8: Green's Functions
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In this chapter we will investigate the solution of nonhomogeneous differential equations using Green's functions. Our goal is to solve the nonhomogeneous differential equation
\[L[u]=f, \nonumber \]
where \(L\) is a differential operator. The solution is formally given by
\[u=L^{-1}[f] \nonumber \]
The inverse of a differential operator is an integral operator, which we seek to write in the form
\[u=\int G(x, \xi) f(\xi) d \xi \nonumber \]
The function \(G(x, \xi)\) is referred to as the kernel of the integral operator and is called the Green's function.
The history of the Green's function dates back to 1828 , when George Green published work in which he sought solutions of Poisson's equation \(\nabla^{2} u=f\) for the electric potential \(u\) defined inside a bounded volume with specified boundary conditions on the surface of the volume. He introduced a function now identified as what Riemann later coined the "Green's function".
We will restrict our discussion to Green's functions for ordinary differential equations. Extensions to partial differential equations are typically one of the subjects of a PDE course. We will begin our investigations by examining solutions of nonhomogeneous second order linear differential equations using the Method of Variation of Parameters, which is typically seen in a first course on differential equations. We will identify the Green's function for both initial value and boundary value problems. We will then focus on boundary value Green's functions and their properties. Determination of Green's functions is also possible using Sturm-Liouville theory. This leads to series representation of Green's functions, which we will study in the last section of this chapter.