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''of a linear partial differential equation''

A solution of a partial differential equation  $  Lu ( x) = 0 $, 
$  x \in \mathbf R  ^ {n} $, 
with coefficients of class  $  C  ^  \infty  $, 
in the form of a function  $  E ( x, y) $
that satisfies, for fixed  $  y \in \mathbf R  ^ {n} $, 
the equation

$$ 
L E ( x, y)  =  \delta ( x - y),\ \ 
x \neq y,
$$

which is interpreted in the sense of the theory of generalized functions, where  $  \delta $
is the [[Delta-function|delta-function]]. There is a fundamental solution for every partial differential equation with constant coefficients, and also for arbitrary elliptic equations. For example, for the elliptic equation

$$ 
\sum _ {i, j = 1 } ^ { n } 
a _ {ij} 
\frac{\partial   ^ {2} u }{\partial  x _ {i} \partial  x _ {j} }
  =  0
$$

with constant coefficients  $  a _ {ij} $
forming a positive-definite matrix  $  a $, 
a fundamental solution is provided by the function

$$ 
E ( x, y)  = 
\begin{cases} \displaystyle
\left [ \sum _ {i, j = 1 } ^ { n } 
A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} )
\right ] ^ {( 2 - n)/2 } , &  n > 2 ,
\\

\displaystyle \log \left [ \sum _ {i, j = 1 } ^ { n } 
A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} ) \right ] , &  n = 2 , 
\end{cases}
$$

where  $  A _ {ij} $
is the cofactor of  $  a _ {ij} $
in the matrix  $  a $.

Fundamental solutions are widely used in the study of boundary value problems for elliptic equations.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,   "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Bers,   F. John,   M. Schechter,   "Partial differential equations" , Interscience  (1964)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. John,   "Plane waves and spherical means: applied to partial differential equations" , Interscience  (1955)</TD></TR></table>

====Comments====
Fundamental solutions are also used in the study of Cauchy problems (cf. [[Cauchy problem|Cauchy problem]]) for hyperbolic and parabolic equations. The name  "elementary solution of a linear partial differential equation" is also used.

See also [[Green function|Green function]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O.A. Ladyzhenskaya,   N.N. Ural'tseva,   "Linear and quasilinear elliptic equations" , Acad. Press  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O.A. Ladyzhenskaya,   V.A. Solonnikov,   N.N. Ural'tseva,   "Linear and quasilinear parabolic equations" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Schwartz,   "Théorie des distributions" , '''1–2''' , Hermann  (1957–1959)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I.M. Gel'fand,   G.E. Shilov,   "Generalized functions" , Acad. Press  (1964)  (Translated from Russian)</TD></TR></table>