In
mathematics, a fundamental solution for a linear
partial differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
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 differentia ...
(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 varia ...
.
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
Marcel Riesz ( hu, Riesz Marcell ; 16 November 1886 – 4 September 1969) was a Hungarian mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, ...
.
The existence of a fundamental solution for any operator with
constant coefficients
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
— 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
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 pre ...
. 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 defined on ...
, 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 th ...
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 given ...
.
Example
Consider the following differential equation with
The fundamental solutions can be obtained by solving , explicitly,
Since for the
Heaviside function we have
there is a solution
Here is an arbitrary constant introduced by the integration. For convenience, set .
After integrating
and choosing the new integration constant as zero, one has
Motivation
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.
Application to the example
Consider the operator 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 , while the above integral diverges for all . The two expressions for are, however, equal as distributions.
An example that more clearly works
where is the
characteristic (indicator) function of the unit interval . In that case, it can be verified that the convolution of with is
which is a solution, i.e., has second derivative equal to .
Proof that the convolution is a solution
Denote the
convolution 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
provided has constant coefficients.
If is the fundamental solution, the right side of the equation reduces to
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 ...
for convolution, this is simply . Summing up,
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 = \nab ...
,
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
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 t ...
,
the biharmonic equation has the fundamental solutions
Signal processing
In
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, d ...
, 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 react ...
of a filter.
See also
*
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 differentia ...
*
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 react ...
*
Parametrix
In mathematics, and specifically the field of partial differential equations (PDEs), a parametrix is an approximation to a fundamental solution of a PDE, and is essentially an approximate inverse to a differential operator.
A parametrix for a dif ...
References
*
* For adjustment to Green's function on the boundary se
Shijue Wu notes
{{DEFAULTSORT:Fundamental Solution
Partial differential equations
Generalized functions