Concept in the solution of linear partial differential equations
In
mathematics, a fundamental solution for a linear
partial differential operator L is a formulation in the language of
distribution theory of the older idea of a
Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions).
In terms of the
Dirac delta "function" δ(x), a fundamental solution F is a solution of the
inhomogeneous equation
LF = δ(x).
Here F is a priori only assumed to be a
distribution.
This concept has long been utilized for the
Laplacian in two and three dimensions. It was investigated for all dimensions for the Laplacian by
Marcel Riesz.
The existence of a fundamental solution for any operator with
constant coefficients — the most important case, directly linked to the possibility of using
convolution to solve an
arbitrary
right hand side — was shown by
Bernard Malgrange and
Leon Ehrenpreis. In the context of
functional analysis, fundamental solutions are usually developed via the
Fredholm alternative and explored in
Fredholm theory.
Example
Consider the following differential equation Lf = sin(x) with
The fundamental solutions can be obtained by solving LF = δ(x), explicitly,
Since for the unit step function (also known as the
Heaviside function) H we have
there is a solution
Here
C is an arbitrary constant introduced by the integration. For convenience, set
C = −1/2.
After integrating and choosing the new integration constant as zero, one has
Motivation
Once the fundamental solution is found, it is straightforward to find a solution of the original equation, through
convolution of the fundamental solution and the desired right hand side.
Fundamental solutions also play an important role in the numerical solution of partial differential equations by the
boundary element method.
Application to the example
Consider the operator L and the differential equation mentioned in the example,
We can find the solution of the original equation by
convolution (denoted by an asterisk) of the right-hand side with the fundamental solution :
This shows that some care must be taken when working with functions which do not have enough regularity (e.g. compact support, L1 integrability) since, we know that the desired solution is f(x) = −sin(x), while the above integral diverges for all x. The two expressions for f are, however, equal as distributions.
An example that more clearly works
where
I is the
characteristic (indicator) function of the unit interval
[0,1]. In that case, it can be verified that the convolution of
I with
F(x) = |x|/2 is
which is a solution, i.e., has second derivative equal to
I.
Proof that the convolution is a solution
Denote the
convolution of functions F and g as F ∗ g. Say we are trying to find the solution of Lf = g(x). We want to prove that F ∗ g is a solution of the previous equation, i.e. we want to prove that L(F ∗ g) = g. When applying the differential operator, L, to the convolution, it is known that
provided
L has constant coefficients.
If F is the fundamental solution, the right side of the equation reduces to
But since the delta function is an
identity element for convolution, this is simply g(x). Summing up,
Therefore, if F is the fundamental solution, the convolution F ∗ g is one solution of Lf = g(x). This does not mean that it is the only solution. Several solutions for different initial conditions can be found.
Fundamental solutions for some partial differential equations
The following can be obtained by means of Fourier transform:
Laplace equation
For the
Laplace equation,
the fundamental solutions in two and three dimensions, respectively, are
Screened Poisson equation
For the
screened Poisson equation,
the fundamental solutions are
where
is a
modified Bessel function of the second kind.
In higher dimensions the fundamental solution of the screened Poisson equation is given by the
Bessel potential.
Biharmonic equation
For the
Biharmonic equation,
the biharmonic equation has the fundamental solutions
Signal processing
In
signal processing, the analog of the fundamental solution of a differential equation is called the
impulse response of a filter.
See also
References