Subsections

• Introduction
• Background
• Exercises

Laplace, Heat, and Wave Equations

Introduction

The purpose of this lab is to aquaint you with partial differential equations.

Background

Second-order partial derivatives show up in many physical models such as heat, wave, or electrical potential equations. For example, the one-dimensional wave equation

$$\frac{\partial^2 y}{\partial t^2}=a^2\frac{\partial^2 y}{\partial x^2}$$

can be solved by the displacement equation $f(x,t)=\sin(x+at)$, or $g(x,t)=\sin(kx)\cos(kat)$, or even $h(x,t)=\sin(x+at)+37.8$. To show this, enter the displacement equation and then show that both sides of the differential equation are equal.

>f:=(x,t)->sin(x+a*t);
>g:=(x,t)->sin(k*x)*cos(k*a*t);
>h:=(x,t)->sin(x+a*t)+37.8;
>diff(f(x,t),t,t)-a^2*diff(f(x,t),x,x);
>diff(g(x,t),t,t)=a^2*diff(g(x,t),x,x);
>diff(h(x,t),t,t)=a^2*diff(h(x,t),x,x);

Exercises

1. The function
   
   $$u(x,t)=T+ae^{-\sqrt{\frac{\omega}{2k}}}\cos(\omega t-x\sqrt{\frac{\omega}{2k}})$$
   
   
   denotes temperature at a depth $x$ and time $t$ where the seasonal variation of the surface $(x=0)$ temperature is
   
   $$u(0,t)=T+a\cos(\omega t)$$
   
   
   Where $T$ is the annual average surface temperature and $\omega$ is chosen such that the period is one year.

   A)

   Enter the function $u(x,t)$

   B)

   Show that the function satisfies the surface condition (i.e. $x=0$).

   C)

   Show that the function $u(x,t)$ satisfies that one-dimensional heat equation $\frac{\partial^2 u}{\partial t^2}=k\frac{\partial^2 u}{\partial x^2}$.

   D)

   Enter the following data into memory (Do this after completeing parts A, B, and C, or you will need to re-initialize the constants).

   $\bullet$

   The thermal conductivity of brick is $k=0.0015$.

   $\bullet$

   The average temperature of the surface temperature is $22$ degrees celsius, please enter as $22+273$ as the conductivity uses units of Kelvin.

   $\bullet$

   The surface area of a brick is $a = 72 cm^2$

   $\bullet$

   $\omega = \frac{2 \pi}{365}$

   E)

   Plot the surface temperature $(x=0)$ of a brick for one year ($t=0..365$). To make the temperature axis celsius, plot the function minus 273.

   F)

   From your graph, what is the approximate temperature variation of the surface of the brick?

   G)

   Plot the temperature of the inside of the brick with dimensions six by six by twelve (i.e. depth = $x=6$). Remember to plot using celsius.

   H)

   From your graph, what is the temperature variation of the inside of the brick?

2. Determine which of the following functions satisfy Laplace's equation
   
   $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$
   
   

   A)

   $u1(x,y)=ln(\sqrt{x^2+y^2})$

   B)

   $u2(x,y)=\sqrt{x^2+y^2}$

   C)

   $u3(x,y)=arctan(\frac{y}{x})$

   D)

   $u4(x,y)=e^{-x}\sin(y)$

3. Using a function from exercise 2 that you found satisfies the Laplace equation, answer the following without calculating the differential equation.

   A)

   Will the function plus $10$ satisfy the Laplace equation? Why or why not?

   B)

   Will the function plus $7x$ satisfy the Laplace equation? Why or why not?

   C)

   Will the function plus $5x^3$ satisfy the Laplace equation? Why or why not?