A kernel function G(x, s) that represents the response of a linear differential operator L to a point source (delta distribution) located at s, so that L[G(·,s)]=δ(·−s); used to construct particular solutions of linear inhomogeneous boundary-value problems by convolution or integration against source terms.

Principle

Superposition for linear operators: because L is linear, the response to a general source f(x) can be represented as the integral of the Green's function against f, yielding the solution as u(x)=∫G(x,s)f(s)ds plus homogeneous solutions imposed by boundary conditions.

Demonstration

Solving Poisson's equation −Δu=f in R^n: the fundamental solution (Green's function) for −Δ yields u as the convolution of f with the Green's kernel; in bounded domains one modifies the kernel to satisfy boundary conditions and obtains u(x)=∫_Ω G(x,s)f(s)ds.

Misapplication

Using a Green's function derived for a different operator or for free space without adjusting for boundary conditions or operator coefficients; applying the linear superposition formula to a nonlinear PDE where the operator depends on u.

Consequence

When correctly identified and combined with the homogeneous solution, the Green's function converts an inhomogeneous PDE into an explicit integral representation; it clarifies the influence of localized sources and facilitates analytical and numerical evaluation of solutions.

Reversal

Instead of representing the response to a point source (impulse-to-response), the inverse problem asks for the source distribution given observed responses; inverting the integral relation yields source reconstruction rather than forward Green's representation.

Boundary

Applies only to linear differential operators (possibly with distributions) and usually requires specifying boundary conditions; does not exist or is not useful for generic nonlinear operators or when no fundamental solution satisfying the physical boundary conditions can be found.

Semantic Tension

Green's function as a distributional fundamental solution versus as a kernel tailored to boundary conditions: the former is a universal inverse of the operator in free space, the latter is a problem-specific construction that enforces boundaries; both notions compete in terminology.

Synthesis

A Green's function is the operator inverse's kernel for linear PDEs: it encapsulates the point-source response and, via superposition and integration, yields particular solutions appropriate to the operator and chosen boundary conditions.