Problem Statement:

A wealthy and successful businessman decides to showcase his wealth by wearing a tailor made 'gilded' silk shirt. However, in order to ward off potential muggers, he hires four (4) bodyguards to follow him around as he flaunts his way about town.

The guards are well compensated for their services and, in time, they make enough money to afford their own gilded silk shirts. And so that's what they did. Now we have the boss plus his four guards all wearing gilded silk shirts. Now that the guards are well off, they each decide to hire four (4) bodyguards of their own to ward off potential muggers.

If the above scenario perpetuates, then we can envision each successive generation of employees (guards) becoming wealthy enough so that they can afford their own gilded shirts and bodyguards. Guards in the n-th generation will concurrently become wealthy enough to guy their gilded silk shirt and each will, concurrently, hire 4 of their own bodyguards.

Assume that the population of India is 1.3 billion. Answer the following questions:

$(a)$
What proportion (\rho) of the total population have gilded silk shirts?

$(b)$
Find ratio (R) of gilded silk shirt-wearers to 'ungilded' bodyguards.

$(c)$
What proportion (\epsilon) of the population are neither gilded shirt-wears or bodyguards?



Suggested Solution: 

Let $\zeta(t)$ be the number of gilded silk shirt wearers at time $t$ and note that $\zeta(0)=1$.

Let $\eta(t)$ be the number of guards without gilded silk shirts where $eta(0)=4$.

Let $L=1.3\cdot10^{9}$ be the total population.

The value of $\zeta(t)$ and $\eta(t)$ respectively grows as follows:

$$\zeta(t)=\sum_{k=0}^{t}4^{k}$$ and $$\eta(t)=4^{t+1}$$

This is a deterministic branching process whereby each instance of $\zeta$ in the current iteration spawns four new instances of $\eta$ in the next iteration. Other examples of (probabilistic) branching processes include nuclear fission, population growth and cellular division.

We choose some time $t=T$ such that $\zeta(T) + \eta(T) \leq L$ and $\zeta(T+1) + \eta(T+1) > L$

$(a)$
The proportion of the total population have gilded silk shirts:

$$\rho=\frac{\zeta(T)}{L}=\frac{\sum_{k=0}^{T}4^{k}}{L}$$

$(b)$
The ratio of gilded silk shirt-wearers to 'ungilded' bodyguards.

$$R=\frac{\zeta(T)}{\eta(T)}=\frac{\sum_{k=0}^{T}4^{k}}{4^{T+1}}$$

$(c)$
The proportion of the population who are neither gilded nor bodyguards?

$$\epsilon=\frac{L-\zeta(T)-\eta(T)}{L}=\frac{L-\sum_{k=0}^{T+1}4^{k}}{L}$$

Now solve the above using recursion.

To my MEC2010 group members,

The test function you should implement in your expression is as follows:

$$\theta(i,j)=\varepsilon \cdot \sin \bigg( \frac{ij\pi}{n_{x}} \bigg)$$

The test case was the diffusion (heat) equation:

$$\frac{\partial U}{\partial t}=\kappa\frac{\partial^{2}U}{\partial x^2}$$

In your response, you need to discuss the conditions that ensure numerical stability and produce a plot using meshgrid() to show the stable and unstable cases respectively.

Email me if further clarification is required.