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Difference between revisions of "Fundamental solution"

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m (use E for fundamental solution)
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$  x \in \mathbf R  ^ {n} $,  
 
$  x \in \mathbf R  ^ {n} $,  
 
with coefficients of class  $  C  ^  \infty  $,  
 
with coefficients of class  $  C  ^  \infty  $,  
in the form of a function  $  I ( x, y) $
+
in the form of a function  $  E ( x, y) $
 
that satisfies, for fixed  $  y \in \mathbf R  ^ {n} $,  
 
that satisfies, for fixed  $  y \in \mathbf R  ^ {n} $,  
 
the equation
 
the equation
  
 
$$  
 
$$  
L I ( x, y)  =  \delta ( x - y),\ \  
+
L E ( x, y)  =  \delta ( x - y),\ \  
 
x \neq y,
 
x \neq y,
 
$$
 
$$
Line 40: Line 40:
  
 
$$  
 
$$  
I ( x, y)  = \  
+
E ( x, y)  = \  
 
\left \{
 
\left \{
 
\begin{array}{l}
 
\begin{array}{l}

Revision as of 08:44, 13 May 2022


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. 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

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

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=52373
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article