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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

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.

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

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.

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, *L*^{1} 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.

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

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:

For the Laplace equation,

For the screened Poisson equation,

In higher dimensions the fundamental solution of the screened Poisson equation is given by the Bessel potential.

For the Biharmonic equation,

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

- "Fundamental solution",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - For adjustment to Green's function on the boundary see Shijue Wu notes.