Create a 2D grid and solve a PDE with NumPy and SciPy

NumPy: Integration with SciPy Exercise-17 with Solution

Write a NumPy program to create a 2D grid of data points and solve a partial differential equation (PDE) using SciPy's integrated module.

Sample Solution:

Python Code:

import numpy as np
from scipy.integrate import solve_bvp
import matplotlib.pyplot as plt

# Define the PDE as a system of first-order ODEs
def pde_system(x, y):
    return np.vstack((y[1], -np.pi**2 * y[0]))

# Define the boundary conditions
def boundary_conditions(ya, yb):
    return np.array([ya[0], yb[0]])

# Create a 2D grid of data points
x = np.linspace(0, 1, 100)
y_initial = np.zeros((2, x.size))

# Solve the boundary value problem (BVP)
solution = solve_bvp(pde_system, boundary_conditions, x, y_initial)

# Plot the solution
plt.plot(solution.x, solution.y[0])
plt.xlabel('x')
plt.ylabel('y')
plt.title('Solution of the PDE')
plt.grid(True)
plt.show()

Output:

Explanation:

• Import libraries:
• Import the necessary modules from NumPy, SciPy, and Matplotlib.
• Define PDE:
• Define the partial differential equation as a system of first-order ordinary differential equations (ODEs).
• Boundary conditions:
• Define the boundary conditions for the PDE.
• Create 2D grid:
• Create a 2D grid of data points using NumPy.
• Initial guess:
• Set an initial guess for the solution.
• Solve BVP:
• Use SciPy's solve_bvp function to solve the boundary value problem.
• Plot solution:
• Plot the solution using Matplotlib for visualization.

Python-Numpy Code Editor:

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