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
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.
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.
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