Chapter 6.6, Problem 10P

Diffusion of particles on a lattice with reflecting boundaries was described in Example 3, Section 6.3. In this problem, we introduce a known particle source $x_0\left(t\right)=f\left(t\right)$ at the left end of the lattice as shown in Figure $\mathrm{6.6.3}$. Treating locations $1$ and $2$ as interior points and locations $3$ as a reflecting boundary, the system of differential equations for $\text{x=}\left(x_1,x_2,x_3\right)$, the number of particles at lattice points $1$, $2$, and $3$, respectively, is

$\text{x'}=k\left(\begin{array}{ccc}-2& 1& 0\\ 1& -2& 1\\ 0& 1& -1\end{array}\right)\text{x}+k\left(\begin{array}{c}f\left(t\right)\\ 0\\ 0\end{array}\right)$. (i)

(a) Find a numerical approximation to the solution of Eq. (i) subject to the initial condition $\text{x}\left(0\right)={\left(0,0,0\right)}^T$ if $f\left(t\right)=1$ and $k=1$. Draw component plots of the solution and determine $\underset{t\to \infty }{\lim }\text{x}\left(t\right)$.

(b) Find a numerical approximation to the solution of Eq. (i) subject to the initial condition $\text{x}\left(0\right)={\left(0,0,0\right)}^T$ if $k=1$ and $f\left(t\right)=1-\cos \left(\omega t\right)$ for each of the cases $\omega =1$ and $\omega =4$. Draw component plots of the solutions in each of the two cases.