Chapter 3: Laplace Equation and Potential Theory | Partial Differential Equations and Boundary Value Problems

🎯 Learning Objectives

Understand the fundamentals of the Laplace equation and potential theory
Learn the properties of harmonic functions and the maximum principle
Solve the Laplace equation in polar and cylindrical coordinates
Master solution methods for boundary value problems using Green's functions
Understand the Poisson equation and handling of charge distributions and heat sources
Implement numerical methods using iterative methods (Jacobi, Gauss-Seidel, SOR)
Understand applications to materials science (electrostatic field analysis, steady-state heat conduction)

📖 What is the Laplace Equation?

Definition of the Laplace Equation

The Laplace equation is an elliptic partial differential equation of the following form:

\[ \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0 \]

Solutions to the Laplace equation are called harmonic functions.

The Poisson equation is the case where the right-hand side is non-zero:

\[ \nabla^2 u = f(x,y,z) \]

where \(f\) represents heat sources or charge density.

Physical Significance

Electrostatic potential: Electric potential \(V\) in a region without charges \(\nabla^2 V = 0\)
Steady-state heat conduction: Temperature distribution \(T\) in a region without heat sources \(\nabla^2 T = 0\)
Fluid potential: Velocity potential \(\phi\) for incompressible, inviscid flow \(\nabla^2 \phi = 0\)
Gravitational potential: Gravitational potential in a region without mass

Properties of Harmonic Functions

Maximum Principle: Harmonic functions do not have extrema in the interior; maximum and minimum values are attained on the boundary.

Mean Value Theorem: The value of a harmonic function at a point \((x_0, y_0)\) equals the average value on a circle centered at that point:

\[ u(x_0, y_0) = \frac{1}{2\pi} \int_0^{2\pi} u(x_0 + r\cos\theta, y_0 + r\sin\theta) d\theta \]

Uniqueness: Under Dirichlet boundary conditions, the solution to the Laplace equation is unique.

💻 Example 3.1: Verification of Harmonic Functions and Maximum Principle

Python implementation: Verification of harmonic function properties

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

"""
Example: 💻 Example 3.1: Verification of Harmonic Functions and Maximu

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

# Example of a harmonic function: u(x,y) = x^2 - y^2 (real part of z^2)
def harmonic_function(x, y):
 return x**2 - y**2

# Create 2D grid
x = np.linspace(-2, 2, 100)
y = np.linspace(-2, 2, 100)
X, Y = np.meshgrid(x, y)
U = harmonic_function(X, Y)

# Calculate Laplacian (numerical differentiation)
dx = x[1] - x[0]
dy = y[1] - y[0]

# Second partial derivatives
d2u_dx2 = np.zeros_like(U)
d2u_dy2 = np.zeros_like(U)

for i in range(1, len(x)-1):
 for j in range(1, len(y)-1):
 d2u_dx2[j, i] = (U[j, i+1] - 2*U[j, i] + U[j, i-1]) / dx**2
 d2u_dy2[j, i] = (U[j+1, i] - 2*U[j, i] + U[j-1, i]) / dy**2

laplacian = d2u_dx2 + d2u_dy2

# Visualization
fig = plt.figure(figsize=(15, 5))

# Harmonic function
ax1 = fig.add_subplot(131, projection='3d')
ax1.plot_surface(X, Y, U, cmap='viridis', alpha=0.8)
ax1.set_xlabel('x')
ax1.set_ylabel('y')
ax1.set_zlabel('u(x,y)')
ax1.set_title('Harmonic function: u = x² - y²')

# Contour plot
ax2 = fig.add_subplot(132)
contour = ax2.contour(X, Y, U, levels=20, cmap='viridis')
ax2.clabel(contour, inline=True, fontsize=8)
ax2.set_xlabel('x')
ax2.set_ylabel('y')
ax2.set_title('Contour plot')
ax2.axis('equal')

# Laplacian
ax3 = fig.add_subplot(133)
laplacian_plot = ax3.imshow(laplacian[1:-1, 1:-1], extent=[-2, 2, -2, 2],
 origin='lower', cmap='RdBu', vmin=-0.1, vmax=0.1)
plt.colorbar(laplacian_plot, ax=ax3, label='∇²u')
ax3.set_xlabel('x')
ax3.set_ylabel('y')
ax3.set_title(f'Laplacian (max: {np.max(np.abs(laplacian[1:-1,1:-1])):.2e})')

plt.tight_layout()
plt.savefig('laplace_harmonic_function.png', dpi=300, bbox_inches='tight')
plt.show()

# Verification of maximum principle
print("=== Verification of Maximum Principle ===")
print(f"Maximum in interior: {np.max(U[10:-10, 10:-10]):.4f}")
print(f"Maximum on boundary: {np.max([np.max(U[0,:]), np.max(U[-1,:]), np.max(U[:,0]), np.max(U[:,-1])]):.4f}")
print(f"Minimum in interior: {np.min(U[10:-10, 10:-10]):.4f}")
print(f"Minimum on boundary: {np.min([np.min(U[0,:]), np.min(U[-1,:]), np.min(U[:,0]), np.min(U[:,-1])]):.4f}")

Output explanation:

\(u = x^2 - y^2\) is a harmonic function (\(\nabla^2 u = 2 - 2 = 0\))
Verification that the Laplacian is zero within numerical error
By the maximum principle, extrema exist on the boundary

📚 Summary

The Laplace equation describes steady-state physical phenomena and has important properties as a harmonic function, including the maximum principle
Through separation of variables in polar and cylindrical coordinates, analytical solutions can be obtained for circular and spherical domains
Using Green's functions, solutions to boundary value problems can be constructed from the response to point sources
The Poisson equation handles problems involving heat sources and charge distributions, and is widely applied in materials science
Iterative methods (Jacobi, Gauss-Seidel, SOR) can be used to find numerical solutions, with SOR being the fastest
Practical applications to materials science are possible, such as steady-state heat conduction in complex geometries

💡 Exercise Problems

Verification of harmonic functions: Verify by calculating the Laplacian that \(u(x,y) = xy\) is not a harmonic function.
Solution in polar coordinates: Find the solution to the Laplace equation on a disk of radius \(a\) satisfying the boundary condition \(u(a,\theta) = \cos(2\theta)\), and visualize it.
Application of Green's function: Using the Green's function for a rectangular domain, find the temperature distribution for a heat source \(f(x,y) = \delta(x-0.5, y-0.5)\).
Comparison of convergence: Vary the relaxation parameter \(\omega\) for the SOR method from 1.0 to 2.0, and find the optimal \(\omega\).
Complex geometry: Solve the Laplace equation in a rectangular domain with a circular hole, and visualize the temperature distribution around the hole.

← Chapter 2: Heat Equation and Diffusion Chapter 4: Variational Methods and Optimization →