can you fix this code so that the graphs appear properly with all the lines, and so that the last graph is not a straight line? :import numpy as np

import matplotlib.pyplot as plt

# Material properties

rho $=138$ # Density $($kg$/$m$^3)$

c $=608$ # Specific heat capacity $($J$/$kg$-$K$)$

k$\_$room $=0\mathrm{.}0512$ # Thermal conductivity at room temperature $($W$/$m$-$K$)$

k$\_1100=0\mathrm{.}1462$ # Thermal conductivity at $1100\backslash$deg C $($W$/$m$-$K$)$

emissivity $=0\mathrm{.}8$ # Emissivity of the coating

# Heat flux data

time $=$ np$.$array$([0,100,200,300,400,500,600,700,800,900,1000,1100,1200,1300,1400,1500])$

heat$\_$flux $=$ np$.$array$([0,10,25,50,75,80,78,80,80,82,70,50,30,20,10,0])*1000$ # Convert to W$/$m$^2$

# Function to calculate temperature$-$dependent thermal conductivity

def thermal$\_$conductivity$($T$)$:

return k$\_$room $+($k$\_1100-$ k$\_$room$)*($T $-273\mathrm{.}15)/(1100-273\mathrm{.}15)$ # Linear interpolation

# Function to calculate surface radiation

def surface$\_$radiation$($T$)$:

return emissivity $*5\mathrm{.}67$e$-8*$ T$**4$ # Stefan$-$Boltzmann law

# Finite difference method implementation

def solve$\_$heat$\_$equation$($L$,$ dx$,$ dt$,$ initial$\_$temp$)$:

nx $=$ int$($L $/$ dx$)+1$ # Number of spatial nodes

nt $=$ len$($time$)$ # Number of time steps based on the provided heat$\_$flux data length

T $=$ np$.$zeros$(($nt$,$ nx$))$ # Initialize temperature array

T$[0,$ :$]=$ initial$\_$temp # Set initial temperature

alpha $=$ k$\_$room $/($rho $*$ c$)$ # Thermal diffusivity at room temperature

r $=$ alpha $*$ dt $/$ dx$**2$ # Stability condition

for j in range$($nt $-1)$: # Iterate over time steps

heat$\_$flux$\_$value $=$ heat$\_$flux$[$min$($j$,$ len$($heat$\_$flux$)-1)]$

# Update temperature profile using finite difference method

T$[$j $+1,0]=$ T$[$j$,0]+(2*$ r $*$ thermal$\_$conductivity$($T$[$j$,1])*($T$[$j$,1]-$ T$[$j$,0])+$

dt $*$ heat$\_$flux$\_$value $/($rho $*$ c$)-$ dt $*$ surface$\_$radiation$($T$[$j$,0])/($rho $*$ c $*$ dx$))$

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

T$[$j $+1,$ i$]=$ T$[$j$,$ i$]+$ r $*($thermal$\_$conductivity$($T$[$j$,$ i $+1])*($T$[$j$,$ i $+1]-$ T$[$j$,$ i$])-$

thermal$\_$conductivity$($T$[$j$,$ i$])*($T$[$j$,$ i$]-$ T$[$j$,$ i $-1]))$

T$[$j $+1,-1]=$ T$[$j$,-1]$ # Adiabatic boundary condition at the back surface

# Debugging print statement

print$($f$"$Time step $\{$j $+1\}$: Heat flux value $=\{$heat$\_$flux$\_$value$\}")$

return T

# $($a$)$ Plot temperature vs$.$ depth for different grid resolutions

# Validate lengths of time and heat$\_$flux arrays

print$("$Length of time array:", len$($time$))$

print$("$Length of heat$\_$flux array:", len$($heat$\_$flux$))$

# Check the time step calculation

alpha $=$ k$\_$room $/($rho $*$ c$)$

print$("$Alpha $($thermal diffusivity$)$:$",$ alpha$)$

# Define grid resolutions

dx$\_$values $=[0\mathrm{.}01,0\mathrm{.}005,0\mathrm{.}002,0\mathrm{.}001]$

# Plot temperature vs$.$ depth for different grid resolutions

L $=0\mathrm{.}1$ # Tile thickness $($m$)$

t $=1000$ # Time $($s$)$

for dx in dx$\_$values:

alpha $=$ k$\_$room $/($rho $*$ c$)$ # Thermal diffusivity at room temperature

dt $=$ dx$**2/(2*$ alpha$)$ # Calculate time step based on stability condition

T $=$ solve$\_$heat$\_$equation$($L$,$ dx$,$ dt$,0)$

# Determine the time step index corresponding to time t

time$\_$step$\_$index $=$ np$.$where$($time $==$ t$)[0]$ # Find the index in time array closest to t

if len$($time$\_$step$\_$index$)>0$:

time$\_$step$\_$index $=$ time$\_$step$\_$index$[0]$ # Take the first index if multiple found

else:

time$\_$step$\_$index $=$ len$($time$)-1$ # Use the last time step index if t is beyond the range

# Plot temperature profile at the specified time t

x $=$ np$.$linspace$(0,$ L$,$ len$($T$[0]))$

plt$.$plot$($x$,$ T$[$time$\_$step$\_$index, :$],$ label$=$f$\text{'}$dx $=\{$dx:$\mathrm{.}3$f$\}$ m$\text{'})$

# label plot

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

plt$.$ylabel$(\text{'}$Temperature $(\backslash$deg C$)\text{'})$

plt$.$title$($f$\text{'}$Temperature vs$.$ Depth at t $=\{$t$\}$ s$\text{'})$

plt$.$legend$()$

plt$.$show$()$

# $($b$)$ Plot temperature vs$.$ time at different depths

L $=0\mathrm{.}1$ # Tile thickness $($m$)$

dx $=0\mathrm{.}001$ # Grid resolution based on part $($a$)$

dt $=$ dx$**2/(2*$ k$\_$room $/($rho $*$ c$))$ # Stability condition

T $=$ solve$\_$heat$\_$equation$($L$,$ dx$,$ dt$,0)$

x$\_$values $=[0,0\mathrm{.}05,0\mathrm{.}1]$ # Depths $($m$)$

for x in x$\_$values:

i $=$ int$($x $/$ dx$)$

plt$.$plot$($time$,$ T$[$:$,$ i$],$ label$=$f$\text{'}$x $=\{$x:$\mathrm{.}2$f$\}$ m$\text{'})$

plt$.$xlabel$(\text{'}$Time $($s$)\text{'})$

plt$.$ylabel$(\text{'}$Temperature $(\backslash$deg C$)\text{'})$

plt$.$title$(\text{'}$Temperature vs$.$ Time'$)$

plt$.$legend$()$

plt$.$show$()$

# $($c$)$ Plot temperature vs$.$ depth at different times

# Plot temperature vs$.$ depth at the specified times

L $=0\mathrm{.}1$ # Tile thickness $($m$)$

dx $=0\mathrm{.}001$ # Grid resolution based on part $($a$)$

dt $=$ dx$**2/(2*$ k$\_$room $/($rho $*$ c$))$ # Stability condition

T $=$ solve$\_$heat$\_$equation$($L$,$ dx$,$ dt$,0)$

x $=$ np$.$linspace$(0,$ L$,$ len$($T$[0]))$

# Define the times of interest for plotting

t$\_$values $=[500,1000,1500]$

for t in t$\_$values:

# Determine the time step index corresponding to time t

time$\_$step$\_$index $=$ np$.$where$($time $==$ t$)[0]$

if len$($time$\_$step$\_$index$)>0$:

time$\_$step$\_$index $=$ time$\_$step$\_$index$[0]$ # Take the first index if multiple found

else:

continue # Skip if t is beyond