$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ###[Problems to Section 3.2](id:sect-3.2.P) > 1. [Problem 1](#problem-3.2.P.1) > 2. [Problem 2](#problem-3.2.P.2) > 3. [Problem 3](#problem-3.2.P.3) > 4. [Problem 4](#problem-3.2.P.4) > 5. [Problem 5](#problem-3.2.P.5) > 6. [Problem 6](#problem-3.2.P.6) > 7. [Problem 7](#problem-3.2.P.7) > 8. [Problem 8](#problem-3.2.P.8) Crucial in many problems is formula ([3.2.14](./S3.2.html#mjx-eqn-eq-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.](id:problem-3.2.P.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 a. Dirichlet boundary condition $u|\_{x=0}=0$; b. Neumann boundary condition $u\_x|\_{x=0}=0$; **[Problem 2.](id:problem-3.2.P.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 a. Dirichlet boundary condition on both ends $u|\_{x=0}=u|\_{x=L}=0$; b. Neumann boundary condition on both ends $u\_x|\_{x=0}=u\_x|\_{x=L}=0$; c. Dirichlet boundary condition on one end and Neumann boundary condition on another $u|\_{x=0}=u\_x|\_{x=L}=0$. **[Problem 3.](id:problem-3.2.P.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} a. 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 $c$. b. 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 $u (x,t)$. c. 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. d. Plugging $u(x,t)= v(x,t)e^{\alpha x +\beta t}$ with appropriate constants $\alpha,\beta$ reduce (\ref{d}) to ordinary heat equation. e. 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.](id:problem-3.2.P.4)** Using either formula (\ref{a})-(\ref{b}) or its modification (if needed) a. Solve IVP for a heat equation (\ref{c}) with $g(x)=e^{-\varepsilon |x|}$; what happens as $\varepsilon \to +0$? b. Solve IVP for a heat equation with convection (\ref{d}) with $g(x)=e^{-\varepsilon |x|}$; what happens as $\varepsilon \to +0$? c. Solve IBVP with the Dirichlet boundary condition for a heat equation (\ref{d}) with $g(x)=e^{-\varepsilon |x|}$; what happens as $\varepsilon \to +0$? d. 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.](id:problem-3.2.P.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\*} a. Does $M(T)$ increase or decrease as a function of $T$? b. Does $m(T)$ increase or decrease as a function of $T$? **[Problem 6.](id:problem-3.2.P.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. a. Verify that $u=-2xt-x^2$ is a solution. b. Find the location of its maximum in the closed rectangle $[-2\le x\le 2, 0\le t\le 1]$. 3. Where precisely does our proof of the maximum principle break down for this equation? **[Problem 7.](id:problem-3.2.P.7)** a. 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 $u(x,t)=\const$; b. 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 $u(x,t)=\const$; c. 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.](id:remark-3.2.P.1)** In the case of heat (or diffusion) equation an *energy* given by (\ref{eq-e}) is rather mathematical artefact. **[Problem 8.](id:problem-3.2.P.8)** Find a self–similar solution $u$ of \begin{equation} u\_t = (u u\_x)\_x \qquad -\infty\< x \<\infty , t>0 \label{f} \end{equation} with finite $\int_{-\infty}^\infty u\,dx$. ______________ [$\Uparrow$](../contents.html) [$\uparrow$](./S3.2.html) [$\Rightarrow$](../Chapter4/S4.1.html)