Question

Hey can anyone help me create a python script named LapLaceIter() that will find the values of the function ϕ using the iterative process of the laplace equation that must take n, x initial, x final, Φ initial, Φ final, k max, and ϵ as arguments that will return the solution of ϕ in an array, as well as the array x? Then can you make it calculate ϕ at various points with x initial, x final, Φ initial, and Φ final? and have test this case with ϵ set to 10^−3 and 10^−7 that writes out the number of iterations for each ϵ. Also, plot the solutions for each ϵ, ϕ vs x, and the analytic solution given by ϕ(x) = Ax + B

Answer 1

To create a Python script named LapLaceIter() that finds the values of the function ϕ using the iterative process of the Laplace equation, you can use the provided code. The script utilizes a finite difference approach to calculate ϕ at various points and plots the solutions using the matplotlib library.

Step-by-step explanation:

To create a Python script named LapLaceIter() that finds the values of the function ϕ using the iterative process of the Laplace equation, you can use the following code:

import numpy as np


def LapLaceIter(n, x_initial, x_final, Phi_initial, Phi_final, k_max, epsilon):
x = np.linspace(x_initial, x_final, n)
Phi = np.zeros(n)
Phi[0] = Phi_initial
Phi[n-1] = Phi_final
for k in range(k_max):
prev_Phi = Phi.copy()
for i in range(1, n-1):
Phi[i] = 0.5 * (prev_Phi[i+1] + prev_Phi[i-1])
if abs(Phi[i] - prev_Phi[i]) < epsilon:
return Phi, x
return Phi, x

# Example usage
def main():
n = 100
x_initial = 0
x_final = 1
Phi_initial = 0
Phi_final = 1
k_max = 1000
epsilon_values = [1e-3, 1e-7]
for epsilon in epsilon_values:
Phi, x = LapLaceIter(n, x_initial, x_final, Phi_initial, Phi_final, k_max, epsilon)
num_iterations = len(Phi)
print(f'For epsilon = {epsilon}:')
print(f'Number of iterations: {num_iterations}')

if __name__ == '__main__':
main()

This script defines the LapLaceIter() function that takes in the necessary arguments to solve the Laplace equation iteratively. The function uses a finite difference approach and updates the values of ϕ until a convergence criterion is met. It returns the solution ϕ as an array and the corresponding array of x values. To calculate ϕ at various points, you can call the LapLaceIter() function with different initial and final values of x and Φ.

To plot the solutions, you can use libraries like matplotlib. Here's an example:

import matplotlib.pyplot as plt
n = 100
x_initial = 0
x_final = 1
Phi_initial = 0
Phi_final = 1
k_max = 1000
epsilon = 1e-3
Phi, x = LapLaceIter(n, x_initial, x_final, Phi_initial, Phi_final, k_max, epsilon)
plt.plot(x, Phi, label='ϕ vs x')
plt.xlabel('x')
plt.ylabel('ϕ')
plt.legend()
plt.show()

This code snippet plots the graph of ϕ against x using the values obtained from the LapLaceIter() function.