Problem $12$:

Use the difference equation

$u_{i,j+1}=u_{i+1,j}+(1-2)u_{ij}+u_{i-1,j}$

to approximate the solution of the boundary$-$value problem

$n=5m=25\frac{C}{C}++\frac{C}{C_{++}}\frac{de{l}^{2}u}{del{x}^2}=\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}},0$

Use $n=5$ and $m=25.$ Solve this using Matlab, Python, $or$ another programming language using built$-in$ functions $(you$ can also use NumPy and SciPy for Python, and you can use BLAS, LAPACK, and Eigen for$\frac{C}{C}++).$ Print out your scripts and attach them with your solution for submission. $If$ your chosen programming language needs compilation $(e.g.\frac{C}{C_{++}}),$ please also include the compilation commands.