Abstract
The chapter starts by identifying a Green's function as the contribution to the solution of a linear differential equation that results from the inclusion of a point-source inhomogeneous term to an otherwise homogeneous equation subject to given boundary conditions. Green's functions therefore necessarily depend upon two sets of coordinates: those of the inhomogeneity and those of the point at which the solution is described. The point-source inhomogeneity causes Green's functions to have derivatives that are discontinuous when these two points coincide. General linear inhomogeneous differential equations (with boundary conditions) have solutions that can be constructed as integrals involving the applicable Green's functions. The symmetry and other properties of Green's functions are surveyed, and Green's functions are illustrated for ODEs (problems in one dimension) and for PDEs (problems in more than one dimension).
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