$\newcommand{\const}{\mathrm{const}}$ $\newcommand{\erf}{\operatorname{erf}}$
Deadline Thursday, February 5.
Crucial in many problems is formula (14) rewritten as \begin{equation} u(x,t)=\int _{-\infty}^\infty G(x,y,t) g(y)\,dy. \label{eq-HA3.1} \end{equation} with \begin{equation} G(x,y,t)=\frac{1}{2\sqrt{k\pi t}}e^{-\frac{(x-y)^2}{4kt}} \label{eq-HA3.2} \end{equation} This formula solves IVP for a heat equation \begin{equation} u_t=ku_{xx} \label{eq-HA3.3} \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)=\sqrt{\frac{2}{\pi}}\int_0^ze^{-z^2/2}\,dz \tag{Erf}\label{eq-HA3.Erf} \end{equation*} as a standard function.
Using method of continuation obtain formula similar to (\ref{eq-HA3.1})-(\ref{eq-HA3.2}) for solution of IBVP for a heat equation on ${x>0,t>0}$ with the initial function $g(x)$ and with
Consider heat equation with a convection term \begin{equation} u_t+\underbracket{v u_x}_{\text{convection term}} =ku_{xx}. \label{eq-HA3.4} \end{equation}
Using either formula (\ref{eq-HA3.1})-(\ref{eq-HA3.2}) or its modification (if needed)
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*}
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 variable coefficient.
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-HA3.5} \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_lu(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-HA3.5}) is rather mathematical artefact.