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Difference between revisions of "Delta-function method"

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A method for finding the [[Green function|Green function]] of a linear differential equation in mathematical physics (i.e. a method for determining the function of the effect of a point source) with the aid of the [[Delta-function|delta-function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d0309601.png" />. The Green function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d0309602.png" /> of a linear differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d0309603.png" /> is defined by the equation
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A method for finding the [[Green function|Green function]] of a linear differential equation in mathematical physics (i.e. a method for determining the function of the effect of a point source) with the aid of the [[Delta-function|delta-function]] $\delta(x)$. The Green function $G(x,x')$ of a linear differential operator $L$ is defined by the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d0309604.png" /></td> </tr></table>
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$$L(x)G(x,x')=\delta(x-x'),$$
  
or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d0309605.png" />, i.e. it expresses the effect of a point source located at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d0309606.png" /> on the value of the resulting perturbation at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d0309607.png" />. The form of the inverse operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d0309608.png" /> is most simply determined in the frequently occurring case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d0309609.png" /> is a differential operator with constant coefficients (independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d03096010.png" />). The solution of a non-homogeneous linear differential equation of a general type for a perturbation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d03096011.png" /> with source <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d03096012.png" />,
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or $G(x,x')=-L^{-1}(x)\delta(x-x')$, i.e. it expresses the effect of a point source located at the point $x'$ on the value of the resulting perturbation at the point $x$. The form of the inverse operator $L^{-1}$ is most simply determined in the frequently occurring case when $L$ is a differential operator with constant coefficients (independent of $x$). The solution of a non-homogeneous linear differential equation of a general type for a perturbation $\phi$ with source $\rho$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d03096013.png" /></td> </tr></table>
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$$L(x)\phi(x)=-\rho(x),$$
  
is described with the aid of the Green function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d03096014.png" /> as the convolution
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is described with the aid of the Green function $G(x,x')$ as the convolution
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d03096015.png" /></td> </tr></table>
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$$\phi(x)=\int G(x,x')\rho(x')dx',$$
  
the integration being effected over the entire domain of action of the source <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030960/d03096016.png" />.
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the integration being effected over the entire domain of action of the source $\rho$.
  
 
====References====
 
====References====

Revision as of 09:56, 26 April 2014

A method for finding the Green function of a linear differential equation in mathematical physics (i.e. a method for determining the function of the effect of a point source) with the aid of the delta-function $\delta(x)$. The Green function $G(x,x')$ of a linear differential operator $L$ is defined by the equation

$$L(x)G(x,x')=\delta(x-x'),$$

or $G(x,x')=-L^{-1}(x)\delta(x-x')$, i.e. it expresses the effect of a point source located at the point $x'$ on the value of the resulting perturbation at the point $x$. The form of the inverse operator $L^{-1}$ is most simply determined in the frequently occurring case when $L$ is a differential operator with constant coefficients (independent of $x$). The solution of a non-homogeneous linear differential equation of a general type for a perturbation $\phi$ with source $\rho$,

$$L(x)\phi(x)=-\rho(x),$$

is described with the aid of the Green function $G(x,x')$ as the convolution

$$\phi(x)=\int G(x,x')\rho(x')dx',$$

the integration being effected over the entire domain of action of the source $\rho$.

References

[1] D. Ivanenko, A. Sokolov, "Classical field theory" , Mosow-Leningrad (1951) (In Russian)


Comments

References

[a1] P.M. Morse, H. Feshbach, "Methods of theoretical physics" , 1–2 , McGraw-Hill (1953)
How to Cite This Entry:
Delta-function method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Delta-function_method&oldid=12041
This article was adapted from an original article by V.D. Kukin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article