Bhavin Moriya, Ph.D’s Post

Solving the 1D Poisson Equation with Finite Differences in Python 🚀 Ever wondered how to numerically solve the 1D Poisson equation using finite differences? Here’s a concise implementation using NumPy: import numpy as np n = 10 x = np.linspace(0, 1, n+1) h = x[1] - x[0] # Construct the stiffness matrix K and load vector f K = np.zeros((n-1, n-1)) f = np.zeros(n-1) def rhs(x): return np.sin(x) for i in range(n-1): K[i, i] = 2 / h if i > 0: K[i, i-1] = -1 / h if i < n-2: K[i, i+1] = -1 / h f[i] = rhs(x[i+1]) * h # Solve the linear system u = np.linalg.solve(K, f) # Extend solution to full grid u_full = np.zeros(n+1) u_full[1:-1] = u # Exact solution for comparison y_exact = np.sin(x) - x * np.sin(1) # Compute error error = np.linalg.norm(u_full - y_exact) print("Error:", error) Key Takeaways: ✅ Finite Difference Method: Discretizes the Poisson equation into a linear system. ✅ NumPy Efficiency: Uses np.linalg.solve for fast matrix inversion. ✅ Error Analysis: Compares numerical and exact solutions to validate accuracy. Why This Matters: This is a foundational technique for solving PDEs in scientific computing, finance, and engineering. How would you extend this to 2D or 3D problems? #NumericalMethods #Python #ScientificComputing #FiniteDifferences #DataScience

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