3.2.P. Problems to Section 3.2

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

### Problems to Section 3.2

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. Solve Cauchy problem for a heat equation \begin{align} &u_t =ku_{xx}&&t>0,\ -\infty< x <\infty \label{A}\\ &u|_{t=0}= g(x) \label{B} \end{align} with

1. g(x)=\left\{\begin{aligned} &1 &&|x|<1, \&0 &&|x|\ge 1;\end{aligned}\right.;
2. g(x)=\left\{\begin{aligned} &1-|x| &&|x|<1, \&0 &&|x|\ge 1\end{aligned}\right.;
3. $g(x)=e^{-a |x|}$;
4. $g(x)=x e^{-a |x|}$;
5. $g(x)=|x| e^{-a|x|}$;
6. $g(x)=x^2 e^{-a|x|}$;
7. $g(x)=e^{-ax^2}$;
8. $g(x)=xe^{-ax^2}$;
9. $g(x)=x^2 e^{-ax^2}$.

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

1. Dirichlet boundary condition $u|_{x=0}=0$;
2. Neumann boundary condition $u_x|_{x=0}=0$.

Problem 3. Solve IBVP problem for a heat equation \begin{align} &u_t =ku_{xx} &&t>0,\ 0< x <\infty \label{C}\\ &u|_{t=0}= g(x) \label{D}\\ &u|_{x=0}=h(t) \label{E} \end{align} with

1. $g(x)=1$,     $h(t)=0$;
2. g(x)=\left\{\begin{aligned} &1 &&x<1, \&0 &&x\ge 1,\end{aligned}\right.     $h(t)=0$;
3. g(x)=\left\{\begin{aligned} &1-x &&x<1, \&0 &&x\ge 1,\end{aligned}\right.     $h(t)=0$;
4. $g(x)=e^{-ax}$,     $h(t)=0$;
5. $g(x)=x e^{-ax}$,     $h(t)=0$;
6. $g(x)= e^{-ax^2}$,     $h(t)=0$;
7. $g(x)= x e^{-ax^2}$,     $h(t)=0$;
8. $g(x)= x^2e^{-ax^2}$,     $h(t)=0$;
9. $g(x)=0$,     $h(t)=1$;
10. $g(x)=0$,     h(t)=\left\{\begin{aligned} &1 &&t<1, \&0 &&t\ge 1.\end{aligned}\right.

Problem 4. \begin{align} &u_t =ku_{xx} &&t>0,\ 0< x <\infty \label{F}\\ &u|_{t=0}= g(x) \label{G}\\ &u_x|_{x=0}=h(t) \end{align} with

1. $g(x)=1$,     $h(t)=0$;
2. g(x)=\left\{\begin{aligned} &1 &&x<1, \&0 &&x\ge 1,\end{aligned}\right.     $h(t)=0$;
3. g(x)=\left\{\begin{aligned} &1-x &&x<1, \&0 &&x\ge 1,\end{aligned}\right.     $h(t)=0$;
4. $g(x)=e^{-x}$,     $h(t)=0$;
5. $g(x)=x e^{-x}$,     $h(t)=0$;
6. $g(x)= e^{-ax^2}$,     $h(t)=0$;
7. $g(x)= x e^{-ax^2}$,     $h(t)=0$;
8. $g(x)= x^2e^{-ax^2}$,     $h(t)=0$;
9. $g(x)=0$,     $h(t)=1$;
10. $g(x)=0$,     h(t)=\left\{\begin{aligned} &1 &&t<1, \&0 &&t\ge 1.\end{aligned}\right.

Problem 5. 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

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

Problem 6. Consider heat equation with a convection term \begin{equation} u_t+\underbracket{c u_x}_{\text{convection term}} =ku_{xx}. \label{d} \end{equation}

1. 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 $c$.
2. 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 $u (x,t)$.
3. 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.
4. Plugging $u(x,t)= v(x,t)e^{\alpha x +\beta t}$ with appropriate constants $\alpha,\beta$ reduce (\ref{d}) to ordinary heat equation.
5. 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 7. Using either formula (\ref{a})-(\ref{b}) or its modification (if needed)

1. Solve IVP for a heat equation (\ref{c}) with $g(x)=e^{-\varepsilon |x|}$; what happens as $\varepsilon \to +0$?2. Solve IVP for a heat equation with convection (\ref{d}) with $g(x)=e^{-\varepsilon |x|}$; what happens as $\varepsilon \to +0$?3. Solve IBVP with the Dirichlet boundary condition for a heat equation (\ref{d}) with $g(x)=e^{-\varepsilon |x|}$; what happens as $\varepsilon \to +0$?4. 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 8. 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*}

1. Does $M(T)$ increase or decrease as a function of $T$?
2. Does $m(T)$ increase or decrease as a function of $T$?

Problem 9. 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.

1. Verify that $u=-2xt-x^2$ is a solution.
2. 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 10.

1. 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 $u(x,t)=\const$;
2. 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 $u(x,t)=\const$;
3. 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 11. Find a self–similar solution $u$ of \begin{equation} u_t = (u^m u_x)_x \qquad -\infty< x <\infty ,\ t>0 \label{f} \end{equation} with finite $\int_{-\infty}^\infty u\,dx$ and $m>0$.

Problem 12. Find a self–similar solution $u$ of \begin{equation} u_t = iku_{xx} \qquad -\infty< x <\infty ,\ -\infty < t < \infty. \label{h} \end{equation} with constant $I(t)=\int_{-\infty}^\infty u\,dx =1$.

Hint. Some complex variables required.