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Difference between revisions of "Inverse parabolic partial differential equation"

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An equation of the form
 
An equation of the form
  
$$ \tag{* }
+
\begin{equation}
 +
\label{eq:1}
 
u _ {t} +
 
u _ {t} +
 
\sum _ {i , j = 1 } ^ { n }  
 
\sum _ {i , j = 1 } ^ { n }  
 
a _ {ij} ( x , t ) u _ {x _ {i}  x _ {j} } -
 
a _ {ij} ( x , t ) u _ {x _ {i}  x _ {j} } -
\sum _ { i= } 1 ^ { n }  
+
\sum _ { i= 1 }^ { n }  
a _ {i} ( x , t ) u _ {x _ {i}  } - a ( x , t ) u =
+
a _ {i} ( x , t ) u _ {x _ {i}  } - a ( x , t ) u = f ( x , t ) ,
$$
+
\end{equation}
 
 
$$
 
= \
 
f ( x , t ) ,
 
$$
 
  
 
where the form  $  \sum a _ {ij} \xi _ {i} \xi _ {j} $
 
where the form  $  \sum a _ {ij} \xi _ {i} \xi _ {j} $
Line 35: Line 31:
  
 
====Comments====
 
====Comments====
The [[Cauchy problem|Cauchy problem]] for an equation (*) is a well-known example of an ill-posed problem (cf. [[Ill-posed problems|Ill-posed problems]]). For a discussion of the backward heat equation (cf. also [[Thermal-conductance equation|Thermal-conductance equation]])
+
The [[Cauchy problem]] for an equation \eqref{eq:1} is a well-known example of an ill-posed problem (cf. [[Ill-posed problems]]). For a discussion of the backward heat equation (cf. also [[Thermal-conductance equation]])
  
 
$$  
 
$$  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.E. Payne,   "Improperly posed problems in partial differential equations" , SIAM  (1975)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> L.E. Payne, "Improperly posed problems in partial differential equations" , SIAM  (1975)</TD></TR>
 +
</table>

Latest revision as of 19:49, 4 November 2023


An equation of the form

\begin{equation} \label{eq:1} u _ {t} + \sum _ {i , j = 1 } ^ { n } a _ {ij} ( x , t ) u _ {x _ {i} x _ {j} } - \sum _ { i= 1 }^ { n } a _ {i} ( x , t ) u _ {x _ {i} } - a ( x , t ) u = f ( x , t ) , \end{equation}

where the form $ \sum a _ {ij} \xi _ {i} \xi _ {j} $ is positive definite. The variable $ t $ plays the role of "inverse" time. The substitution $ t = - t ^ \prime $ reduces equation (*) to the usual parabolic form. Parabolic equations of "mixed" type occur, for example, $ u _ {t} = x u _ {xx} $ is a direct parabolic equation for $ x > 0 $ and an inverse parabolic equation for $ x < 0 $, with degeneracy of the order for $ x = 0 $.

Comments

The Cauchy problem for an equation \eqref{eq:1} is a well-known example of an ill-posed problem (cf. Ill-posed problems). For a discussion of the backward heat equation (cf. also Thermal-conductance equation)

$$ u _ {t} + \Delta u = 0 $$

( $ \Delta $ being the Laplace operator) see [a1].

References

[a1] L.E. Payne, "Improperly posed problems in partial differential equations" , SIAM (1975)
How to Cite This Entry:
Inverse parabolic partial differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inverse_parabolic_partial_differential_equation&oldid=54247
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article