}) with $g(x)=e^{-\varepsilon |x|}$; what happens as $\varepsilon \to +0$?
- Solve IBVP with the Dirichlet boundary condition for a heat equation (\ref{d}) with $g(x)=e^{-\varepsilon |x|}$; what happens as $\varepsilon \to +0$?
- Solve IBVP with the Neumann boundary condition for a heat equation (\ref{c}) with $g(x)=e^{-\varepsilon |x|}$; what happens as $\varepsilon \to +0$?

**Problem 5.**
Consider a solution of the diffusion equation $u_t=u_{xx}$ in
$[0\le x \le L, 0\le t <\infty]$.

Let \begin{gather*} M(T)= \max _{[0\le x \le L, 0\le t \le T]} u(x,t),\\ m(T)= \min _{[0\le x \le L, 0\le t \le T]} u(x,t). \end{gather*}

- Does $M(T)$ increase or decrease as a function of $T$?
- Does $m(T)$ increase or decrease as a function of $T$?

**Problem 6.**
The purpose of this exercise is to show that the maximum principle is
not true for the equation $u_t=xu_{xx}$ which has a coefficient which changes sign.

- Verify that $u=-2xt-x^2$ is a solution.
- Find the location of its maximum in the closed rectangle $[-2\le x\le 2, 0\le t\le 1]$.
- Where precisely does our proof of the maximum principle break down for this equation?

**Problem 7.**

Consider the heat equation on $J=(-\infty,\infty)$ and prove that an

*energy*\begin{equation} E(t)=\int_J u^2 (x,t)\,dx \label{eq-e} \end{equation} does not increase; further, show that it really decreases unless $u(x,t)=\const$;Consider the heat equation on $J=(0,l)$ with the Dirichlet or Neumann boundary conditions and prove that an $E(t)$ does not increase; further, show that it really decreases unless $u(x,t)=\const$;

Consider the heat equation on $J=(0,l)$ with the Robin boundary conditions \begin{gather} u_x(0,t)-a_0u(0,t)=0,\\ u_x(L,t)+a_L u(L,t)=0. \end{gather} If $a_0>0$ and $a_l>0$, show that the endpoints contribute to the decrease of $E(t)=\int_0^L u^2 (x,t)\,dx$.

This is interpreted to mean that part of the *energy* is lost at the
boundary, so we call the boundary conditions *radiating* or *dissipative*.

**Hint.** To prove decrease of $E(t)$ consider it derivative by $t$, replace $u_t$ by $ku_{xx}$ and integrate by parts.

**Remark 1.**

In the case of heat (or diffusion) equation an *energy* given by (\ref{eq-e}) is rather mathematical artefact.

**Problem 8.**
Find a selfâ€“similar solution $u$ of
\begin{equation}
u_t = (u u_x)_x \qquad -\infty< x <\infty , t>0
\label{f}
\end{equation}
with finite $\int_{-\infty}^\infty u\,dx$.