ABSTRACT
A Green’s function for a partial differential equation is the solution of its adjoint equation, where the forcing term is the Dirac delta function due to a unit point source in a given domain Ω. This solution enables us to generate solutions of partial differential equations subject to a range of boundary conditions and internal sources. This technique is important in a variety of physical problems. For the derivation of Green’s functions, we can assume the presence of an internal source or a certain boundary condition which results in the same effect as the point source. If L is a linear differential operator, andLu(x) = f(x) is a linear differential equation, where x denotes a point in the domain Ω ∈ Rn, and f(x) is the nonhomogeneous term in the differential equation, then the solution of this equation subject to homogeneous boundary conditions can be written in the form u(x) =
∫ · · ·∫ Ω G(x,y) f(y) dy,
where G(x,y) is the Green’s function, which depends only on the adjoint operator L∗ of L and on the geometry of the problem for homogeneous adjoint initial and boundary conditions.
