(Solved): Problem 1 ( 5 points): The governing ODE for the evaporation of liquid that is sitting at the botto ...

Problem 1 ( 5 points): The governing ODE for the evaporation of liquid that is sitting at the bottom of a vertical tube can be approximated by the following equation: $\frac{dy}{dt}=-\frac{2D}{L-y}\exp \left(-\frac{{\pi }^{2}Dt}{(L-y{)}^2}\right)$ where $\mathrm{dy}/\mathrm{dt}$ is the rate of change of the liquid level $(\mathrm{y}$, units $\mathrm{cm})$ with respect to time $(\mathrm{t}$, units seconds). The tube has a length $L$, which is $5\mathrm{cm}$ in this example. $D$ is the diffusion coefficient (units ${\mathrm{cm}}^2/\mathrm{s}$ ) of the evaporated liquid in the gas phase. The value of $D$ is $0.15{\mathrm{cm}}^2/\mathrm{s}$. The liquid level has an initial value of $1\mathrm{cm}$ at $\mathrm{t}=0\sec$. Write a MATLAB script $m$-file that does the following: - Numerically solves the ODE using the explicit and implicit Euler's methods. Each ODE solution must be solved in the range $t=[0,60]\mathrm{s}$. Use a step size of 2 seconds. (Hint: the $ODE$ is very nonlinear, so you might need to use the fzero function for the implicit Euler's method). - Numerically solves the ODE using the built-in MATLAB function ode 45 . - Plots all 3 numerical solutions on the same graph. Comment on the performance of the Euler methods. Do the numerical solutions obtained make sense? (i.e., are the values physically possible?) If not, what can you conclude about the performance of the implicit and explicit Euler methods for this problem?