Can someone help me step by step with part D$?$ Consider the following lumped element model of a single vessel with the terminal resistance model.

The parameters are given by;

Vessel diameter, $D=0\mathrm{.}4cm,$ Vessel length, $L=20cm,$ Vessel wall thickness, $h=0\mathrm{.}08D$ Vessel Young's modulus, E$=2$ MPa, Blood density, $=1\frac{g}{c}{m}^3,$ and blood viscosity, $=0\mathrm{.}004Pa*sec.$

The upstream pressure, $P{A}_A$ and the terminal resistance $Z_1$ are time dependent and modeled by

$P_A=(P_{\text{m}\text{a}\text{x}}-P_{\text{m}\text{i}\text{n}})\frac{1+cos(2ft)}{2}+P_{\text{m}\text{i}\text{n}}$

$Z_1=(Z_{\text{m}\text{a}\text{x}}-Z_{\text{m}\text{i}\text{n}})\frac{1+cos(2ft)}{2}+Z_{\text{m}\text{i}\text{n}}$

where $P_{\text{m}\text{a}\text{x}}=12000[Pa],P_{\text{m}\text{i}\text{n}}=8000[Pa],Z_{\text{m}\text{a}\text{x}}=60000[Pas\frac{ec}{mL}],$

Pa sec$/$mL$],$ and $f=1[Hz](=60$ BPM$).$

a$.$ Derive a system of ODEs for $P_1(t)$ and $Q_1(t).$

b$.$ Write a computer program using the finite difference approximation and time marching method to solve for $P_1(t)$ and $Q_1(t).$ Plot $P_1(t)$ and $Q_1(t).$

c$.$ For the Runge$-$Kutta $4$ stage $($RK$4)$ time marching method, find the maximum time step size, $t$ allowing a stable solution.

d$.$ Decrease the vessel diameter to $0\mathrm{.}3,0\mathrm{.}2,0\mathrm{.}1,$ and $0\mathrm{.}05$ cm $,$ and plot the time averaged flow rate $Q_1$ versus the vessel cross sectional area.