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''to an ordinary linear differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a0108101.png" />''
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The ordinary linear differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a0108102.png" />, where
+
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 +
{{TEX|done}}
  
<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/a/a010/a010810/a0108103.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
''to an ordinary linear differential equation  $  l (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/a/a010/a010810/a0108104.png" /></td> </tr></table>
+
The ordinary linear differential equation  $  l  ^ {*} ( \xi ) = 0 $,
 +
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/a/a010/a010810/a0108105.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
l (y)  \equiv  a _ {0} (t) y  ^ {(n)} + \dots + a _ {n} (t) y ,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a0108106.png" /> is the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a0108107.png" />-times continuously-differentiable complex-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a0108108.png" />, and
+
$$
 +
y ^ {( \nu ) }  =
 +
\frac{d  ^  \nu  y }{d t  ^  \nu  }
 +
,\  y \in C  ^ {n} (I) ,\  a _ {k} \in C  ^ {n-k} (I) ,
 +
$$
  
<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/a/a010/a010810/a0108109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$
 +
a _ {0} (t) \neq  0 ,\  t \in I ;
 +
$$
  
<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/a/a010/a010810/a01081010.png" /></td> </tr></table>
+
$  C  ^ {m} (I) $
 +
is the space of  $  m $-
 +
times continuously-differentiable complex-valued functions on  $  I = ( \alpha , \beta ) $,
 +
and
 +
 
 +
$$ \tag{2 }
 +
l  ^ {*} ( \xi )  \equiv \
 +
( - 1 )  ^ {n} ( \overline{a}\; _ {0} \xi )  ^ {(n)} + ( - 1 )
 +
^ {n-1} ( \overline{a}\; _ {1} \xi )  ^ {(n-1)} + \dots + \overline{a}\; _ {n} \xi ,
 +
$$
 +
 
 +
$$
 +
\xi  \in  C  ^ {n} (I)
 +
$$
  
 
(the bar denotes complex conjugation). It follows at once that
 
(the bar denotes complex conjugation). It follows at once that
  
<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/a/a010/a010810/a01081011.png" /></td> </tr></table>
+
$$
 +
( l _ {1} + l _ {2} )  ^ {*}  = \
 +
l _ {1}  ^ {*} + l _ {2}  ^ {*} ,\ \
 +
( \lambda l )  ^ {*}  = \
 +
\overline \lambda \; l  ^ {*} ,
 +
$$
  
for any scalar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081012.png" />. The adjoint of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081013.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081014.png" />. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081015.png" />-times continuously-differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081017.png" />, Lagrange's identity holds:
+
for any scalar $  \lambda $.  
 +
The adjoint of the equation $  l  ^ {*} ( \xi ) = 0 $
 +
is $  l (y) = 0 $.  
 +
For all $  n $-
 +
times continuously-differentiable functions $  y (t) $
 +
and $  \xi (t) $,  
 +
Lagrange's identity holds:
  
<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/a/a010/a010810/a01081018.png" /></td> </tr></table>
+
$$
 +
\overline{ {\xi l }}\; (y) - \overline{ {l  ^ {*} ( \xi ) }}\; y  = 
 +
\frac{d}{dt}
 +
 
 +
\left \{
 +
\sum _ { k=1 } ^ { n }  \sum _ { j=0 } ^ { k-1 }
 +
( - 1 )  ^ {j} ( a _ {n-k} \overline \xi \; )  ^ {(j)} y ^ {( k - j - 1 ) }
 +
\right \} .
 +
$$
  
 
It implies Green's formula
 
It implies Green's formula
  
<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/a/a010/a010810/a01081019.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { s } ^  \tau 
 +
[ \overline \xi \; l (y) -
 +
\overline{ {l  ^ {*} ( \xi ) }}\;
 +
y ]  d t =
 +
$$
  
<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/a/a010/a010810/a01081020.png" /></td> </tr></table>
+
$$
 +
= \
 +
\left . \sum _ { k=1 } ^ { n }  \sum _ { j=0 } ^ { k-1 }  ( - 1 )  ^ {j} (
 +
a _ {n-k} \overline \xi \; )  ^ {(j)} y ^ {( k - j - 1 ) } \right | _ {t=s} ^ {t = \tau } .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081022.png" /> are arbitrary solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081024.png" />, respectively, then
+
If $  y (t) $
 +
and $  \xi (t) $
 +
are arbitrary solutions of $  l (y) = 0 $
 +
and $  l  ^ {*} ( \xi ) = 0 $,  
 +
respectively, then
  
<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/a/a010/a010810/a01081025.png" /></td> </tr></table>
+
$$
 +
\sum _ { k=1 } ^ { n }  \sum _ { j=0 } ^ { k-1 }
 +
( - 1 )  ^ {j} ( a _ {n-k} \overline \xi \; )  ^ {(j)}
 +
y  ^ {(k-j-1)}
 +
\equiv  \textrm{ const } ,\ \
 +
t \in I .
 +
$$
  
A knowledge of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081026.png" /> linearly independent solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081027.png" /> enables one to reduce the order of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081028.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081029.png" /> (see [[#References|[1]]]–[[#References|[3]]]).
+
A knowledge of $  m  ( \leq  n ) $
 +
linearly independent solutions of the equation $  l  ^ {*} ( \xi ) = 0 $
 +
enables one to reduce the order of the equation $  l (y) = 0 $
 +
by $  m $(
 +
see [[#References|[1]]]–[[#References|[3]]]).
  
 
For a system of differential equations
 
For a system of differential equations
  
<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/a/a010/a010810/a01081030.png" /></td> </tr></table>
+
$$
 +
L (x)  = 0 ,\ \
 +
L (x)  \equiv \
 +
\dot{x} + A (t) x ,\ \
 +
t \in I ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081031.png" /> is a continuous complex-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081032.png" />-matrix, the adjoint system is given by
+
where $  A (t) $
 +
is a continuous complex-valued $  ( n \times n ) $-
 +
matrix, the adjoint system is given by
  
<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/a/a010/a010810/a01081033.png" /></td> </tr></table>
+
$$
 +
L  ^ {*} ( \psi )
 +
\equiv  - \dot \psi 
 +
+ A  ^ {*} (t) \psi  = \
 +
0 ,\  t \in I
 +
$$
  
(see [[#References|[1]]], [[#References|[4]]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081034.png" /> is the Hermitian adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081035.png" />. The Lagrange identity and the Green formula take the form
+
(see [[#References|[1]]], [[#References|[4]]]), where $  A  ^ {*} (t) $
 +
is the Hermitian adjoint of $  A (t) $.  
 +
The Lagrange identity and the Green formula take 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/a/a010/a010810/a01081036.png" /></td> </tr></table>
+
$$
 +
( \overline \psi \; , L (x) ) -
 +
( \overline{ {L  ^ {*} ( \psi ) }}\; , x )  =
 +
\frac{d}{dt}
  
<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/a/a010/a010810/a01081037.png" /></td> </tr></table>
+
( \overline \psi \; , x ) ,\ \
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081038.png" /> is the standard scalar product (the sum of the products of coordinates with equal indices). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081040.png" /> are arbitrary solutions of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081042.png" />, then
+
$$
 +
\left . \int\limits _ { s } ^  \tau  [ ( \overline \psi \; , L (x) ) - (
 +
\overline{ {L  ^ {*} ( \psi ) }}\; , x ) ]  d t  = \
 +
( \overline \psi \; , x ) \right | _ {t=s} ^ {t = \tau } ;
 +
$$
  
<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/a/a010/a010810/a01081043.png" /></td> </tr></table>
+
where  $  ( \cdot , \cdot ) $
 +
is the standard scalar product (the sum of the products of coordinates with equal indices). If  $  x (t) $
 +
and  $  \psi (t) $
 +
are arbitrary solutions of the equations  $  L (x) = 0 $
 +
and  $  L  ^ {*} ( \psi ) = 0 $,
 +
then
  
The concept of an adjoint differential equation is closely connected with the general concept of an [[Adjoint operator|adjoint operator]]. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081044.png" /> is a linear differential operator acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081045.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081046.png" /> in accordance with (1), then its adjoint differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081047.png" /> maps the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081048.png" /> adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081049.png" /> into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081050.png" /> adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081051.png" />. The restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081052.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081053.png" /> is given by formula (2) (see [[#References|[5]]]).
+
$$
 +
( \overline \psi \; (t) , x (t) )  \equiv  \textrm{ const } ,\  t \in I .
 +
$$
 +
 
 +
The concept of an adjoint differential equation is closely connected with the general concept of an [[Adjoint operator|adjoint operator]]. Thus, if $  l $
 +
is a linear differential operator acting on $  C  ^ {n} (I) $
 +
into $  C (I) $
 +
in accordance with (1), then its adjoint differential operator $  l  ^ {*} $
 +
maps the space $  C  ^ {*} (I) $
 +
adjoint to $  C (I) $
 +
into the space $  C  ^ {n*} (I) $
 +
adjoint to $  C  ^ {n} (I) $.  
 +
The restriction of $  l  ^ {*} $
 +
to $  C  ^ {n} (I) $
 +
is given by formula (2) (see [[#References|[5]]]).
  
 
Adjoints are also defined for linear partial differential equations (see [[#References|[6]]], [[#References|[5]]]).
 
Adjoints are also defined for linear partial differential equations (see [[#References|[6]]], [[#References|[5]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081054.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081055.png" /> be linearly independent linear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081056.png" />. Then the boundary value problem adjoint to the linear boundary value problem
+
Let $  \Delta = [ t _ {0} , t _ {1} ] \subset  I $,  
 +
and let $  U _ {k} $
 +
be linearly independent linear functionals on $  C  ^ {n} ( \Delta ) $.  
 +
Then the boundary value problem adjoint to the linear boundary value problem
  
<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/a/a010/a010810/a01081057.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
l (y)  = 0 ,\ \
 +
t \in \Delta ,\  \
 +
U _ {k} (y)  = 0 ,\ \
 +
k = 1 \dots m ,\ \
 +
m < 2 n ,
 +
$$
  
 
is defined by the equations
 
is defined by the equations
  
<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/a/a010/a010810/a01081058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
l  ^ {*} ( \xi )  = 0 ,\ \
 +
U _ {j}  ^ {*} ( \xi )  = 0 ,\ \
 +
j = 1 \dots 2 n - m .
 +
$$
  
Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081059.png" /> are linear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081060.png" /> describing the adjoint boundary conditions, that is, they are defined in such a way that the equation (see [[Green formulas|Green formulas]])
+
Here the $  U _ {j}  ^ {*} $
 +
are linear functionals on $  C  ^ {n} ( \Delta ) $
 +
describing the adjoint boundary conditions, that is, they are defined in such a way that the equation (see [[Green formulas|Green formulas]])
  
<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/a/a010/a010810/a01081061.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {t _ {0} } ^ { {t } _ {1} }
 +
[ \overline \xi \; l (y) -
 +
\overline{ {l  ^ {*} ( \xi ) }}\;
 +
y ]  d t  = 0
 +
$$
  
holds for any pair of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081062.png" /> that satisfy the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081064.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081066.png" />.
+
holds for any pair of functions $  y , \xi \in C  ^ {n} ( \Delta ) $
 +
that satisfy the conditions $  U _ {k} (y) = 0 $,  
 +
$  k = 1 \dots m $;  
 +
$  U _ {j}  ^ {*} ( \xi ) = 0 $,  
 +
$  j = 1 \dots 2 n - m $.
  
 
If
 
If
  
<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/a/a010/a010810/a01081067.png" /></td> </tr></table>
+
$$
 +
U _ {k} (y)  \equiv \
 +
\sum _ { p=1 } ^ { n }
 +
[ \alpha _ {kp} y  ^ {(p-1)} ( t _ {0} ) +
 +
\beta _ {kp} y  ^ {(p-1)} ( t _ {1} ) ]
 +
$$
  
 
are linear forms in the variables
 
are linear forms in the variables
  
<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/a/a010/a010810/a01081068.png" /></td> </tr></table>
+
$$
 +
y  ^ {(p-1)} ( t _ {0} ) ,\ \
 +
y  ^ {(p-1)} ( t _ {1} ) ,\ \
 +
p = 1 \dots n ,
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081069.png" /> are linear forms in the variables
+
then $  U _ {j}  ^ {*} ( \xi ) $
 +
are linear forms in the variables
  
<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/a/a010/a010810/a01081070.png" /></td> </tr></table>
+
$$
 +
\xi  ^ {(p-1)} ( t _ {0} ) ,\ \
 +
\xi  ^ {(p-1)} ( t _ {1} ) ,\ \
 +
p = 1 \dots n .
 +
$$
  
 
Examples. For the problem
 
Examples. For the problem
  
<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/a/a010/a010810/a01081071.png" /></td> </tr></table>
+
$$
 +
\dot{y} dot + a (t) y  = 0 ,\ \
 +
0 \leq  t \leq  1 ,
 +
$$
  
<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/a/a010/a010810/a01081072.png" /></td> </tr></table>
+
$$
 +
y (0) + \alpha y (1) + \beta \dot{y} (1)  = 0 ,
 +
$$
  
<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/a/a010/a010810/a01081073.png" /></td> </tr></table>
+
$$
 +
\dot{y} (0) + \gamma y (1) + \delta \dot{y} (1)  = 0 ,
 +
$$
  
with real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081074.png" />, the adjoint boundary value problem has the form
+
with real a (t) , \alpha , \beta , \gamma , \delta $,
 +
the adjoint boundary value problem 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/a/a010/a010810/a01081075.png" /></td> </tr></table>
+
$$
 +
\dot \xi  dot + a (t) \xi  = 0 ,\ \
 +
0 \leq  t \leq  1 ,
 +
$$
  
<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/a/a010/a010810/a01081076.png" /></td> </tr></table>
+
$$
 +
\alpha \xi (0) + \gamma \dot \xi  (0) + \xi (1)  = 0 ,
 +
$$
  
<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/a/a010/a010810/a01081077.png" /></td> </tr></table>
+
$$
 +
\beta \xi (0) + \delta \dot \xi  (0) + \dot \xi  (1)  = 0 .
 +
$$
  
If problem (3) has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081078.png" /> linearly independent solutions (in this case the rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081079.png" /> of the boundary value problem is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081080.png" />), then problem (4) has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081081.png" /> linearly independent solutions (its rank is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081082.png" />). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081083.png" />, problems (3) and (4) have an equal number of linearly independent solutions. Therefore, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081084.png" />, problem (3) has only a trivial solution if and only if the adjoint boundary value problem (4) has the same property. The Fredholm alternative holds: The semi-homogeneous boundary value problem
+
If problem (3) has $  k $
 +
linearly independent solutions (in this case the rank $  r $
 +
of the boundary value problem is equal to $  n-k $),  
 +
then problem (4) has $  m - n + k $
 +
linearly independent solutions (its rank is $  r  ^  \prime  = 2n - m - k $).  
 +
When $  m = n $,  
 +
problems (3) and (4) have an equal number of linearly independent solutions. Therefore, when $  m = n $,  
 +
problem (3) has only a trivial solution if and only if the adjoint boundary value problem (4) has the same property. The Fredholm alternative holds: The semi-homogeneous boundary value problem
  
<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/a/a010/a010810/a01081085.png" /></td> </tr></table>
+
$$
 +
l (y)  = f (t) ,\ \
 +
U _ {k} ( y )  = 0 ,\ \
 +
k = 1 \dots n ,
 +
$$
  
has a solution if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081086.png" /> is orthogonal to all non-trivial solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081087.png" /> of the adjoint boundary value problem (4), i.e. if
+
has a solution if $  f ( t ) $
 +
is orthogonal to all non-trivial solutions $  \xi ( t ) $
 +
of the adjoint boundary value problem (4), i.e. if
  
<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/a/a010/a010810/a01081088.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {t _ {0} } ^ { {t } _ {1} } \
 +
\overline \xi \; ( t ) f ( t )  dt  = 0
 +
$$
  
 
(see [[#References|[1]]]–[[#References|[3]]], [[#References|[7]]]).
 
(see [[#References|[1]]]–[[#References|[3]]], [[#References|[7]]]).
Line 111: Line 293:
 
For the eigen value problem
 
For the eigen value problem
  
<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/a/a010/a010810/a01081089.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
l ( y )  = \lambda y ,\ \
 +
U _ {k} (y)  = 0 ,\ \
 +
k = 1 \dots n ,
 +
$$
  
 
the adjoint eigen value problem is defined as
 
the adjoint eigen value problem is defined as
  
<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/a/a010/a010810/a01081090.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
l  ^ {*} ( \xi )  = \mu \xi ,\ \
 +
U _ {j}  ^ {*} ( \xi )  = 0 ,\ \
 +
j = 1 \dots n .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081091.png" /> is an eigen value of (5), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081092.png" /> is an eigen value of (6). The eigen functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081093.png" /> corresponding to eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081094.png" /> of (5), (6), respectively, are orthogonal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081095.png" /> (see [[#References|[1]]]–[[#References|[3]]]):
+
If $  \lambda $
 +
is an eigen value of (5), then $  \mu = \overline \lambda \; $
 +
is an eigen value of (6). The eigen functions $  y (t) , \xi ( t ) $
 +
corresponding to eigen values $  \lambda , \mu $
 +
of (5), (6), respectively, are orthogonal if $  \lambda \neq \mu $(
 +
see [[#References|[1]]]–[[#References|[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/a/a010/a010810/a01081096.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {t _ {0} } ^ { {t } _ {1} } \overline{y}\; ( t ) \xi ( t )  dt  = 0 .
 +
$$
  
 
For the linear boundary value problem
 
For the linear boundary value problem
  
<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/a/a010/a010810/a01081097.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
L (x)  \equiv  \dot{x} + A (t) x  = 0 ,\ \
 +
U (x) =  0 ,\  t \in \Delta ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081098.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a01081099.png" />-dimensional vector functional on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a010810100.png" /> of continuously-differentiable complex-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a010810101.png" />-dimensional vector functions with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a010810102.png" />, the adjoint boundary value problem is defined by
+
where $  U $
 +
is an $  m $-
 +
dimensional vector functional on the space $  C _ {n} ( \Delta ) $
 +
of continuously-differentiable complex-valued $  n $-
 +
dimensional vector functions with $  m < 2n $,  
 +
the adjoint boundary value problem is defined by
  
<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/a/a010/a010810/a010810103.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
$$ \tag{8 }
 +
L  ^ {*} ( \psi )  = 0 ,\ \
 +
U  ^ {*} ( \psi ) =  0 ,\ \
 +
t \in \Delta
 +
$$
  
(see [[#References|[1]]]). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a010810104.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a010810105.png" />-dimensional vector functional defined such that the equation
+
(see [[#References|[1]]]). Here $  U  ^ {*} $
 +
is a $  ( 2n - m ) $-
 +
dimensional vector functional defined such that the 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/a/a010/a010810/a010810106.png" /></td> </tr></table>
+
$$
 +
\left .
 +
( \psi ( t ) , x ( t ) )
 +
\right | _ {t = t _ {0}  } ^ {t = t _ {1} }
 +
= 0
 +
$$
  
holds for any pair of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010810/a010810107.png" /> satisfying the conditions
+
holds for any pair of functions $  x , \psi \in C _ {n}  ^ {1} ( \Delta ) $
 +
satisfying the conditions
  
<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/a/a010/a010810/a010810108.png" /></td> </tr></table>
+
$$
 +
U (x)  = 0 ,\  U  ^ {*} ( \psi )  = 0.
 +
$$
  
 
The problems (7), (8) possess properties analogous to those listed above (see [[#References|[1]]]).
 
The problems (7), (8) possess properties analogous to those listed above (see [[#References|[1]]]).

Revision as of 16:09, 1 April 2020


to an ordinary linear differential equation $ l (y) = 0 $

The ordinary linear differential equation $ l ^ {*} ( \xi ) = 0 $, where

$$ \tag{1 } l (y) \equiv a _ {0} (t) y ^ {(n)} + \dots + a _ {n} (t) y , $$

$$ y ^ {( \nu ) } = \frac{d ^ \nu y }{d t ^ \nu } ,\ y \in C ^ {n} (I) ,\ a _ {k} \in C ^ {n-k} (I) , $$

$$ a _ {0} (t) \neq 0 ,\ t \in I ; $$

$ C ^ {m} (I) $ is the space of $ m $- times continuously-differentiable complex-valued functions on $ I = ( \alpha , \beta ) $, and

$$ \tag{2 } l ^ {*} ( \xi ) \equiv \ ( - 1 ) ^ {n} ( \overline{a}\; _ {0} \xi ) ^ {(n)} + ( - 1 ) ^ {n-1} ( \overline{a}\; _ {1} \xi ) ^ {(n-1)} + \dots + \overline{a}\; _ {n} \xi , $$

$$ \xi \in C ^ {n} (I) $$

(the bar denotes complex conjugation). It follows at once that

$$ ( l _ {1} + l _ {2} ) ^ {*} = \ l _ {1} ^ {*} + l _ {2} ^ {*} ,\ \ ( \lambda l ) ^ {*} = \ \overline \lambda \; l ^ {*} , $$

for any scalar $ \lambda $. The adjoint of the equation $ l ^ {*} ( \xi ) = 0 $ is $ l (y) = 0 $. For all $ n $- times continuously-differentiable functions $ y (t) $ and $ \xi (t) $, Lagrange's identity holds:

$$ \overline{ {\xi l }}\; (y) - \overline{ {l ^ {*} ( \xi ) }}\; y = \frac{d}{dt} \left \{ \sum _ { k=1 } ^ { n } \sum _ { j=0 } ^ { k-1 } ( - 1 ) ^ {j} ( a _ {n-k} \overline \xi \; ) ^ {(j)} y ^ {( k - j - 1 ) } \right \} . $$

It implies Green's formula

$$ \int\limits _ { s } ^ \tau [ \overline \xi \; l (y) - \overline{ {l ^ {*} ( \xi ) }}\; y ] d t = $$

$$ = \ \left . \sum _ { k=1 } ^ { n } \sum _ { j=0 } ^ { k-1 } ( - 1 ) ^ {j} ( a _ {n-k} \overline \xi \; ) ^ {(j)} y ^ {( k - j - 1 ) } \right | _ {t=s} ^ {t = \tau } . $$

If $ y (t) $ and $ \xi (t) $ are arbitrary solutions of $ l (y) = 0 $ and $ l ^ {*} ( \xi ) = 0 $, respectively, then

$$ \sum _ { k=1 } ^ { n } \sum _ { j=0 } ^ { k-1 } ( - 1 ) ^ {j} ( a _ {n-k} \overline \xi \; ) ^ {(j)} y ^ {(k-j-1)} \equiv \textrm{ const } ,\ \ t \in I . $$

A knowledge of $ m ( \leq n ) $ linearly independent solutions of the equation $ l ^ {*} ( \xi ) = 0 $ enables one to reduce the order of the equation $ l (y) = 0 $ by $ m $( see [1][3]).

For a system of differential equations

$$ L (x) = 0 ,\ \ L (x) \equiv \ \dot{x} + A (t) x ,\ \ t \in I , $$

where $ A (t) $ is a continuous complex-valued $ ( n \times n ) $- matrix, the adjoint system is given by

$$ L ^ {*} ( \psi ) \equiv - \dot \psi + A ^ {*} (t) \psi = \ 0 ,\ t \in I $$

(see [1], [4]), where $ A ^ {*} (t) $ is the Hermitian adjoint of $ A (t) $. The Lagrange identity and the Green formula take the form

$$ ( \overline \psi \; , L (x) ) - ( \overline{ {L ^ {*} ( \psi ) }}\; , x ) = \frac{d}{dt} ( \overline \psi \; , x ) ,\ \ $$

$$ \left . \int\limits _ { s } ^ \tau [ ( \overline \psi \; , L (x) ) - ( \overline{ {L ^ {*} ( \psi ) }}\; , x ) ] d t = \ ( \overline \psi \; , x ) \right | _ {t=s} ^ {t = \tau } ; $$

where $ ( \cdot , \cdot ) $ is the standard scalar product (the sum of the products of coordinates with equal indices). If $ x (t) $ and $ \psi (t) $ are arbitrary solutions of the equations $ L (x) = 0 $ and $ L ^ {*} ( \psi ) = 0 $, then

$$ ( \overline \psi \; (t) , x (t) ) \equiv \textrm{ const } ,\ t \in I . $$

The concept of an adjoint differential equation is closely connected with the general concept of an adjoint operator. Thus, if $ l $ is a linear differential operator acting on $ C ^ {n} (I) $ into $ C (I) $ in accordance with (1), then its adjoint differential operator $ l ^ {*} $ maps the space $ C ^ {*} (I) $ adjoint to $ C (I) $ into the space $ C ^ {n*} (I) $ adjoint to $ C ^ {n} (I) $. The restriction of $ l ^ {*} $ to $ C ^ {n} (I) $ is given by formula (2) (see [5]).

Adjoints are also defined for linear partial differential equations (see [6], [5]).

Let $ \Delta = [ t _ {0} , t _ {1} ] \subset I $, and let $ U _ {k} $ be linearly independent linear functionals on $ C ^ {n} ( \Delta ) $. Then the boundary value problem adjoint to the linear boundary value problem

$$ \tag{3 } l (y) = 0 ,\ \ t \in \Delta ,\ \ U _ {k} (y) = 0 ,\ \ k = 1 \dots m ,\ \ m < 2 n , $$

is defined by the equations

$$ \tag{4 } l ^ {*} ( \xi ) = 0 ,\ \ U _ {j} ^ {*} ( \xi ) = 0 ,\ \ j = 1 \dots 2 n - m . $$

Here the $ U _ {j} ^ {*} $ are linear functionals on $ C ^ {n} ( \Delta ) $ describing the adjoint boundary conditions, that is, they are defined in such a way that the equation (see Green formulas)

$$ \int\limits _ {t _ {0} } ^ { {t } _ {1} } [ \overline \xi \; l (y) - \overline{ {l ^ {*} ( \xi ) }}\; y ] d t = 0 $$

holds for any pair of functions $ y , \xi \in C ^ {n} ( \Delta ) $ that satisfy the conditions $ U _ {k} (y) = 0 $, $ k = 1 \dots m $; $ U _ {j} ^ {*} ( \xi ) = 0 $, $ j = 1 \dots 2 n - m $.

If

$$ U _ {k} (y) \equiv \ \sum _ { p=1 } ^ { n } [ \alpha _ {kp} y ^ {(p-1)} ( t _ {0} ) + \beta _ {kp} y ^ {(p-1)} ( t _ {1} ) ] $$

are linear forms in the variables

$$ y ^ {(p-1)} ( t _ {0} ) ,\ \ y ^ {(p-1)} ( t _ {1} ) ,\ \ p = 1 \dots n , $$

then $ U _ {j} ^ {*} ( \xi ) $ are linear forms in the variables

$$ \xi ^ {(p-1)} ( t _ {0} ) ,\ \ \xi ^ {(p-1)} ( t _ {1} ) ,\ \ p = 1 \dots n . $$

Examples. For the problem

$$ \dot{y} dot + a (t) y = 0 ,\ \ 0 \leq t \leq 1 , $$

$$ y (0) + \alpha y (1) + \beta \dot{y} (1) = 0 , $$

$$ \dot{y} (0) + \gamma y (1) + \delta \dot{y} (1) = 0 , $$

with real $ a (t) , \alpha , \beta , \gamma , \delta $, the adjoint boundary value problem has the form

$$ \dot \xi dot + a (t) \xi = 0 ,\ \ 0 \leq t \leq 1 , $$

$$ \alpha \xi (0) + \gamma \dot \xi (0) + \xi (1) = 0 , $$

$$ \beta \xi (0) + \delta \dot \xi (0) + \dot \xi (1) = 0 . $$

If problem (3) has $ k $ linearly independent solutions (in this case the rank $ r $ of the boundary value problem is equal to $ n-k $), then problem (4) has $ m - n + k $ linearly independent solutions (its rank is $ r ^ \prime = 2n - m - k $). When $ m = n $, problems (3) and (4) have an equal number of linearly independent solutions. Therefore, when $ m = n $, problem (3) has only a trivial solution if and only if the adjoint boundary value problem (4) has the same property. The Fredholm alternative holds: The semi-homogeneous boundary value problem

$$ l (y) = f (t) ,\ \ U _ {k} ( y ) = 0 ,\ \ k = 1 \dots n , $$

has a solution if $ f ( t ) $ is orthogonal to all non-trivial solutions $ \xi ( t ) $ of the adjoint boundary value problem (4), i.e. if

$$ \int\limits _ {t _ {0} } ^ { {t } _ {1} } \ \overline \xi \; ( t ) f ( t ) dt = 0 $$

(see [1][3], [7]).

For the eigen value problem

$$ \tag{5 } l ( y ) = \lambda y ,\ \ U _ {k} (y) = 0 ,\ \ k = 1 \dots n , $$

the adjoint eigen value problem is defined as

$$ \tag{6 } l ^ {*} ( \xi ) = \mu \xi ,\ \ U _ {j} ^ {*} ( \xi ) = 0 ,\ \ j = 1 \dots n . $$

If $ \lambda $ is an eigen value of (5), then $ \mu = \overline \lambda \; $ is an eigen value of (6). The eigen functions $ y (t) , \xi ( t ) $ corresponding to eigen values $ \lambda , \mu $ of (5), (6), respectively, are orthogonal if $ \lambda \neq \mu $( see [1][3]):

$$ \int\limits _ {t _ {0} } ^ { {t } _ {1} } \overline{y}\; ( t ) \xi ( t ) dt = 0 . $$

For the linear boundary value problem

$$ \tag{7 } L (x) \equiv \dot{x} + A (t) x = 0 ,\ \ U (x) = 0 ,\ t \in \Delta , $$

where $ U $ is an $ m $- dimensional vector functional on the space $ C _ {n} ( \Delta ) $ of continuously-differentiable complex-valued $ n $- dimensional vector functions with $ m < 2n $, the adjoint boundary value problem is defined by

$$ \tag{8 } L ^ {*} ( \psi ) = 0 ,\ \ U ^ {*} ( \psi ) = 0 ,\ \ t \in \Delta $$

(see [1]). Here $ U ^ {*} $ is a $ ( 2n - m ) $- dimensional vector functional defined such that the equation

$$ \left . ( \psi ( t ) , x ( t ) ) \right | _ {t = t _ {0} } ^ {t = t _ {1} } = 0 $$

holds for any pair of functions $ x , \psi \in C _ {n} ^ {1} ( \Delta ) $ satisfying the conditions

$$ U (x) = 0 ,\ U ^ {*} ( \psi ) = 0. $$

The problems (7), (8) possess properties analogous to those listed above (see [1]).

The concept of an adjoint boundary value problem is closely connected with that of an adjoint operator [5]. Adjoint boundary value problems are also defined for linear boundary value problems for partial differential equations (see [6], [7]).

References

[1] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971)
[2] M.A. Naimark, "Linear differential operators" , 1–2 , F. Ungar (1967–1968) (Translated from Russian)
[3] E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[4] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[5] N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963)
[6] V.P. Mikhailov, "Partial differential equations" , MIR (1978) (Translated from Russian)
[7] V.S. Vladimirov, "Gleichungen der mathematischen Physik" , MIR (1984) (Translated from Russian)
How to Cite This Entry:
Adjoint differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_differential_equation&oldid=45036
This article was adapted from an original article by E.L. Tonkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article