Fundamental solution

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. In terms of the Dirac delta "function" δ(x), a fundamental solution F is the 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.

Example

Consider the following differential equation Lf = sin(x) with

$L=\frac{d^2}{d x^2}$.

The fundamental solutions can be obtained by solving LF = δ(x), explicitly,

$\frac{d^2}{d x^2} F(x) = \delta(x) ~.$

Since for the Heaviside function H we have

$\frac{d}{d x} H(x) = \delta(x) ~,$

there is a solution

$\frac{d}{d x} F(x) = H(x) + C ~.$

Here C is an arbitrary constant introduced by the integration. For convenience, set C = − 1/2.

After integrating dFdx and choosing the new integration constant as zero, one has

$F(x) = x H(x) - \frac{1}{2}x = \frac{1}{2} |x| ~.$

Motivation

Once the fundamental solution is found, it is easy to find the desired solution of the original equation. In fact, this process is achieved by convolution.

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,

$\frac{d^2}{d x^2} f(x) = \sin(x) ~.$

We can find the solution of the original equation by convolving the right-hand side sin(x) with the fundamental solution F(x) = |x |/2,

$f(x) = \int_{-\infty}^{\infty} \frac{1}{2}|x - y|\sin(y)dy ~.$

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) = −sinx, while the above integral diverges for all x. The two expressions for f are, however, equal as distributions.

An example that more clearly works

$\frac{d^2}{d x^2} f(x) = I(x) ~,$

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

$L(F*g) = (LF)*g ~,$

provided L has constant coefficients.

If F is the fundamental solution, the right side of the equation reduces to

$\delta * g~.$

But since the delta function is an identity element for convolution, this is simply g(x). Summing up,

$L(F*g)=(LF)*g=\delta(x)*g(x)=\int_{-\infty}^{\infty} \delta (x-y) g(y) dy=g(x) ~.$

Therefore, if F is the fundamental solution, the convolution Fg 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

Laplace equation

For the Laplace equation,

$[-\Delta] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')$

the fundamental solutions in two and three dimensions are

$\Phi_{2D}(\mathbf{x},\mathbf{x}')= -\frac{1}{2\pi}\ln|\mathbf{x}-\mathbf{x}'|,\quad \Phi_{3D}(\mathbf{x},\mathbf{x}')= \frac{1}{4\pi|\mathbf{x}-\mathbf{x}'|} ~.$

Screened Poisson equation

For the Screened Poisson equation, where the parameter k is real and the fundamental solution a modified Bessel function,

$[-\Delta+k^2] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}') ~,$

the two- and three-dimensional Helmholtz equations have the fundamental solutions

$\Phi_{2D}(\mathbf{x},\mathbf{x}')= \frac{1}{2\pi}K_0(k|\mathbf{x}-\mathbf{x}'|),\quad \Phi_{3D}(\mathbf{x},\mathbf{x}')= \frac{1}{4\pi|\mathbf{x}-\mathbf{x}'|}\exp(-k|\mathbf{x}-\mathbf{x}'|) ~.$

Biharmonic equation

For the Biharmonic equation,

$[-\Delta^2] \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}')$

the biharmonic equation has the fundamental solutions

$\Phi_{2D}(\mathbf{x},\mathbf{x}')= -\frac{|\mathbf{x}-\mathbf{x}'|^2}{8\pi}(\ln|\mathbf{x}-\mathbf{x}'| - 1),\quad \Phi_{3D}(\mathbf{x},\mathbf{x}')= \frac{|\mathbf{x}-\mathbf{x}'|}{8\pi} ~.$

Signal processing

Main article: Impulse response

In signal processing, the analog of the fundamental solution of a differential equation is called the impulse response of a filter.