Concept

# Green's function

Summary
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname{L} is the linear differential operator, then
• the Green's function G is the solution of the equation \operatorname{L} G = \delta, where \delta is Dirac's delta function;
• the solution of the initial-value problem \operatorname{L} y = f is the convolution (G \ast f).
Through the superposition principle, given a linear ordinary differential equation (ODE), \operatorname{L} y = f, one can first solve \operatorname{L} G = \delta_s, for each s, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L. Green's functions are named after the British mathematician Geo
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