or equivalently where p(x) > 0 in (a, b) and p'(x) and q(x) are all continuous functions in interval [a, b].
Despite its appearance, the self-adjoint form is general enough to embrace virtually all second-order linear ordinary differential equations. For example,
We can easily transform (1.3) into self-adjoint form by multiplying through by the function
producing
Equation (1.5) is now in self-adjoint form provided. We pick p(x) such that
By solving this first-order ordinary differential equation for p(x), we find
Thus, we can rewrite (1.3) into the form as the definition (1.1). And it is notationally convenient to introduce the special differential operator
where
L is called a self-adjoint operator.
For the case of two-dimensional problem, we could also introduce the
concept of self-adjoint operator. Since we will concentrate on the Laplace
differential operator in this report, the concept will be described under
the consideration of this operator. Suppose 
be the Laplace differential operator on the domain 
. Consider the integral 
where 
is a differential element of area of the region
under consideration. Using the integration by parts, therefore, we could
have
where 
is the outward unit normal vector, ds is a differential element
of arc-length along 
. Since
where 
denotes the gradient operator, defined in Cartesian coordinates as 
, 
and 
. The equation (1.10) becomes
As we mentioned in the introduction, the boundary condition of the problems which will be discussed in this report is homogeneous, so that the Laplace differential operator plus the imposed boundary condition must be self-adjoint.
On the other hand, we will try to use Green's function to solve the differential equations in this report, but the method of Green's function involved an important function, that is Dirac delta function. Therefore, we will discuss the definition and some important results in next section.
