24.3 Exercises
Exercise 24.1 Through direct computation, verify the following calculations:
- When \(\displaystyle f(t)=\frac{1}{\sqrt{4 \pi Dt} }\), then \(\displaystyle f'(t)=-2\pi D (4 \pi D t)^{-3/2}\)
- When \(\displaystyle g(t)=\frac{-x^{2}}{4Dt}\), then \(\displaystyle g'(t)=\frac{x^{2}}{4Dt^{2}}\)
- Verify that \(\displaystyle \left( -\frac{1}{2t} + \frac{x^{2}}{4Dt^{2}} \right)= \left( \frac{x^{2}-2Dt}{(2Dt)^{2}}\right)\)
Exercise 24.3 For this problem you will investigate \(p(x,t)\) (Equation @ref(eq:diffusion-equation}) with \(\displaystyle D=\frac{1}{2}\).
- Evaluate \(\displaystyle \int_{-1}^{1} p(x,10) \; dx\). Write a one sentence description what this quantity represents.
- Using desmos or some other numerical integrator, complete the following table:
| Equation | Result |
|---|---|
| \(\displaystyle \int_{-1}^{1} p(x,10) \; dx=\) | |
| \(\displaystyle \int_{-1}^{1} p(x,5) \; dx=\) | |
| \(\displaystyle \int_{-1}^{1} p(x,2.5) \; dx=\) | |
| \(\displaystyle \int_{-1}^{1} p(x,1) \; dx=\) | |
| \(\displaystyle \int_{-1}^{1} p(x,0.1) \; dx=\) | |
| \(\displaystyle \int_{-1}^{1} p(x,0.01) \; dx=\) | |
| \(\displaystyle \int_{-1}^{1} p(x,0.001) \; dx=\) |
- Based on the evidence from your table, what would you say is the value of \(\displaystyle \lim_{t \rightarrow 0^{+}} \int_{-1}^{1} p(x,t) \; dx\)?
- 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 \(x=1\).
- Explain in your own words what the graph \(p(1,t)\) represents as a function of \(t\).
- 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 \(\displaystyle p(x,t) = \frac{1}{\sqrt{\pi t}} e^{-x^{2}/t}\):
- Using your differentiation skills compute the partial derivatives \(p_{t}\), \(p_{x}\), and \(p_{xx}\).
- 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