Green's Functions

PDEs · Axiom Academy

Understanding the fundamental solution method for solving Laplace and Poisson equations 1. The Green's Function Concept A Green's function G(x, y) is the fundamental solution to a PDE with a point source at y. For the Laplace equation, it satisfies: where δ(x - y) is the Dirac delta function representing a point source. Think of it as the response of the system to a single "impulse" at point y. In two dimensions (R²), the fundamental solution to Laplace's equation is logarithmic: where r = |x - y| is the distance from the source point y to the observation point x. Verification: Computing ∇²G in polar coordinates centered at y gives -δ(x - y). In three dimensions (R³), the fundamental solution has inverse distance form: where r = |x - y| is again the distance from source to observation point. Physical Analogy: This is exactly the electrostatic potential from a point charge, or the gravitational potential from a point mass. Green's identities relate volume integrals to surface integrals and are crucial for deriving solution formulas. These identities follow from the divergence theorem and integration by parts. They connect interior values to boundary data. 5. Solving the Poisson Equation For the inhomogeneous Poisson equation ∇²u = f in a domain, the solution can be expressed using the Green's function: This is a convolution of the source term f with the Green's function. Each point source f(y)dy at y contributes G(x,y)f(y)dy to the solution at x.

This is the written version of the interactive lesson above. See the full PDEs course.