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''Tricomi–Bitsadze–Lavrent'ev problem''
 
''Tricomi–Bitsadze–Lavrent'ev problem''
  
The problem of finding a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105901.png" /> which satisfies
+
The problem of finding a function $  u = u ( x,y ) $
 +
which satisfies
  
<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/b/b110/b110590/b1105902.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
{ \mathop{\rm sgn} } ( y ) u _ {xx }  + u _ {yy }  = 0
 +
$$
  
in a mixed domain that is simply connected and bounded by a Jordan (non-self-intersecting)  "elliptic"  arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105903.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105904.png" />) with end-points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105906.png" /> and by the  "real"  characteristics (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b1105907.png" />)
+
in a mixed domain that is simply connected and bounded by a Jordan (non-self-intersecting)  "elliptic"  arc $  g _ {1} $(
 +
for $  y > 0 $)  
 +
with end-points $  O = ( 0,0 ) $
 +
and $  A = ( 1,0 ) $
 +
and by the  "real"  characteristics (for $  y < 0 $)
  
<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/b/b110/b110590/b1105908.png" /></td> </tr></table>
+
$$
 +
g _ {2} : x - y = 1, \quad g _ {3} : x + y = 0
 +
$$
  
 
of the Bitsadze–Lavrent'ev equation (a1), which satisfy the characteristic equation
 
of the Bitsadze–Lavrent'ev equation (a1), which satisfy the characteristic 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/b/b110/b110590/b1105909.png" /></td> </tr></table>
+
$$
 +
- ( d y )  ^ {2} + ( d x )  ^ {2} = 0
 +
$$
  
and meet at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059010.png" />, and which assumes prescribed continuous boundary values
+
and meet at the point $  P = ( {1 / 2 } , - {1 / 2 } ) $,  
 +
and which assumes prescribed continuous boundary values
  
<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/b/b110/b110590/b11059011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
u = p ( s )  \textrm{ on  }  g _ {1} , \quad u = q ( x ) \textrm{ on  }  g _ {3} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059012.png" /> is the arc length reckoned from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059013.png" /> and
+
where $  s $
 +
is the arc length reckoned from the point $  A $
 +
and
  
<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/b/b110/b110590/b11059014.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm sgn} } ( y ) = \left \{
 +
\begin{array}{l}
 +
{1 \  \textrm{ for  }  y > 0, } \\
 +
{0 \  \textrm{ for  }  y = 0, } \\
 +
{-1 \  \textrm{ for  }  y < 0. }
 +
\end{array}
 +
\right .
 +
$$
  
Consider the aforementioned domain (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059015.png" />). Then a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059016.png" /> is a regular solution of the Bitsadze–Lavrent'ev problem if it satisfies the following conditions:
+
Consider the aforementioned domain (denoted by $  D $).  
 +
Then a function $  u = u ( x,y ) $
 +
is a regular solution of the Bitsadze–Lavrent'ev problem if it satisfies the following conditions:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059017.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059018.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059020.png" />;
+
1) $  u $
 +
is continuous in $  {\overline{D}\; } $
 +
= D \cup \partial  D $,  
 +
$  \partial  D = g _ {1} \cup g _ {2} \cup g _ {3} $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059021.png" /> are continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059022.png" /> (except, possibly, at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059024.png" />, where they may have poles of order less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059025.png" />, i.e., they may tend to infinity with order less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059026.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059028.png" />);
+
2) $  u _ {x} ,u _ {y} $
 +
are continuous in $  {\overline{D}\; } $(
 +
except, possibly, at the points $  O $
 +
and $  A $,  
 +
where they may have poles of order less than $  1 $,  
 +
i.e., they may tend to infinity with order less than $  1 $
 +
as $  x \rightarrow 0 $
 +
and $  x \rightarrow 1 $);
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059030.png" /> are continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059031.png" /> (except possibly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059032.png" />, where they need not exist);
+
3) $  u _ {xx }  $,  
 +
$  u _ {yy }  $
 +
are continuous in $  D $(
 +
except possibly on $  OA $,  
 +
where they need not exist);
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059033.png" /> satisfies (a1) at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059034.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059035.png" /> without <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059036.png" />);
+
4) $  u $
 +
satisfies (a1) at all points $  D \setminus  OA $(
 +
i.e., $  D $
 +
without $  OA $);
  
5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059037.png" /> satisfies the boundary conditions (a2).
+
5) $  u $
 +
satisfies the boundary conditions (a2).
  
 
Consider the normal curve (of Bitsadze–Lavrent'ev)
 
Consider the normal curve (of Bitsadze–Lavrent'ev)
  
<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/b/b110/b110590/b11059038.png" /></td> </tr></table>
+
$$
 +
g _ {1}  ^ {0} : \left ( x - {
 +
\frac{1}{2}
 +
} \right )  ^ {2} + y  ^ {2} = {
 +
\frac{1}{4}
 +
} ,  y > 0.
 +
$$
  
 
Note that it is the upper semi-circle and can also be given by (the upper part of)
 
Note that it is the upper semi-circle and can also be given by (the upper part of)
  
<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/b/b110/b110590/b11059039.png" /></td> </tr></table>
+
$$
 +
g _ {1}  ^ {0} : \left | {z - {
 +
\frac{1}{2}
 +
} } \right | = {
 +
\frac{1}{2}
 +
} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059040.png" />. The curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059041.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059042.png" /> in its interior.
+
where $  z = x + iy $.  
 +
The curve $  g _ {1} $
 +
contains $  g _ {1}  ^ {0} $
 +
in its interior.
  
The idea of A.V. Bitsadze and M.A. Lavrent'ev for finding regular solutions of the above problem is as follows. First, solve the problem N (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059044.png" />). That is, find a regular solution of equation (a1) satisfying the boundary conditions:
+
The idea of A.V. Bitsadze and M.A. Lavrent'ev for finding regular solutions of the above problem is as follows. First, solve the problem N (in $  D $,
 +
$  y > 0 $).  
 +
That is, find a regular solution of equation (a1) satisfying the boundary conditions:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059046.png" />;
+
$  u = p ( s ) $
 +
on $  g _ {1} $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059049.png" /> is continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059051.png" />, and may tend to infinity of order less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059052.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059054.png" />.
+
$  u _ {y} = r ( x ) $
 +
on $  OA $,  
 +
where $  r = r ( x ) $
 +
is continuous for $  x $,
 +
$  0 < x < 1 $,  
 +
and may tend to infinity of order less than $  1 $
 +
as $  x \rightarrow 0 $
 +
and $  x \rightarrow 1 $.
  
Secondly, solve the Cauchy–Goursat problem (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059056.png" />). That is, find a regular solution of (a1) satisfying the boundary conditions:
+
Secondly, solve the Cauchy–Goursat problem (in $  D $,  
 +
$  y < 0 $).  
 +
That is, find a regular solution of (a1) satisfying the boundary conditions:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059058.png" />;
+
$  u = t ( x ) $
 +
on $  OA $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059059.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059061.png" /> is continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059063.png" />, and may tend to infinity of order less that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059064.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059066.png" />.
+
$  u _ {y} = r ( x ) $
 +
on $  OA $,  
 +
where $  t = t ( x ) $
 +
is continuous for $  x $,
 +
$  0 < x < 1 $,  
 +
and may tend to infinity of order less that $  1 $
 +
as $  x \rightarrow 0 $
 +
and $  x \rightarrow 1 $.
  
 
Finally, take into account the boundary condition
 
Finally, take into account the boundary condition
  
<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/b/b110/b110590/b11059067.png" /></td> </tr></table>
+
$$
 +
u = q ( x )  \textrm{ on  }  g _ {3} .
 +
$$
  
Therefore, one has a [[Goursat problem|Goursat problem]] (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059069.png" />) for (a1) with boundary conditions:
+
Therefore, one has a [[Goursat problem|Goursat problem]] (in $  D $,  
 +
$  y < 0 $)  
 +
for (a1) with boundary conditions:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059070.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059071.png" />;
+
$  u = t ( x ) $
 +
on $  OA $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059072.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110590/b11059073.png" />.
+
$  u = q ( x ) $
 +
on $  g _ {3} $.
  
 
Several extensions and generalizations of the above boundary value problem of mixed type have been established [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]]. These problems are important in fluid mechanics (aerodynamics and hydrodynamics, [[#References|[a1]]], [[#References|[a2]]]).
 
Several extensions and generalizations of the above boundary value problem of mixed type have been established [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]]. These problems are important in fluid mechanics (aerodynamics and hydrodynamics, [[#References|[a1]]], [[#References|[a2]]]).

Latest revision as of 10:59, 29 May 2020


Tricomi–Bitsadze–Lavrent'ev problem

The problem of finding a function $ u = u ( x,y ) $ which satisfies

$$ \tag{a1 } { \mathop{\rm sgn} } ( y ) u _ {xx } + u _ {yy } = 0 $$

in a mixed domain that is simply connected and bounded by a Jordan (non-self-intersecting) "elliptic" arc $ g _ {1} $( for $ y > 0 $) with end-points $ O = ( 0,0 ) $ and $ A = ( 1,0 ) $ and by the "real" characteristics (for $ y < 0 $)

$$ g _ {2} : x - y = 1, \quad g _ {3} : x + y = 0 $$

of the Bitsadze–Lavrent'ev equation (a1), which satisfy the characteristic equation

$$ - ( d y ) ^ {2} + ( d x ) ^ {2} = 0 $$

and meet at the point $ P = ( {1 / 2 } , - {1 / 2 } ) $, and which assumes prescribed continuous boundary values

$$ \tag{a2 } u = p ( s ) \textrm{ on } g _ {1} , \quad u = q ( x ) \textrm{ on } g _ {3} , $$

where $ s $ is the arc length reckoned from the point $ A $ and

$$ { \mathop{\rm sgn} } ( y ) = \left \{ \begin{array}{l} {1 \ \textrm{ for } y > 0, } \\ {0 \ \textrm{ for } y = 0, } \\ {-1 \ \textrm{ for } y < 0. } \end{array} \right . $$

Consider the aforementioned domain (denoted by $ D $). Then a function $ u = u ( x,y ) $ is a regular solution of the Bitsadze–Lavrent'ev problem if it satisfies the following conditions:

1) $ u $ is continuous in $ {\overline{D}\; } $ $ = D \cup \partial D $, $ \partial D = g _ {1} \cup g _ {2} \cup g _ {3} $;

2) $ u _ {x} ,u _ {y} $ are continuous in $ {\overline{D}\; } $( except, possibly, at the points $ O $ and $ A $, where they may have poles of order less than $ 1 $, i.e., they may tend to infinity with order less than $ 1 $ as $ x \rightarrow 0 $ and $ x \rightarrow 1 $);

3) $ u _ {xx } $, $ u _ {yy } $ are continuous in $ D $( except possibly on $ OA $, where they need not exist);

4) $ u $ satisfies (a1) at all points $ D \setminus OA $( i.e., $ D $ without $ OA $);

5) $ u $ satisfies the boundary conditions (a2).

Consider the normal curve (of Bitsadze–Lavrent'ev)

$$ g _ {1} ^ {0} : \left ( x - { \frac{1}{2} } \right ) ^ {2} + y ^ {2} = { \frac{1}{4} } , y > 0. $$

Note that it is the upper semi-circle and can also be given by (the upper part of)

$$ g _ {1} ^ {0} : \left | {z - { \frac{1}{2} } } \right | = { \frac{1}{2} } , $$

where $ z = x + iy $. The curve $ g _ {1} $ contains $ g _ {1} ^ {0} $ in its interior.

The idea of A.V. Bitsadze and M.A. Lavrent'ev for finding regular solutions of the above problem is as follows. First, solve the problem N (in $ D $, $ y > 0 $). That is, find a regular solution of equation (a1) satisfying the boundary conditions:

$ u = p ( s ) $ on $ g _ {1} $;

$ u _ {y} = r ( x ) $ on $ OA $, where $ r = r ( x ) $ is continuous for $ x $, $ 0 < x < 1 $, and may tend to infinity of order less than $ 1 $ as $ x \rightarrow 0 $ and $ x \rightarrow 1 $.

Secondly, solve the Cauchy–Goursat problem (in $ D $, $ y < 0 $). That is, find a regular solution of (a1) satisfying the boundary conditions:

$ u = t ( x ) $ on $ OA $;

$ u _ {y} = r ( x ) $ on $ OA $, where $ t = t ( x ) $ is continuous for $ x $, $ 0 < x < 1 $, and may tend to infinity of order less that $ 1 $ as $ x \rightarrow 0 $ and $ x \rightarrow 1 $.

Finally, take into account the boundary condition

$$ u = q ( x ) \textrm{ on } g _ {3} . $$

Therefore, one has a Goursat problem (in $ D $, $ y < 0 $) for (a1) with boundary conditions:

$ u = t ( x ) $ on $ OA $;

$ u = q ( x ) $ on $ g _ {3} $.

Several extensions and generalizations of the above boundary value problem of mixed type have been established [a3], [a4], [a5], [a6], [a7], [a8]. These problems are important in fluid mechanics (aerodynamics and hydrodynamics, [a1], [a2]).

References

[a1] A.V. Bitsadze, "Equations of mixed type" , Macmillan (1964) (In Russian)
[a2] C. Ferrari, F.G. Tricomi, "Transonic aerodynamics" , Acad. Press (1968) (Translated from Italian)
[a3] J.M. Rassias, "Mixed type equations" , 90 , Teubner (1986)
[a4] J.M. Rassias, "Lecture notes on mixed type partial differential equations" , World Sci. (1990)
[a5] J.M. Rassias, "The Bitsadze–Lavrentjev problem" Bull. Soc. Roy. Sci. Liège , 48 (1979) pp. 424–425
[a6] J.M. Rassias, "The bi-hyperbolic Bitsadze–Lavrentjev–Rassias problem in three-dimensional Euclidean space" C.R. Acad. Sci. Bulg. Sci. , 39 (1986) pp. 29–32
[a7] J.M. Rassias, "The mixed Bitsadze–Lavrentjev–Tricomi boundary value problem" , Texte zur Mathematik , 90 , Teubner (1986) pp. 6–21
[a8] J.M. Rassias, "The well posed Tricomi–Bitsadze–Lavrentjev problem in the Euclidean plane" Atti. Accad. Sci. Torino , 124 (1990) pp. 73–83
How to Cite This Entry:
Bitsadze-Lavrent'ev problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bitsadze-Lavrent%27ev_problem&oldid=15296
This article was adapted from an original article by J.M. Rassias (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article