gistfile1.txt

import numpy as np
import matplotlib.pyplot as plt

def heat_equation_1d(L, nx, nt, alpha, T_initial, T_left, T_right):
    """
    Solve the 1D heat equation using explicit finite difference method.
    
    Parameters:
    L : float
        Length of the domain
    nx : int
        Number of spatial points
    nt : int
        Number of time steps
    alpha : float
        Thermal diffusivity
    T_initial : float
        Initial temperature
    T_left : float
        Left boundary temperature
    T_right : float
        Right boundary temperature
    
    Returns:
    T : ndarray
        Temperature distribution over time and space
    """
    # Calculate step sizes
    dx = L / (nx - 1)
    dt = 0.5 * dx**2 / alpha  # Ensure stability

    # Initialize temperature array
    T = np.ones((nt, nx)) * T_initial
    
    # Set boundary conditions
    T[:, 0] = T_left
    T[:, -1] = T_right
    
    # Compute finite difference
    for n in range(1, nt):
        for i in range(1, nx-1):
            T[n, i] = T[n-1, i] + alpha * dt / dx**2 * \
                      (T[n-1, i+1] - 2*T[n-1, i] + T[n-1, i-1])
    
    return T

# Set parameters
L = 1.0         # Length of the domain
nx = 50         # Number of spatial points
nt = 1000       # Number of time steps
alpha = 0.01    # Thermal diffusivity
T_initial = 0   # Initial temperature
T_left = 100    # Left boundary temperature
T_right = 0     # Right boundary temperature

# Solve the heat equation
T = heat_equation_1d(L, nx, nt, alpha, T_initial, T_left, T_right)

# Plot results
x = np.linspace(0, L, nx)
plt.figure(figsize=(10, 6))
plt.plot(x, T[0, :], label='t = 0')
plt.plot(x, T[int(nt/4), :], label=f't = {int(nt/4)}')
plt.plot(x, T[int(nt/2), :], label=f't = {int(nt/2)}')
plt.plot(x, T[-1, :], label=f't = {nt-1}')
plt.xlabel('Position')
plt.ylabel('Temperature')
plt.title('1D Heat Equation - Finite Difference Solution')
plt.legend()
plt.grid(True)
plt.show()