The Green's Functions
Introduction to Green's Functions
The basic idea of a Green's function is very familiar to students of mathematical physics. However, most engineers are not typically exposed to the concept of a Green's function until possibly at the upper-graduate level. Green's functions play an important role in the solution of linear ordinary and partial differential equations, and are a key component to the development of boundary integral equation methods.
Here we present:
Consider a linear differential equation written in the general form
where L(x) is a linear, self-adjoint differential operator, u(x) is the unknown function, and f(x) is a known non-homogeneous term. For a discussion of the concept of self-adjoint and non self-adjoint differential operators please click here. Opertationally, we can write a solution to equation (1) as
where L-1 is the inverse of the differential operator L. Since L is a differential operator, it is reasonable to expect its inverse to be an integral operator. We expect the usual properties of inverses to hold,
where I is the identity operator. More specifically, we define the inverse operator as
where the kernel G(x;x') is the Green's Function associated with the differential operator L. Note that G(x;x') is a two-point function which depends on x and x'. To complete the idea of the inverse operator L, we introduce the Dirac delta function as the identity operator I. Recall the properties of the Dirac delta function are
The Green's function G(x;x') then satisfies
The solution to equation (1) can then be written directly in terms of the Green's function as
To prove that equation (7) is indeed a solution to equation (1), simply substitute as follows:
Note that we have used the linearity of the differential and inverse operators in addition to equations (4), (5), and (6) to arrive at the final answer.
The Green's function can be interpreted physically for a variety of differential operators encountered in mathematical physics. For example, consider the two-dimensional Laplace's equation,
In this case the operator . The Green's functions for this particular differential operator is known to be
From basic physics, we know the Green's function gives the potential at the point
Other physical interpretations of Green's functionsa can also be made. In elastostatics, the Green's function represents the displacement in the solid due to the application of a unit force. In heat transfer, the Green's function represents the temperature at the field point due to a unit heat source applied at the source point.
Free-Space and Region Dependent Green's Functions
In the discuassion above concerning the solution of a differential equation with a Green's function, no mention was made of boundary conditions for the problem. This is true when we are seeking a particular solution to equation (6),
The particualr solution is, of course, independent of any boundary conditions for the problem. However, we can always add homogeneous solutions to the Green's function,
Go, the particular solution, is termed the free-space Green's
function and is also referred to as the fundamental solution for the
differential operator L(x). As we have seen with the example of Laplace's equation
given in the previous section, the free-space Green's function is singular. The homogenous
solution GR is non-singular. Since GR is a homogenous
solution, it will contain constants, which can be evaluated to satisfy any boundary
conditions for the problem. We term the full Green's functions G(x;x') a region-dependent
Green's function since, in general, it contains not only the particular solution, but
also the necessary terms to satisfy any boundary conditions for the problem. As shown by
Martin and Rizzo [Int. Jnl. Num. Meth. Engr.,
Consider the Helmholtz equation in three dimensions,
where D is the Laplace operator, . In this case,
and we seek the Greens function
Note that the three-dimensional Dirac delta function is simply a compact representation for the product of delta functions in each coordinate,
To obtain the free-space Green's function for this example problem, we will use a Fourier transform method. Since we will only be calculating the free-space component of the Green's function, we can use a single variable r = x-x' , as the free-space Green's function will only depend on the relative distance between the source and field points and not their absolute positions. The Fourier transform pair we will use is:
Applying the forward transform to the differential equation for the Greens function we have
Now, let . Then
In transform space the Greens function is then
In physical space the Green's function is then given through the inversion integral
The integral is an isotropic Fourier integral since it depends only on the magnitude of q, which is q, but does depend on the direction of q. Barton [Elements of Greens Functions and Propagation, Oxford Science Press, 1989] gives the general result for isotropic Fourier integrals in three dimensions as
where R is the magnitude of r. Utilizing this result, the inversion integral we seek is then
Since the integrand is even,
This integral can be evaluated by contour integration. First, the sin term is written in terms of complex exponentials,
and the integral is written as
The first integral will be evaluated by considering a contour in the complex q plane. Since the denominator of the integrand has poles on the real axis, we introduce a small imaginary part to offset the poles from the real q axis,
We next take a contour in the upper half-plane due to the behavior of the numerator of the integrand as q becomes large. Using the theory of integration by residues we then have
Taking the limit as e tends to zero we then have
Similarly, for I2, we take a contour in the lower half plane and obtain
The Green's function is then
Texts whose primary focus is Green's functions:
Barton, G., Elements of Green's Functions and Propagation, Oxford University Press, New York, USA, 1989.
Kellog, O. D., Foundations of Potential Theory, Dover Science Publications, New York, USA, 1969.
Melnikov, Y., Green's Functions in Applied Mechanics, Computational Mechanics Publications, Southampton, Great Britain, 1994.
Roach, G. F., Green's Functions, 2nd Edition, Cambridge University Press, Cambridge, Great Britain, 1982.
Stakgold, I., Green's Functions and Boundary Value Problems, Wiley-Interscience Publications, New York, USA, 1979.
Texts with an emphasis on mathematical physics with substantial sections on Greens functions
Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, 2nd Edition, Clarendon Press, Oxford, Great Britain, 1959.
Courant, D., and Hilbert, D., Methods of Mathematical Physics, Wiley-Interscience Publications, New York, USA, 1953.
Morse, P. M. and Feshbach, H., Methods of Theoretical Physics, McGraw-Hill, New
York, USA, 1953.