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The same as [[Poisson integral|Poisson integral]].
 
The same as [[Poisson integral|Poisson integral]].
  
A formula giving an integral representation for the solution of the [[Cauchy problem|Cauchy problem]] for the [[Wave equation|wave equation]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073320/p0733201.png" />:
+
A formula giving an integral representation for the solution of the [[Cauchy problem|Cauchy problem]] for the [[Wave equation|wave equation]] in $  \mathbf R  ^ {3} $:
 +
 
 +
$$
  
<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/p/p073/p073320/p0733202.png" /></td> </tr></table>
+
\frac{\partial  ^ {2} u }{\partial  t  ^ {2} }
  
<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/p/p073/p073320/p0733203.png" /></td> </tr></table>
+
- a  ^ {2} \Delta u  = 0 ,\ \
 +
t > 0 ,\  M = ( x , y , z ) ,
 +
$$
  
<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/p/p073/p073320/p0733204.png" /></td> </tr></table>
+
$$
 +
- \infty  < x , y , z  < \infty ,
 +
$$
 +
 
 +
$$
 +
u ( M , 0 )  = \phi ( M) ,\ 
 +
\frac{\partial  u ( M , 0 ) }{\partial  t }
 +
  = \psi ( M) .
 +
$$
  
 
This solution has the form
 
This solution has the form
  
<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/p/p073/p073320/p0733205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
u ( M , t )  = \
 +
 
 +
\frac \partial {\partial  t }
 +
 
 +
\{ t \Gamma _ {at} ( \phi ) \} + t \Gamma _ {at} ( \psi ) ,
 +
$$
  
 
where
 
where
  
<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/p/p073/p073320/p0733206.png" /></td> </tr></table>
+
$$
 +
\Gamma _ {at} ( \phi )  = \
 +
 
 +
\frac{1}{4 \pi }
 +
 
 +
\int\limits _ {S _ {at} } \phi ( P)  d \Omega
 +
$$
  
is the mean value of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073320/p0733207.png" /> on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073320/p0733208.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073320/p0733209.png" />-space of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073320/p07332010.png" /> and centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073320/p07332011.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073320/p07332012.png" /> is the area element on the unit sphere. In the case of the inhomogeneous wave equation a third term is added to formula (1) (see ).
+
is the mean value of the function $  \phi $
 +
on the sphere $  S _ {at} $
 +
in the $  ( x , y , z ) $-
 +
space of radius $  a t $
 +
and centre at the point $  M $,  
 +
and $  d \Omega $
 +
is the area element on the unit sphere. In the case of the inhomogeneous wave equation a third term is added to formula (1) (see ).
  
 
From formula (1), by the method of descent (cf. [[Descent, method of|Descent, method of]]) formulas are obtained for solving the Cauchy problem in two- (Poisson's formula) and one- (d'Alembert formula) dimensional space (see ). See also [[Kirchhoff formula|Kirchhoff formula]].
 
From formula (1), by the method of descent (cf. [[Descent, method of|Descent, method of]]) formulas are obtained for solving the Cauchy problem in two- (Poisson's formula) and one- (d'Alembert formula) dimensional space (see ). See also [[Kirchhoff formula|Kirchhoff formula]].
  
Sometimes the phrase  "Poisson formula"  is used for the integral representation of the solution to the Cauchy problem for the [[Heat equation|heat equation]] in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073320/p07332013.png" />:
+
Sometimes the phrase  "Poisson formula"  is used for the integral representation of the solution to the Cauchy problem for the [[Heat equation|heat equation]] in the space $  \mathbf R  ^ {3} $:
 +
 
 +
$$
  
<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/p/p073/p073320/p07332014.png" /></td> </tr></table>
+
\frac{\partial  u }{\partial  t }
 +
- a  ^ {2} \Delta u  = 0 ,\ \
 +
t > 0 ,\  M = ( x , y , z ) ,
 +
$$
  
<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/p/p073/p073320/p07332015.png" /></td> </tr></table>
+
$$
 +
- \infty  < x , y , z  < \infty ,
 +
$$
  
<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/p/p073/p073320/p07332016.png" /></td> </tr></table>
+
$$
 +
u ( M , 0 = \phi ( M) .
 +
$$
  
 
This solution has the form
 
This solution has the form
  
<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/p/p073/p073320/p07332017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
u ( M , t )  = \
 +
 
 +
\frac{1}{( 2 \sqrt {\pi a  ^ {2} t } ) ^ {3} }
  
Formula (2) immediately generalizes to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073320/p07332018.png" />-dimensional space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073320/p07332019.png" />.
+
\int\limits _ {\mathbf R  ^ {3} }
 +
\phi ( P) e ^ {| M P |  ^ {2} / 4 a  ^ {2} t }  d \sigma ( P) .
 +
$$
 +
 
 +
Formula (2) immediately generalizes to an $  n $-
 +
dimensional space, $  n \geq  1 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.D. Poisson,  ''Mém. Acad. Sci. Paris'' , '''3'''  (1818)  pp. 121–176</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Tikhonov,  A.A. Samarskii,  "Equations of mathematical physics" , Pergamon  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.D. Poisson,  ''Mém. Acad. Sci. Paris'' , '''3'''  (1818)  pp. 121–176</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Tikhonov,  A.A. Samarskii,  "Equations of mathematical physics" , Pergamon  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.B. Whitham,  "Linear and non-linear waves" , Wiley (Interscience)  (1974)  pp. 229ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.V. Bitsadze,  "Equations of mathematical physics" , MIR  (1980)  pp. 179  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.B. Whitham,  "Linear and non-linear waves" , Wiley (Interscience)  (1974)  pp. 229ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.V. Bitsadze,  "Equations of mathematical physics" , MIR  (1980)  pp. 179  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


The same as Poisson integral.

A formula giving an integral representation for the solution of the Cauchy problem for the wave equation in $ \mathbf R ^ {3} $:

$$ \frac{\partial ^ {2} u }{\partial t ^ {2} } - a ^ {2} \Delta u = 0 ,\ \ t > 0 ,\ M = ( x , y , z ) , $$

$$ - \infty < x , y , z < \infty , $$

$$ u ( M , 0 ) = \phi ( M) ,\ \frac{\partial u ( M , 0 ) }{\partial t } = \psi ( M) . $$

This solution has the form

$$ \tag{1 } u ( M , t ) = \ \frac \partial {\partial t } \{ t \Gamma _ {at} ( \phi ) \} + t \Gamma _ {at} ( \psi ) , $$

where

$$ \Gamma _ {at} ( \phi ) = \ \frac{1}{4 \pi } \int\limits _ {S _ {at} } \phi ( P) d \Omega $$

is the mean value of the function $ \phi $ on the sphere $ S _ {at} $ in the $ ( x , y , z ) $- space of radius $ a t $ and centre at the point $ M $, and $ d \Omega $ is the area element on the unit sphere. In the case of the inhomogeneous wave equation a third term is added to formula (1) (see ).

From formula (1), by the method of descent (cf. Descent, method of) formulas are obtained for solving the Cauchy problem in two- (Poisson's formula) and one- (d'Alembert formula) dimensional space (see ). See also Kirchhoff formula.

Sometimes the phrase "Poisson formula" is used for the integral representation of the solution to the Cauchy problem for the heat equation in the space $ \mathbf R ^ {3} $:

$$ \frac{\partial u }{\partial t } - a ^ {2} \Delta u = 0 ,\ \ t > 0 ,\ M = ( x , y , z ) , $$

$$ - \infty < x , y , z < \infty , $$

$$ u ( M , 0 ) = \phi ( M) . $$

This solution has the form

$$ \tag{2 } u ( M , t ) = \ \frac{1}{( 2 \sqrt {\pi a ^ {2} t } ) ^ {3} } \int\limits _ {\mathbf R ^ {3} } \phi ( P) e ^ {| M P | ^ {2} / 4 a ^ {2} t } d \sigma ( P) . $$

Formula (2) immediately generalizes to an $ n $- dimensional space, $ n \geq 1 $.

References

[1] S.D. Poisson, Mém. Acad. Sci. Paris , 3 (1818) pp. 121–176
[2] A.N. Tikhonov, A.A. Samarskii, "Equations of mathematical physics" , Pergamon (1963) (Translated from Russian)
[3] V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)

Comments

References

[a1] G.B. Whitham, "Linear and non-linear waves" , Wiley (Interscience) (1974) pp. 229ff
[a2] A.V. Bitsadze, "Equations of mathematical physics" , MIR (1980) pp. 179 (Translated from Russian)
How to Cite This Entry:
Poisson formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_formula&oldid=48218
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article