Please use python

Consider the following differencing of the heat equation:

$\frac{u_j^{n+1}-u_j^n}{t}=D[\frac{u_{j+1}^{n+1}+u_{j-1}^{n+1}-2u_j^{n+1}}{x^2}]$

Spot the difference. What has changed with respect to the FTCS scheme?

This scheme is known as a "fully implicit" scheme, or "backward time" as the spatial derivatives are evaulated at timestep $n+1,$ i$.$e$.$ the timestep in the future.

We can rearrange the equation to the following form:

$-u_{j-1}^{n+1}+(1+2)u_j^{n+1}-u_{j+1}^{n+1}=u_j^n,$

where:

$=\frac{Dt}{(x{)}^2}.$

This is in the form of a matrix equation:

$Ax=b,$

where $A$ is a constant coefficient matrix, $b$ is known $($the values $u_j^n)$ and $x$ are our unknowns $($the values $u_j^{n+1}).$

We therefore have three steps:

$1$Set up the coefficient matrix $A$

$2$Set up the initial conditions $u(x,0)($as we did for FTCS$)$

$3$At each time step solve the linear algebra problem $Ax=b$ for $x.$

a$$sketch a diagram showing how the n$+1$ row depends on row n$.$