24.3 Exercises

Exercise 24.1 Through direct computation, verify the following calculations:

1. When $\displaystyle f(t)=\frac{1}{\sqrt{4 \pi Dt} }$, then $\displaystyle f'(t)=-2\pi D (4 \pi D t)^{-3/2}$
2. When $\displaystyle g(t)=\frac{-x^{2}}{4Dt}$, then $\displaystyle g'(t)=\frac{x^{2}}{4Dt^{2}}$
3. Verify that $\displaystyle \left( -\frac{1}{2t} + \frac{x^{2}}{4Dt^{2}} \right)= \left( \frac{x^{2}-2Dt}{(2Dt)^{2}}\right)$

 

Exercise 24.2 The equation for the normal distribution is $\displaystyle f(x)=\frac{1}{\sqrt{2 \pi} \sigma } e^{-(x-\mu)^{2}/(2 \sigma^{2})}$, with mean $\mu$ and variance $\sigma^{2}$. Examine the formula for the diffusion equation (Equation (24.3)) and compare it to the formula for the normal distribution. If Equation (24.3) represents a normal distribution, what does $\mu$ equal? $\sigma^{2}$?

 

Exercise 24.3 For this problem you will investigate $p(x,t)$ (Equation @ref(eq:diffusion-equation}) with $\displaystyle D=\frac{1}{2}$.

1. Evaluate $\displaystyle \int_{-1}^{1} p(x,10) \; dx$. Write a one sentence description what this quantity represents.
2. Using desmos or some other numerical integrator, complete the following table:

Equation	Result
$\displaystyle \int_{-1}^{1} p(x,10) \; dx=$
$\displaystyle \int_{-1}^{1} p(x,5) \; dx=$
$\displaystyle \int_{-1}^{1} p(x,2.5) \; dx=$
$\displaystyle \int_{-1}^{1} p(x,1) \; dx=$
$\displaystyle \int_{-1}^{1} p(x,0.1) \; dx=$
$\displaystyle \int_{-1}^{1} p(x,0.01) \; dx=$
$\displaystyle \int_{-1}^{1} p(x,0.001) \; dx=$

3. Based on the evidence from your table, what would you say is the value of $\displaystyle \lim_{t \rightarrow 0^{+}} \int_{-1}^{1} p(x,t) \; dx$?
4. Now make graphs of $p(x,t)$ at each of the values of $t$ in your table. What would you say is occuring in the graph as $\displaystyle \lim_{t \rightarrow 0^{+}} p(x,t)$? Does anything surprise you? (The results you computed here lead to the foundation of what is called the Dirac delta function.)

 

Exercise 24.4 Consider the function $\displaystyle p(x,t) = \frac{1}{\sqrt{4 \pi D t}} e^{-x^{2}/(4 D t)}$. Let $x=1$.

1. Explain in your own words what the graph $p(1,t)$ represents as a function of $t$.
2. Graph several profiles of $p(1,t)$ when $D = 1$, $2$, and $0.1$. How does the value of $D$ affect the profile?

 

Exercise 24.5 Consider the function $\displaystyle p(x,t) = \frac{1}{\sqrt{\pi t}} e^{-x^{2}/t}$:

1. Using your differentiation skills compute the partial derivatives $p_{t}$, $p_{x}$, and $p_{xx}$.
2. Verify $p(x,t)$ is consistent with the diffusion equation $\displaystyle p_{t}=\frac{1}{4} p_{xx}$.

 

Exercise 24.6 For the one-dimensional random walk we discussed where there was an equal chance of moving to the left or the right. Here is a variation on this problem.

Let’s assume there is a chance $v$ that it moves to the left (position $x - \Delta x$), and therefore a chance is $1-v$ that the particle remains at position $x$. The basic equation that describes the particle’s position at position $x$ and time $t + \Delta t$ is:

$$\begin{equation} p(x,t + \Delta t) = (1-v) \cdot p(x,t) + v \cdot p(x- \Delta x,t) \end{equation}$$

Apply the techniques of local linearization in $x$ and $t$ to show that this random walk to derive the following partial differential equation, called the advection equation:

$$\begin{equation} p_{t} = - \left( v \cdot \frac{ \Delta x}{\Delta t} \right) \cdot p_{x} \end{equation}$$

Note: you only need to expand this equation to first order