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''of a linear partial differential equation''
 
''of a linear partial differential equation''
  
A solution of a partial differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042250/f0422501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042250/f0422502.png" />, with coefficients of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042250/f0422503.png" />, in the form of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042250/f0422504.png" /> that satisfies, for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042250/f0422505.png" />, the equation
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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 $  I ( x, y) $
 +
that satisfies, for fixed $  y \in \mathbf R  ^ {n} $,
 +
the equation
 +
 
 +
$$
 +
L I ( x, y)  = \delta ( x - y),\ \
 +
x \neq y,
 +
$$
 +
 
 +
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
  
<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/f/f042/f042250/f0422506.png" /></td> </tr></table>
+
$$
 +
\sum _ {i, j = 1 } ^ { n }
 +
a _ {ij}
 +
\frac{\partial  ^ {2} u }{\partial  x _ {i} \partial  x _ {j} }
 +
  = 0
 +
$$
  
which is interpreted in the sense of the theory of generalized functions, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042250/f0422507.png" /> 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
+
with constant coefficients  $  a _ {ij} $
 +
forming a positive-definite matrix  $  a $,  
 +
a fundamental solution is provided by the function
  
<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/f/f042/f042250/f0422508.png" /></td> </tr></table>
+
$$
 +
I ( x, y)  = \
 +
\left \{
 +
\begin{array}{l}
 +
\left [ \sum _ {i, j = 1 } ^ { n }
 +
A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} )
 +
\right ] ^ {( 2 - n)/2 } ,\  n > 2 ,
 +
\\
  
with constant coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042250/f0422509.png" /> forming a positive-definite matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042250/f04225010.png" />, a fundamental solution is provided by the function
+
\mathop{\rm log} \left [ \sum _ {i, j = 1 } ^ { n }
 +
A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} ) \right ] ,\  n = 2 ,  
 +
\end{array}
  
<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/f/f042/f042250/f04225011.png" /></td> </tr></table>
+
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042250/f04225012.png" /> is the cofactor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042250/f04225013.png" /> in the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042250/f04225014.png" />.
+
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.
 
Fundamental solutions are widely used in the study of boundary value problems for elliptic equations.
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====References====
 
====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>
 
<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====
 
====Comments====

Revision as of 19:40, 5 June 2020


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 $ I ( x, y) $ that satisfies, for fixed $ y \in \mathbf R ^ {n} $, the equation

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

which is interpreted in the sense of the theory of generalized functions, where $ \delta $ is the delta-function. 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

$$ I ( x, y) = \ \left \{ \begin{array}{l} \left [ \sum _ {i, j = 1 } ^ { n } A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} ) \right ] ^ {( 2 - n)/2 } ,\ n > 2 , \\ \mathop{\rm log} \left [ \sum _ {i, j = 1 } ^ { n } A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} ) \right ] ,\ n = 2 , \end{array} $$

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

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

Comments

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

See also Green function.

References

[a1] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)
[a2] O.A. Ladyzhenskaya, N.N. Ural'tseva, "Linear and quasilinear elliptic equations" , Acad. Press (1968) (Translated from Russian)
[a3] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural'tseva, "Linear and quasilinear parabolic equations" , Amer. Math. Soc. (1968) (Translated from Russian)
[a4] L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1957–1959)
[a5] I.M. Gel'fand, G.E. Shilov, "Generalized functions" , Acad. Press (1964) (Translated from Russian)
How to Cite This Entry:
Fundamental solution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_solution&oldid=47027
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article