Hello, I$\text{'}$m doing this problem on $1-$D partial differential equation of heat transfer in math programming class. My code is incorrect because the right boundary question needs to be Neuman's Condition. Please fix my PYTHON code to fit the Neuman Condition and solve the problem. The PYTHON code is below and the picture is the question for which I made the code. import numpy as np

import matplotlib.pyplot as plt

# Parameters

l $=0\mathrm{.}02$ # Length in meters

cp $=850$ # Specific heat capacity in J$/($kg$$K$)$

$\backslash$lambda $=130$ # Thermal conductivity in W$/($m$$K$)$

$\backslash$rho $=2700$ # Density in kg$/$m$^3$

qs $=1100$ # Heat flux in W$/$m$^2$

$\backslash$epsi $=0\mathrm{.}2$ # Emissivity

$\backslash$sigma $=5\mathrm{.}67$e$-8$ # Stefan$-$Boltzmann constant in W$/($m$^2$K$^4)$

T$0=300$ # Initial temperature in K

# Derived parameters

alpha $=\backslash$lambda $/($cp $*\backslash$rho $)$ # Thermal diffusivity in m$^2/$s

# Discretization parameters

nx $=50$ # Number of spatial points

dx $=$ l $/($nx $-1)$ # Spatial step size

# Stability criterion for the explicit method

dt $=$ dx$**2/(2*$ alpha$)$ # Time step size based on stability criterion

nt $=1000$ # Number of time steps

# Initialize temperature array

T $=$ np$.$ones$($nx$)*$ T$0$

# Function to update temperature distribution

def update$\_$temperature$($T$,$ dt$,$ dx$,$ alpha, $\backslash$lambda $,$ qs$,\backslash$epsi $,\backslash$sigma $)$:

T$\_$new $=$ T$.$copy$()$

for i in range$(1,$ nx$-1)$:

T$\_$new$[$i$]=$ T$[$i$]+$ alpha $*$ dt $/$ dx$**2*($T$[$i$+1]-2*$T$[$i$]+$ T$[$i$-1])$

# Boundary conditions

# Left boundary $($x $=0)$

q$\_$rad $=\backslash$epsi $*\backslash$sigma $*$ T$[0]**4-$ qs

T$\_$new$[0]=$ T$[0]+$ alpha $*$ dt $/$ dx$**2*($T$[1]-$ T$[0])-$ dt $*$ q$\_$rad $/\backslash$lambda

# Right boundary $($x $=$ l$)$

T$\_$new$[-1]=$ T$[-2]$

return T$\_$new

# Time$-$stepping loop

T$\_$results $=[$T$.$copy$()]$

for n in range$(1,$ nt$+1)$:

T $=$ update$\_$temperature$($T$,$ dt$,$ dx$,$ alpha, $\backslash$lambda $,$ qs$,\backslash$epsi $,\backslash$sigma $)$

T$\_$results.append$($T$.$copy$())$

# Convert results to numpy array for easier slicing

T$\_$results $=$ np$.$array$($T$\_$results$)$

# Plot temperature distribution at different times

plt$.$figure$($figsize$=(10,6))$

times$\_$to$\_$plot $=[0,50,100,200,500,1000]$

for time in times$\_$to$\_$plot:

plt$.$plot$($np$.$linspace$(0,$ l$,$ nx$),$ T$\_$results$[$time$],$ label$=$f$\text{'}$t$=\{$time$*$dt:$\mathrm{.}4$f$\}$s$\text{'})$

plt$.$xlabel$(\text{'}$Position $($m$)\text{'})$

plt$.$ylabel$(\text{'}$Temperature $($K$)\text{'})$

plt$.$title$(\text{'}$Temperature distribution over time'$)$

plt$.$legend$()$

plt$.$show$()$