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).
Here F is a priori only assumed to be a distribution.
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.
Consider the following differential equation Lf = sin(x) with
The fundamental solutions can be obtained by solving LF = δ(x), explicitly,
Since for 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
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 readily verified that the convolution I ∗ F with F(x) = |x|/2 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:
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.
For the Biharmonic equation,
the biharmonic equation has the fundamental solutions