pandas as pd

Create a dataset to implement finite difference method for a heat equation example

Assume a simple 1D heat equation with fixed boundary conditions

Parameters for the example

length = 1.0 # Length of the rod (m) nodes = 11 # Number of discrete points (including boundaries) dx = length / (nodes - 1) # Spatial step size (m) alpha = 0.01 # Thermal diffusivity (m^2/s) dt = 0.005 # Time step size (s) time_steps = 100 # Number of time steps tolerance = 1e-10 # Tolerance for numerical stability

Initialize temperature distribution (initial condition: T = 0, except boundaries)

temperature = [0] * nodes temperature[0] = 100 # Left boundary condition (Celsius) temperature[-1] = 50 # Right boundary condition (Celsius)

Create a dataframe to store initial and boundary conditions

data = {"Node": range(nodes), "Initial Temperature": temperature} df_initial = pd.DataFrame(data)

Apply finite difference approximation over time

results = {"Node": range(nodes)} for step in range(time_steps): new_temp = temperature[:] for i in range(1, nodes - 1): # Ignore boundaries new_temp[i] = temperature[i] + alpha * dt / (dx**2) * (temperature[i - 1] - 2 * temperature[i] + temperature[i + 1])

# Update temperature and check convergence
if max(abs(new_temp[i] - temperature[i]) for i in range(1, nodes - 1)) < tolerance:
    break
temperature = new_temp[:]
results[f"Time Step {step + 1}"] = temperature

Convert results into a DataFrame for export

df_results = pd.DataFrame(results)

Save both files

initial_file = "/mnt/data/finite_difference_initial.xlsx" results_file = "/mnt/data/finite_difference_results.xlsx"

df_initial.to_excel(initial_file, index=False) df_results.to_excel(results_file, index=False)

Provide file paths for download

initial_file, results_fileI have prepared two Excel files for your request:

1. Initial Setup and Boundary Conditions: Download initial setup file
2. Results from Finite Difference Approximation: Download results file

These files include:

• Initial setup with boundary and initial conditions.
• The results of the finite difference approximation for 100 time steps or until convergence (tolerance < $10^{-10}$).

Would you like an explanation of the setup or further details on any part of the implementation? Here are some related questions to explore:

1. What is the role of the thermal diffusivity ($\alpha$) in the heat equation?
2. How does the finite difference method ensure numerical stability?
3. What changes in the results if the boundary conditions are altered?
4. How does the spatial step size ($\Delta x$) influence accuracy?
5. What are the practical applications of solving the heat equation using numerical methods?

Tip: Always verify the stability condition for the chosen $\Delta t$ and $\Delta x$ using the Courant-Friedrichs-Lewy (CFL) condition to ensure reliable results.