In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory

Distribution (mathematics)

In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

 of the older idea of a Green's function

Green's function

In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...

. In terms of the Dirac delta function

Dirac delta function

The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

 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 Schwartz distribution.

This concept was long known for the Laplacian in two and three dimensions. It was investigated for all dimensions for the Laplacian by Marcel Riesz

Marcel Riesz

Marcel Riesz was a Hungarian mathematician who was born in Győr, Hungary . He moved to Sweden in 1908 and spent the rest of his life there, dying in Lund, where he was a professor from 1926 at Lund University...

. The existence of a fundamental solution for any operator with constant coefficients

Constant coefficients

In mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. In other words, it singles out special operators, within the larger class of...

 — the most important case, directly linked to the possibility of using convolution

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

 to solve an arbitrary

Arbitrary

Arbitrariness is a term given to choices and actions subject to individual will, judgment or preference, based solely upon an individual's opinion or discretion.Arbitrary decisions are not necessarily the same as random decisions...

right hand side

Sides of an equation

In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric...

 — was shown by Malgrange and Leon Ehrenpreis

Leon Ehrenpreis

Leon Ehrenpreis was a mathematician at Temple University who proved the Malgrange–Ehrenpreis theorem, the fundamental theorem about differential operators with constant coefficients....

.

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 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 taking the new integration constant as zero, we get

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

Boundary element method

The boundary element method is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations . It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, and fracture...

.

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 convolving 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, integrability) since, we know that the desired solution is , while the above integral diverges for all x. The two expressions for f are, however, equal as distributions.

Proof that the convolution is a solution

Denote the convolution

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

 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, ie. we want to prove that L(F*g) = g(x). 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

δ*g.

But since the delta function is an identity element

Identity element

In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

 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.

Laplace equation

For the Laplace equation,
the fundamental solutions in two and three dimensions are

Screened Poisson equation

For the Screened Poisson equation

Screened Poisson equation

In Physics, the screened Poisson equation is the following partial differential equation:\left[ \Delta - \lambda^2 \right] u = - f...

, where the parameter k is real and the fundamental solution a modified Bessel function

Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...

,


the two and three dimensional Helmholtz equations have the fundamental solutions

Biharmonic equation

For the Biharmonic equation

Biharmonic equation

In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows...

,


the biharmonic equation has the fundamental solutions

Signal processing

In signal processing

Signal processing

Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

, the analog of the fundamental solution of a differential equation is called the impulse response

Impulse response

In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...

of a filter.