Partial Differential Equations
Partial differential equations (PDEs) involve functions of multiple variables and their partial derivatives. They model fundamental phenomena in physics, engineering, and finance.
Key Points
- Major classes include elliptic, parabolic, and hyperbolic equations.
- Separation of variables and Fourier methods are classical solution techniques.
- Numerical methods include finite difference, finite element, and spectral methods.
Formulas
Heat equation
$$\frac{\partial u}{\partial t} = \alpha \nabla^2 u$$
Wave equation
$$\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$$
Laplace equation
$$\nabla^2 u = 0$$
Code Example
import numpy as np
# Finite difference for 1D heat equation
N = 50
u = np.sin(np.pi * np.linspace(0, 1, N))
alpha = 0.01
dt = 0.001
for _ in range(1000):
u[1:-1] += alpha * dt * (u[2:] - 2*u[1:-1] + u[:-2])