A kernel (distributional or classical) that acts as a right-inverse of a linear differential or integral operator under specified boundary conditions, mapping sources to responses by convolution or integral against the kernel.

Principle

Solving an inhomogeneous linear equation L[u]=f is achieved by u(x)=∫G(x,x')f(x')dx' when G is a Green's function for L with the chosen boundary conditions; G encodes both the operator's local character and the global constraints from boundaries.

Demonstration

For the Poisson operator on ℝ^3, ΔG(x,x')=−δ(x−x') and the solution of Δu=f is u(x)=∫G(x,x')f(x')dx'. On bounded domains G must satisfy the operator equation plus the boundary conditions on its second variable.

Misapplication

Using a free-space Green's function in a bounded domain with nontrivial boundary conditions produces solutions that violate the required constraints and lead to physically incorrect responses.

Consequence

A correct Green's function provides explicit integral formulas for solutions, enables construction of resolvents and propagators, and translates linear PDE problems into kernel operations amenable to analysis and approximation.

Reversal

Instead of using an integral kernel inversion, one may discretize the operator and solve the linear system numerically; this circumvents explicit Green's functions but may obscure analytic structure.

Boundary

Defined only for linear operators (or linearizations) and requires specification of boundary or radiation conditions; nonlinear problems admit Green‑function techniques only after linearization or perturbation.

Semantic Tension

'Fundamental solution' denotes a kernel solving the operator equation in free space without boundary enforcement, whereas 'Green's function' typically enforces particular boundary conditions; conflating them causes misuse.

Synthesis

A Green's function is the operator-specific kernel that converts a given source into its linear response while encoding boundary constraints, turning linear operator inversion into an integral transformation.