Fundamental solution

A fundamental solution is a mathematical object from distribution theory . They are solutions of a certain class of inhomogeneous partial differential equations . With their help and the convolution theorem , special solutions of similar differential equations can be calculated. According to the Malgrange-Ehrenpreis theorem, there is a fundamental solution for every linear partial differential operator with constant coefficients.

Laurent Schwartz , who was the first to describe the theory of distribution , was also the first to define the concept of the fundamental solution. It can be understood as a further development of the older concept of Green's function . In the more general sense, these functions are special solutions to boundary value problems , which can also be transformed into special solutions of corresponding inhomogeneous boundary value problems with the help of convolution .

definition

Let be a linear differential operator with constant complex coefficients. Then the distribution is called the fundamental solution of , if it is a distributional solution of the equation ${\displaystyle L}$ ${\displaystyle G\in {\mathcal {D}}'(\mathbb {R} ^{n})}$ ${\displaystyle L}$

${\displaystyle LG=\delta }$

where the Dirac delta distribution is meant. ${\displaystyle \delta }$

Solving inhomogeneous differential equations

If a fundamental solution is known for a linear differential operator , a solution of the equation is obtained ${\displaystyle L}$ ${\displaystyle G}$ ${\displaystyle u(x)}$

${\displaystyle Lu(x)=f(x)}$

for everyone by folding the fundamental solution with the right side ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle G}$ ${\displaystyle f}$

${\displaystyle u(x)=(G*f)(x)=\int _{\mathbb {R} ^{n}}G(x-y)f(y)dy}$.

Method of determining the fundamental solution

In order to use the fundamental solution to determine an inhomogeneous solution to an initial value or boundary value problem, the fundamental solution itself must be determined. If the differential operator has constant coefficients, this can be done with the help of the Fourier transform

${\displaystyle {\hat {f}}(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{\mathbb {R} ^{n}}f(t)e^{-\mathrm {i} t\omega }\,\mathrm {d} t}$

or their inverse transformation can be achieved. It is true

${\displaystyle {\begin{aligned}{\frac {1}{\sqrt {2\pi }}}\int _{\mathbb {R} ^{n}}{\hat {f}}(\omega )\mathrm {e} ^{\mathrm {i} \omega {t}}\mathrm {d} \omega =f(t)&=Ly(t)\\&=L{\frac {1}{\sqrt {2\pi }}}\int _{\mathbb {R} ^{n}}{\hat {y}}(\omega )\mathrm {e} ^{\mathrm {i} \omega {t}}\mathrm {d} \omega \\&={\frac {1}{\sqrt {2\pi }}}\int _{\mathbb {R} ^{n}}L(-\mathrm {i} \omega ){\hat {y}}(\omega )\mathrm {e} ^{\mathrm {i} \omega {t}}\mathrm {d} \omega \,,\end{aligned}}}$

where is the symbol of . Together with the transfer function, the following applies ${\displaystyle L(\omega )}$ ${\displaystyle L}$ ${\displaystyle Y(-\mathrm {i} \omega ):={\tfrac {1}{L(-\mathrm {i} \omega )}}}$

${\displaystyle {\hat {y}}=Y(-\mathrm {i} \omega ){\hat {f}}}$,

almost everywhere . Since also still applies, follows ${\displaystyle {\hat {y}}=(2\pi )^{\frac {1}{2}}{\hat {G}}{\hat {f}}}$

${\displaystyle {\hat {G}}(\omega )={\frac {1}{\sqrt {2\pi }}}\cdot Y(-\mathrm {i} \omega )}$

respectively

${\displaystyle G(t)={\frac {1}{2\pi }}\int _{\mathbb {R} ^{n}}Y(-\mathrm {i} \omega )\mathrm {e} ^{\mathrm {i} \omega {t}}\mathrm {d} \omega }$.

Table of fundamental solutions

The following table gives an overview of fundamental solutions of frequently occurring differential operators, where the area of the surface of the -dimensional unit sphere and the Heaviside step function . ${\displaystyle \omega _{n}={\tfrac {2\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}})}}}$ ${\displaystyle n}$ ${\displaystyle \theta }$

Differential operator	Fundamental solution	Use case
${\displaystyle \partial _{t}}$ (Time derivative)	${\displaystyle \theta (t)}$	(see Delta distribution # Derivation of the Heaviside distribution )
${\displaystyle \partial _{t}+\gamma }$	${\displaystyle \theta (t)}$ ${\displaystyle \mathrm {e} ^{-\gamma t}}$	conventional Langevin equation
${\displaystyle \left(\partial _{t}+\gamma \right)^{2}}$	${\displaystyle \theta (t)t\mathrm {e} ^{-\gamma t}}$
${\displaystyle \partial _{t}^{2}+2\gamma \partial _{t}+\omega _{0}^{2}}$	${\displaystyle \theta (t)\mathrm {e} ^{-\gamma t}{\frac {1}{\omega }}\sin(\omega t)}$ With ${\displaystyle \omega ={\sqrt {\omega _{0}^{2}-\gamma ^{2}}}}$	one-dimensional damped harmonic oscillator
${\displaystyle \Delta }$( Laplace operator )	${\displaystyle \left\{{\begin{array}{rl}{\frac {1}{2\pi }}\ln {|x|}\ ,&n=2\\-{\frac {1}{(n-2)\,\omega _{n}}}{\frac {1}{|x|^{n-2}}}\ ,&n>2\,,\\\end{array}}\right.}$	Poisson's equation
${\displaystyle \Delta +k^{2}}$( Helmholtz operator )	${\displaystyle {\frac {-\mathrm {e} ^{-ik\|x\|}}{4\pi \|x\|}}}$	stationary Schrödinger equation ( ) ${\displaystyle n=3}$
${\displaystyle \square :={\frac {1}{c^{2}}}\partial _{t}^{2}-\Delta }$( D'Alembert operator )	${\displaystyle {\frac {\delta (t-{\frac {\|x\|}{c}})}{4\pi \|x\|}}}$	Wave equation ( ) ${\displaystyle n=3}$
${\displaystyle \partial _{t}-a\Delta }$( Thermal conduction operator )	${\displaystyle \theta (t)\left({\frac {1}{4\pi at}}\right)^{n/2}\mathrm {e} ^{-r^{2}/(4at)}}$	Thermal equation
${\displaystyle {\bar {\partial }}}$( Cauchy-Riemann operator )	${\displaystyle {\frac {1}{\pi }}{\frac {1}{z}}}$ (as distribution)	Cauchy-Riemann differential equations

theory

A fundamental solution is known for many differential equations, such as the Poisson equation , the heat conduction equation , the wave equation, and the Helmholtz equation .

In general, the Malgrange Ehrenpreis theorem holds that every partial differential equation with constant coefficients has a fundamental solution.

See also

• Parametrix , as a generalization of the fundamental solution
• Green's function , as an analogue to the fundamental solution for initial or boundary value problems

literature

• Lars Hörmander : The Analysis of Linear Partial Differential Operators. Volume 1: Distribution Theory and Fourier Analysis. Second edition. Springer-Verlag, Berlin et al. 1990, ISBN 3-540-52345-6 ( basic teaching of mathematical sciences 256).

Individual evidence

1. ↑ some examples from Schulz, Hermann: Physics with pencil. Frankfurt am Main: German, 2001. ISBN 3-8171-1661-6