View source for Differential equation, partial, of the second order

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An equation containing at least one derivative of the second order of the unknown function  $  u $
and not containing derivatives of higher orders. For instance, a linear equation of the second order has the form

$$ \tag{1 }
\sum _ {i , j= 1 } ^ { n }  a _ {ij} ( x)

\frac{\partial   ^ {2} u ( x) }{\partial  x _ {i} \partial  x _ {j} }
 + \sum _
{i = 1 } ^ { n }  b _ {i} ( x) 
\frac{\partial  u ( x) }{\partial  x _ {i} }
 +
$$

$$ 
+ 
c ( x) u ( x) + f ( x)  =  0 ,
$$

where the point  $  x = ( x _ {1} \dots x _ {n} ) $
belongs to some domain  $  \Omega \subset  \mathbf R  ^ {n} $
in which the real-valued functions  $  a _ {ij} ( x) $, 
$  b _ {i} ( x) $
and  $  c ( x) $
are defined, and at each point  $  x \in \Omega $
at least one of the coefficients  $  a _ {ij} ( x) $
is non-zero. For any point  $  x _ {0} \in \Omega $
there exists a non-singular transformation of the independent variables  $  \xi = \xi ( x) $
such that equation (1) assumes the following form in the new coordinates  $  \xi = ( \xi _ {1} \dots \xi _ {n} ) $:

$$ \tag{2 }
\sum _ {i , j = 1 } ^ { n }  a _ {ij}  ^ {*} ( \xi ) 
\frac{\partial   ^ {2}
u ( \xi ) }{\partial  \xi _ {i} \partial  \xi _ {j} }
 + \sum _ {i = 1
 } ^ { n }  b _ {i}  ^ {*} ( \xi ) 
\frac{\partial  u }{\partial  \xi _ {i} }
 +
$$

$$ 
+ 
c  ^ {*} ( \xi ) u ( \xi ) + f  ^ {*} ( \xi )  =  0 ,
$$

where the coefficients  $  a _ {ij}  ^ {*} ( \xi ) $
at the point  $  \xi _ {0} = \xi ( x _ {0} ) $
are equal to zero if  $  i \neq j $
and are equal to  $  \pm  1 $
or to zero if  $  i = j $. 
Equation (2) is known as the canonical form of equation (1) at the point  $  x _ {0} $.

The number  $  k $
and the number  $  l $
of coefficients  $  a _ {ii}  ^ {*} ( \xi ) $
in equation (2) which are, respectively, positive and negative at the point  $  \xi _ {0} $
depend only on the coefficients  $  a _ {ij} ( x) $
of equation (1). As a consequence, differential equations (1) can be classified as follows. If  $  k = n $
or  $  l = n $, 
equation (1) is called elliptic at the point  $  x _ {0} $; 
if  $  k = n - 1 $
and  $  l = 1 $, 
or if  $  k = 1 $
and  $  l = n - 1 $, 
it is called hyperbolic; if  $  k + l = n $
and  $  1 < k < n - 1 $, 
it is called ultra-hyperbolic. The equation is called parabolic in the wide sense at the point  $  x _ {0} $
if at least one of the coefficients  $  a _ {i i }   ^ {*} ( \xi ) $
is zero at the point  $  \xi _ {0} = \xi ( x _ {0} ) $
and  $  k + l < n $; 
it is called parabolic at the point  $  x _ {0} $
if only one of the coefficients  $  a _ {ii}  ^ {*} ( \xi ) $
is zero at the point  $  \xi _ {0} $(
say  $  a _ {11}  ^ {*} ( \xi _ {0} ) = 0 $), 
while all the remaining coefficients  $  a _ {ii}  ^ {*} ( \xi ) $
have the same sign and the coefficient  $  b _ {1}  ^ {*} ( \xi _ {0} ) \neq 0 $.

In the case of two independent variables  $  ( n = 2 ) $
it is more convenient to define the type of an equation by the function

$$ 
\Delta ( x)  =  a _ {11} a _ {22} - a _ {12} a _ {21} .
$$

Thus, equation (1) is elliptic at the point  $  x _ {0} $
if  $  \Delta ( x _ {0} ) > 0 $; 
it is hyperbolic if  $  \Delta ( x _ {0} ) < 0 $
and is parabolic in the wide sense if  $  \Delta ( x _ {0} ) = 0 $.

An equation is called elliptic, hyperbolic, etc., in a domain, if it is, respectively, elliptic, hyperbolic, etc., at each point of this domain. For instance, the Tricomi equation  $  yu _ {xx} + u _ {yy} = 0 $
is elliptic if  $  y > 0 $; 
it is hyperbolic if  $  y < 0 $; 
and it is parabolic in the wide sense if  $  y = 0 $.

The transformation of variables  $  \xi = \xi ( x) $
which converts equation (1) to canonical form at the point  $  x _ {0} $
depends on that point. If there are three or more independent variables, there is, in general, no non-singular transformation of equation (1) to canonical form at all points of some neighbourhood of the point  $  x _ {0} $
at the same time, i.e. to the form

$$ 
\sum _ {i = 1 } ^ { k }  
\frac{\partial   ^ {2} u ( \xi ) }{\partial  \xi _ {i}  ^ {2} }
 - \sum _ { i= } k+ 1 ^ { k+ }  l 
\frac{\partial   ^ {2} u ( \xi ) }{\partial 
\xi _ {i}  ^ {2} }
 +
\sum _ { i= } 1 ^ { n }  b _ {i}  ^ {*} ( \xi ) 
\frac{\partial  u ( \xi ) }{
\partial  \xi _ {j} }
 +
$$

$$ 
+ c  ^ {*} ( \xi ) u ( \xi ) + f  ^ {*} ( \xi )  =  0 .
$$

In the case of two independent variables ( $  n = 2 $), 
on the other hand, it is possible to bring equation (1) to canonical form by imposing certain conditions on the coefficients  $  a _ {ij} ( x) $; 
as an example, the functions  $  a _ {ij} ( x) $
must be continuously differentiable up to the second order inclusive, and equation (1) must be of one type in a certain neighbourhood of the point  $  x _ {0} $.

Let

$$ \tag{3 }
\Phi ( x , u , u _ {x _ {1}  } \dots u _ {x _ {n}  } ,
 u _ {x _ {1}  x _ {1} } , u _ {x _ {1}  x _ {2} } \dots u _ {x _ {n}  x _ {n} } )  =  0
$$

be a non-linear equation of the second order, where  $  u _ {x _ {i}  } = \partial  u / \partial  x _ {i} $, 
$  u _ {x _ {i}  x _ {j} } = \partial   ^ {2} u / \partial  x _ {i} \partial  x _ {j} $, 
and let the derivatives  $  \partial  \Phi / \partial  u _ {x _ {i}  x _ {j} } $
exist at each point in the domain of definition of the real-valued function  $  \Phi $; 
further, let the condition

$$ 
\sum _ {i , j = 1 } ^ { n }  \left ( 
\frac{\partial  \Phi }{\partial  u _ {x _ {i}  x _ {j} } }
 \right )  ^ {2}  \neq  0
$$

be satisfied. In the classification of non-linear equations of the type (3) one determines a certain solution  $  u  ^ {*} ( x) $
of this equation and one considers the linear equation

$$ \tag{4 }
\sum _ {i , j = 1 } ^ { n }  a _ {ij} ( x) 
\frac{\partial   ^ {2} u ( x) }{\partial  x _ {i} \partial  x _ {j} }
  =  0
$$

with coefficients

$$ 
a _ {ij} ( x)  =  \left . 
\frac{\partial  \Phi }{\partial  u _ {x _ {i}  x _ {j} }
 }
 \right | _ {u = u  ^ {*}  ( x) } .
$$

For a given solution  $  u  ^ {*} ( x) $, 
equation (3) is said to be elliptic, hyperbolic, etc., at a point  $  x _ {0} $(
or in a domain) if equation (4) is elliptic, hyperbolic, etc., respectively, at this point (or in this domain).

A very wide class of physical problems is reduced to solving differential equations of the second order. See, for example, [[Wave equation|Wave equation]]; [[Telegraph equation|Telegraph equation]]; [[Thermal-conductance equation|Thermal-conductance equation]]; [[Tricomi equation|Tricomi equation]]; [[Laplace equation|Laplace equation]]; [[Poisson equation|Poisson equation]]; [[Helmholtz equation|Helmholtz equation]].

====Comments====
See also [[Differential equation, partial|Differential equation, partial]].