Partial Differential Equations (PDEs)

Ordinary Differential Equations (ODEs) deal with functions of one variable (usually time). Partial Differential Equations (PDEs) deal with functions of multiple variables, such as the temperature $u(x, t)$ at a position $x$ and time $t$.

The Mother of All PDEs: The Laplacian

Most physical PDEs involve the Laplacian ($\nabla^2 u = \Delta u$), which is the divergence of the gradient ($\nabla \cdot \nabla u$). In 1D, this is just $u_{xx}$ (the second spatial derivative). It measures how much the value at a point differs from the average of its neighbors.

Three Classic PDEs

1. The Heat Equation (Diffusion)

$u_t = \alpha \nabla^2 u$ Meaning: The rate of change in temperature ($u_t$) is proportional to how much “different” a point is from its neighbors. Heat flows from hot to cold to “smooth out” the distribution.

2. The Wave Equation

$u_{tt} = c^2 \nabla^2 u$ Meaning: Acceleration ($u_{tt}$) is proportional to the local “curvature” of the medium. This models everything from guitar strings to light waves.

3. Laplace’s Equation (Equilibrium)

$\nabla^2 u = 0$ Meaning: The system has reached a state where every point is the average of its neighbors. This describes electric potentials in vacuum or the steady-state temperature of a plate.

Numerical Solution: The Finite Difference Method

Solving PDEs analytically is hard. Engineers use the Finite Difference Method, which replaces derivatives with differences on a grid.

import numpy as np

# Simulate 1D Heat Diffusion in a rod
L = 1.0       # Length
nx = 50       # Number of points
dx = L / (nx - 1)
alpha = 0.01  # Thermal diffusivity
dt = 0.001    # Time step

# Initial condition: hot spot in the middle
u = np.zeros(nx)
u[int(0.4*nx):int(0.6*nx)] = 100.0 

# Time loop: simplified explicit scheme
for _ in range(100):
  u_new = u.copy()
  for i in range(1, nx-1):
      # u_t = alpha * u_xx
      u_new[i] = u[i] + alpha * dt / dx**2 * (u[i+1] - 2*u[i] + u[i-1])
  u = u_new

print(f"Temperature at middle: {u[nx//2]:.2f}")
print("The heat has started to spread from the center to the edges.")

Exercises