mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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

Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...

(although unlike Green's functions, fundamental solutions do not address boundary conditions). In terms of the Dirac delta "function" , a fundamental solution is a solution of the inhomogeneous equation Here is ''a priori'' only assumed to be a

distribution Distribution may refer to: Mathematics * Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a vari ...

. This concept has long been utilized for the

Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...

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 In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...

to solve an

arbitrary Arbitrariness is the quality of being "determined by chance, whim, or impulse, and not by necessity, reason, or principle". It is also used to refer to a choice made without any specific criterion or restraint. Arbitrary decisions are not necess ...

right hand side In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.Bernard Malgrange and

Leon Ehrenpreis Eliezer 'Leon' Ehrenpreis (May 22, 1930 – August 16, 2010, Brooklyn) was a mathematician at Temple University who proved the Malgrange–Ehrenpreis theorem, the fundamental theorem about differential operators with constant coefficients. He p ...

. In the context of

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...

, fundamental solutions are usually developed via the

Fredholm alternative In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a ...

and explored in

Fredholm theory In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is giv ...

Example

Consider the following differential equation with

L = \frac .

The fundamental solutions can be obtained by solving , explicitly,

\frac F(x) = \delta(x) \,.

Since for the

Heaviside function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argumen ...

we have

\frac H(x) = \delta(x) \,,

there is a solution

\frac F(x) = H(x) + C \,.

Here is an arbitrary constant introduced by the integration. For convenience, set . After integrating

\frac

and choosing the new integration constant as zero, one has

F(x) = x H(x) - \fracx = \frac , x,  ~.

Motivation

Once the fundamental solution is found, it is straightforward to find a solution of the original equation, through

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 and the differential equation mentioned in the example,

\frac f(x) = \sin(x) \,.

We can find the solution

f(x)

of the original equation by

(denoted by an asterisk) of the right-hand side

\sin(x)

with the fundamental solution

F(x) = \frac, x,

f(x) = (F * \sin)(x) := \int_^ \frac, 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, ''L''¹ integrability) since, we know that the desired solution is , while the above integral diverges for all . The two expressions for are, however, equal as distributions.

An example that more clearly works

\frac f(x) = I(x) \,,

where is the characteristic (indicator) function of the unit interval . In that case, it can be verified that the convolution of with is

(I * F)(x) = \begin
  \fracx^2-\fracx+\frac, & 0 \le x \le 1 \\
  , \fracx-\frac, , & \text
\end

which is a solution, i.e., has second derivative equal to .

Proof that the convolution is a solution

Denote the

of functions and as . Say we are trying to find the solution of . We want to prove that is a solution of the previous equation, i.e. we want to prove that . When applying the differential operator, , to the convolution, it is known that

L(F*g) = (LF)*g \,,

provided has constant coefficients. If is the fundamental solution, the right side of the equation reduces to

\delta * g~.

But since the delta function is an

identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

for convolution, this is simply . Summing up,

L(F*g) = (LF)*g = \delta(x)*g(x) = \int_^ \delta (x-y) g(y) \, dy = g(x) \,.

Therefore, if is the fundamental solution, the convolution is one solution of . 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:

Laplace equation

For the

Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \n ...

\Phi(\mathbf,\mathbf') = \delta(\mathbf-\mathbf')

the fundamental solutions in two and three dimensions, respectively, are

\Phi_\textrm(\mathbf,\mathbf') = -\frac\ln, \mathbf-\mathbf', ,\qquad
\Phi_\textrm(\mathbf,\mathbf') = \frac ~.

Screened Poisson equation

For the

screened Poisson equation In physics, the screened Poisson equation is a Poisson equation, which arises in (for example) the Klein–Gordon equation, electric field screening in plasmas, and nonlocal granular fluidity in granular flow. Statement of the equation The equa ...

\Phi(\mathbf,\mathbf') = \delta(\mathbf-\mathbf'), \quad k \in \R,

the fundamental solutions are

\Phi_\textrm(\mathbf,\mathbf') = \fracK_0(k, \mathbf-\mathbf', ),\qquad
\Phi_\textrm(\mathbf,\mathbf') = \frac,

where

K_0

is a

modified Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

of the second kind. In higher dimensions the fundamental solution of the screened Poisson equation is given by the

Bessel potential In mathematics, the Bessel potential is a potential (named after Friedrich Wilhelm Bessel) similar to the Riesz potential but with better decay properties at infinity. If ''s'' is a complex number with positive real part then the Bessel potentia ...

Biharmonic equation

For the

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. Specifically, it is used in the modeling of ...

\Phi(\mathbf,\mathbf') = \delta(\mathbf-\mathbf')

the biharmonic equation has the fundamental solutions

\Phi_\textrm(\mathbf,\mathbf') = -\frac\ln, \mathbf-\mathbf', ,\qquad
\Phi_\textrm(\mathbf,\mathbf') = \frac ~.

Signal processing

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

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

impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...

of a filter.

References

* * For adjustment to Green's function on the boundary se
Shijue Wu notes
{{DEFAULTSORT:Fundamental Solution Partial differential equations Generalized functions

Example

Motivation

Application to the example

An example that more clearly works

Proof that the convolution is a solution

Fundamental solutions for some partial differential equations

Laplace equation

Screened Poisson equation

Biharmonic equation

Signal processing

See also

References