Green's Functions and Boundary Value Problems

The Principle of Superposition as a Tool

All linear differential equations (DEs) possess the principle of superposition: if φ₁ and φ₂ are solutions to the homogeneous equation, then α₁φ₁ + α₂φ₂ is also a solution. The Green's function utilizes this principle to the fullest: we find the solution for a point source, and then obtain the solution for any source via superposition (convolution).

This is a powerful idea: instead of solving the problem anew every time the source changes, one needs to find the Green's function once and then merely integrate.

Definition of the Green's Function

For a linear differential operator L (for example, L = −∇² or L = ∂_t − κ∇²), the Green's function G(r, r') is defined as:

L G(r, r') = δ³(r − r')

with appropriate boundary conditions. Here, δ³(r − r') is the three-dimensional delta function: an infinite "spike" at point r', normalized so that ∫δ³ d³r = 1.

The physical meaning of G(r, r'): it is the field (temperature, potential, pressure) at point r created by a unit point source at point r'.

Application: If the source is distributed with density f(r'), then the equation Lφ = f has the solution:

φ(r) = ∫ G(r, r') f(r') d³r'

This is a convolution—the "weighted superposition" of the Green's function over all sources.

Green's Functions for Classical Equations

Poisson Equation: −∇²G = δ(r − r') in ℝ³ (the equation of electrostatics).

The solution (with G → 0 as r → ∞):

G(r, r') = 1 / (4π|r − r'|)

Physically: the potential of a point unit charge. For ρ(r') — charge density: φ(r) = (1/4πε₀) ∫ ρ(r')/|r − r'| d³r' — Coulomb's formula for distributed charges.

Heat Conduction Equation: (∂_t − κ∇²)G = δ(r−r')δ(t−t')

The solution for t > t':

G(r,t;r',t') = [4πκ(t−t')]^{−3/2} exp(−|r−r'|²/[4κ(t−t')])

This is a "Gaussian packet" that spreads out over time. As t → t'+0: G → δ³(r−r') (an instantaneous thermal "spot"). For large t: heat smears out over the entire space.

Helmholtz Equation: (∇² + k²)G = −δ(r − r')

Outgoing wave (Sommerfeld condition):

G(r, r') = e^{ik|r−r'|} / (4π|r−r'|)

This is a spherical wave emitted by a point source. Application: wave scattering, diffraction, acoustic fields in acoustics.

Numerical Example: Charged Sphere

A uniformly charged sphere of radius R, density ρ₀, in vacuum. Find the potential φ(r).

Step 1. For r > R (outside the sphere): φ = (1/4πε₀) ∫_{|r'|<R} ρ₀/|r−r'| d³r' = ρ₀(4πR³/3)/(4πε₀r) = Q/(4πε₀r) — same as for a point charge! (Gauss's theorem in electrostatics.)

Step 2. For r < R (inside): an additional contribution from charge at r' > r: φ_inside = Q/(4πε₀R) − ρ₀(r² − R²)/(6ε₀). The potential is quadratic in r inside the sphere.

Check: ∇²φ = −ρ₀/ε₀ inside ✓; ∇²φ = 0 outside ✓.

The Method of Images

The Green's function is often found by the method of images: a real charge q near a conducting plane at z = 0 is supplemented by an "image" −q at z → −z. The superposition of the two Green's functions automatically nullifies the potential on the plane—the boundary condition is satisfied!

Similarly: acoustics (reflection from walls), hydrodynamics (motion near a wall), the theory of conformal mappings in 2D.

Real Applications

MRI (magnetic resonance imaging): the Bloch equation for magnetization is linear, and its Green's function (impulse response) allows images to be reconstructed via the inverse Fourier transform.

Acoustic diagnostics: ultrasonic sensors emit a pulse and receive its echo. The received signal is the convolution of the Green's function of the medium with the source function.

Nanophotonics: the electromagnetic field's Green's function is used to calculate the rate of spontaneous emission near nanoparticles (the Purcell effect).

Connection with Fourier theory: the Green's function of the heat conduction equation is a Gaussian kernel—in Fourier space it takes the form e^{−κk²t}. This means: spatial harmonics with large wavenumber k (fine details) decay more quickly than coarse ones (small k). Thermal diffusion is the filtering of high spatial frequencies: any "sharp" temperature profile is smoothed out into a broad Gaussian peak over time.

Green's Functions in Engineering Mechanics and Geophysics

The method of Green's functions is a universal approach to solving boundary value problems with sources. In structural mechanics, the problem of deformation of a beam under a concentrated load is solved via the Green's function of the fourth-order equation: the Green's function represents the displacement of the beam from a unit force and allows calculation of the deflection from any load by integration. In geophysics, seismic waves from earthquakes are described by the Green's tensor—the solution of the elastic wave equation from a point source. Knowing the medium's Green's tensor, one can reconstruct the earthquake source mechanism from seismograms at the surface—a method used by geophysical services worldwide. In acoustics, the boundary element method uses the Green's function of the Helmholtz equation to calculate sound fields in rooms, acoustic propagation from submarines, and urban noise pollution. In electromagnetism, the Green's tensor allows calculation of the field from any source in complex geometry, applied in designing antennas and shielding from electromagnetic interference. In quantum field theory, the Green's function (propagator) describes the amplitude for quantum field propagation and is the principal object computed via Feynman diagrams.

Assignment: (a) Compute φ(r) from a sphere of radius R with volume density ρ₀ via the Poisson Green's function. Compare with Gauss's theorem. (b) Construct the Green's function for the 1D Helmholtz equation u'' + k²u = −δ(x) on the half-line x > 0 with boundary condition u(0) = 0. (Answer: G = sin(kx<) e^{ikx>}/k — where x< = min(x,x'), x> = max(x,x').) (c) How does one obtain the solution for the initial condition u(x,0) = f(x) from the heat conduction equation's Green's function?