Ross Ogilvie      4th November, 2024
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30.

Special solutions of the heat equation

(a)

Solutions of PDEs that are constant in the time variable are called “steady-state” solutions. Describe steady-state solutions of the inhomogeneous heat equation. (1 point)

(b)

Consider the heat equation $\dot{u}-\mathrm{\Delta }u=0$ on ${ℝ}^n\times {ℝ}^{+}$ with smooth initial condition $u(x,0)=h(x)$. Suppose that the Laplacian of $h$ is a constant. Show that there is a solution whose time derivative is constant. (1 point)

(c)

Consider “translational solutions” to the heat equation on $ℝ\times {ℝ}^{+}$ (ie $n=1$). These are solutions of the form $u(x,t)=F(x-\mathit{𝑏𝑡})$. Find all such solutions. (2 points)

(d)

If $u$ is a solution to the heat equation, show for every $\lambda \in ℝ$ that $u_{\lambda }(x,t):=u(\mathit{𝜆𝑥},{\lambda }^{2}t)$ is also a solution to the heat equation. (2 points)

31.

The Fourier transform

In this question we expand on some details from Section 4.1. Recall that the Fourier transform of a function $h(x):{ℝ}^n\to ℝ$ is defined to be a function $ĥ(k):{ℝ}^n\to ℝ$ given by

$$ĥ(k)={\int \; <!--\; nolimits\; -->}_{{ℝ}^n}{e}^{-2\mathit{𝜋𝑖𝑘}\cdot x}h(x)\mathit{𝑑𝑥}.$$

Lemma 4.3 shows that it is well-defined for Schwartz functions.

(a)

Give the definition of a Schwartz function. (1 point)

(b)

Argue that $f:ℝ\to ℝ$ given by $f(x)=\exp <!--\; FUNCTION\; APPLICATION\; -->(-x^2)$ is a Schwartz function. (2 points)

(c)

Consider

$$I^2={\left({\int \; <!--\; nolimits\; -->}_{ℝ}{e}^{-x^2}\mathit{𝑑𝑥}\right)}^2=\left({\int \; <!--\; nolimits\; -->}_{ℝ}{e}^{-x^2}\mathit{𝑑𝑥}\right)\left({\int \; <!--\; nolimits\; -->}_{ℝ}{e}^{-y^2}\mathit{𝑑𝑦}\right)={\int \; <!--\; nolimits\; -->}_{{ℝ}^2}{e}^{-x^2-y^2}\mathit{𝑑𝑥}\mathit{𝑑𝑦}.$$

By changing to polar coordinates, compute this integral. (1 point)

(d)

Prove the rescaling law for Fourier transforms: if $h(x)=g(\mathit{𝑎𝑥})$ then (1 point)

$$ĥ(k)=|a{|}^{-n}ĝ(a^{-1}k).$$

(e)

Prove the shift law for Fourier transforms: if $h(x)=g(x-a)$ then (1 point)

$$ĥ(k)=e^{-2\mathit{𝜋𝑖𝑎}\cdot k}ĝ(k).$$

(f)

Show that $\delta$ is a tempered distribution. (2 points)

(g)

Compute the Fourier transform of $\delta$. (2 points)

(h)

Try to compute the Fourier transform of $1$. What is the difficulty?
(2 points)

32.

One step at a time

Prove the following identity for the fundamental solution in one dimension ($n=1$):

$$\mathrm{\Phi }(x,s+t)={\int \; <!--\; nolimits\; -->}_{ℝ}\mathrm{\Phi }(x-y,t)\mathrm{\Phi }(y,s)\mathit{𝑑𝑦}.$$

(2 points)

Hint. You may use without proof that

$${\int \; <!--\; nolimits\; -->}_{ℝ}\exp <!--\; FUNCTION\; APPLICATION\; -->(-A+\mathit{𝐵𝑦}-C{y}^2)\mathit{𝑑𝑦}=\sqrt{\frac{\pi }{C}}\exp <!--\; FUNCTION\; APPLICATION\; -->\left(\frac{B^2}{4C}-A\right).$$