4 Question 4 (30 points)
Consider the following equation
$\frac{\partial f}{\partial t} - \nu \frac{\partial^2 f}{\partial x^2} = 0$
Between hyperbolic, parabolic, and elliptic, it is classified as ______ equation. We call the
phenomenon related to this equation as ______ phenomenon.
Consider the total concentration $M$ in the system with the following with the following initial condition.
$f(0, x) = \begin{cases} \frac{M}{2\beta}, & -\beta < x < \beta\\ 0, & \text{otherwise} \end{cases}$
Derive the analytical solution when the initial condition become a point source (i.e. $\beta$ is close to zero)
2
(Derivation space here)
Plot the function at $t = 0$ and $t > 0$. Label the axes, peak value, and half width expansion $L_{1/2}$
(Plotting space for case $t = 0$ and $t > 0$ here)
Now, consider the bounded case where the boundary conditions are
$\frac{\partial f(0)}{\partial x} = \frac{\partial f(L)}{\partial x} = 0$
Use Fourier series to derive the analytical solution of this case in three steps. (There will be a space for
writing the derivation in each step)
1. Separate the PDE into 2 ODEs.
2. Solve for the solution for each ODE.
3. Use Fourier series to obtain the solution of the equation.
