The finite difference approximation of heat equation $c{u}_{}=u_t$ is

$u_{i,j+1}=u_{i-1,j}+(1-2)u_{ij}+u_{i+1,j},j0,$

where $=c\frac{k}{h^2}.$ Here $h$ and $k$ are dimensions of a mesh cell.

Assignment: Write a Python program that uses this difference equation to approximate the solution to the boundary value problem

$0\mathrm{.}15u_{}=u_t,0x3,0t50$

$u(0,t)=-5$

$u(3,t)=40-0\mathrm{.}01t$

$u(x,0)=x^3+2x^2-5$

Use $h=0\mathrm{.}3$ and $k=0\mathrm{.}2.$