View source for Delta-function method

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

<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>

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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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

<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>

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" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Ivanenko,   A. Sokolov,   "Classical field theory" , Mosow-Leningrad  (1951)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Morse,   H. Feshbach,   "Methods of theoretical physics" , '''1–2''' , McGraw-Hill  (1953)</TD></TR></table>