Using the Crank$-$Nicolson method and Letting $=\frac{k}{h^2},$ we need to solve

$-\frac{}{2}{U}_{i-1}^{j+1}+(1+)U_i^{j+1}-\frac{}{2}{U}_{i+1}^{j+1}=\frac{}{2}{U}_{i-1}^j+(1-)U_i^j+\frac{}{2}{U}_{i+1}^j+v(x_i,t_j).$ To do this, we first need to rewrite $U_i^j$ as a long, one$-$dimensional vector $y$ of size $n(m-1).$ Note that in this vector we do not include the known values, i$.$e$.,$ initial values $\{U_i^0\}$ and the boundary values $\{U_0^j\},\{U_m^j\}.$ Using the enumeration

$y_{(j-1)(m-1)+i}=U_i^j,i=1,2,$dots,$m-1,j=1,2,$dots,$n-1$and similarly

$v_{(j-1)(m-1)+i}=v(x_i,t_j),i=1,2,$dots,$m-1,j=1,2,$dots,$n-1.$ system $(2)$ can be represented in a matrix equation

$Ay=v$ where $y=(y_1,y_2,$dots,$y_{(n-1)(m-1)}{)}^t,v=(v_1,v_2,$dots,$v_{(n-1)(m-1)}{)}^t,$ and $A$ is the square matrix resulting from the equation. Example $1$ We test the above discretization scheme $(3)$ to find an approximate solution of $(1)$ with

an input $v(x,t)=2.$ Fix $L=T=1$ and $m=10,n=20.$ The analytic solution can be found in this

case and is given by

hat$(y)(x,t)=x(1-x)+{\displaystyle \underset{n=1}{\overset{}{}}}{b}_n{e}^{-n^2{}^{2}t}sinnx$

where

$b_n=2{\displaystyle \underset{0}{\overset{1}{}}}x(x-1)sinnxdx$

We compare the solution obtained from $(3)$ and the analytic solution evaluated at $m+1$ mesh points

at time $t=0\mathrm{.}95.$