Chapter 6.6, Problem 11P

Find numerical approximations to the initial value problems posed in Problem 10 under the assumption that the right boundary is an absorbing boundary (see problem 14, Section 6.3).

10. 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.