write a matlab or python code

We consider the boundary value problem

$u^{\text{'}\text{'}}+u^3=cosx,$xin$(0,1)$

$u(0)=0$

$u(1)=0$

a$)$ By approximating $u^{\text{'}}$ with the central difference formula

$u^{\text{'}}$~~$\frac{u_{m+1}-u_{m-1}}{2h}$

with $h$ being the mesh size and $u_m$ the approximation of $u(t_m).$ Show that the boundary value problem

is reduced to a system of nonlinear equations of the form $Ax+Nx=b$ where $A$ is linear part and $N$

is a nonlinear part, and $x$ is the unknown and $b$ is a vector.

b$)$ write a computer program to solve the nonlinear system obtained in $($a$)$

for $h={10}^{-1},{10}^{-2},{10}^{-3},{10}^{-4},{10}^{-5},{10}^{-6},{10}^{-7},{10}^{-8}$

c$)$ Find the convergence and compute the rate of convergence.

d$)$ repeat the exercise if one considers

$u^{\text{'}}(t_m)$~~$\frac{u_{m+1}-u_m}{h},($ forward scheme $)$

g$)$repeat the exercise if

$u^{\text{'}}(t_m)$~~$\frac{u_m-u_{m-1}}{h},($ backward scheme $)$

We assume that the second derivative in all situations is approximated as follows

$u^{\text{'}\text{'}}(t_m)$~~$\frac{u_{m+1}-2u_m+u_{m-1}}{h^2}$