Subsections

• Introduction
• Background
• Exercises

Laplace, Diffusion, 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. Fick's second law of diffusion satisfies the partial differential equation
   
   $$\frac{\partial N}{\partial t}=D\frac{\partial^2 N}{\partial x^2}.$$
   
   
   The function
   
   $$N(x,t)=\frac{Q_oe^{\frac{-x^2}{4Dt}}}{\sqrt{\pi}\sqrt{Dt}}$$
   
   
   denotes diffusion of a substance at a depth $x$ and time $t$ where $Q_o$ is the initial amount of the substance and $D$ is the diffusion coefficient.

   A)

   Enter the function $N(x,t)$ and show that the function satisfies the partial differential equation given above. (Note: you may need to use the simplify command on both sides of the equal sign.)

   B)

   If the initial amount of $CO_2$ is $3 x 10^9 molecules/mm^2$ and the diffusion coefficients of $CO_2$ is $16mm^2/s$ in the air and $0.0016mm^2/s$ in the water, plot the surface concentration (N(0,t)) for ten seconds. There should be two plots; one titled diffusion through air and the other titled diffussion through water.

   C)

   Does the surface concentration increase or decrease for each as time increases? Explain why this is so. Which concentration is higher over time? Explain why this is so.

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)

   $u1(x,y)=x^3y-xy^3$

   B)

   $u2(x,y)=cos(2x^2-y^2)$

   C)

   $u3(x,y)=e^xcos(y)$

   D)

   $u4(x,y)=cot(xy)$

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?