ABSTRACT
The types of problems considered for the heat, wave, and Laplace equations have solutions that are determined uniquely by the prescribed data (boundary conditions, initial conditions, and any nonhomogeneous term in the equation). It is natural, therefore, to seek a formula that gives the solution directly in terms of the data. Such closed-form solutions are constructed by means of the so-called Green’s function of the given problem, and are of great importance in practical applications. This chapter presents the closed-form solutions for the heat, wave, and Laplace equations. Green’s functions and representation formulas in terms of such functions can also be constructed for initial boundary value problems (IBVPs) with mixed boundary conditions and for IBVPs where the space variable takes values in a semi-infinite or infinite interval.
