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m (tex encoded by computer)
m (fixing spaces)
Line 31: Line 31:
  
 
$  C  ^ {m} (I) $
 
$  C  ^ {m} (I) $
is the space of  $  m $-
+
is the space of  $  m $-times continuously-differentiable complex-valued functions on  $  I = ( \alpha , \beta ) $,  
times continuously-differentiable complex-valued functions on  $  I = ( \alpha , \beta ) $,  
 
 
and
 
and
  
 
$$ \tag{2 }
 
$$ \tag{2 }
 
l  ^ {*} ( \xi )  \equiv \  
 
l  ^ {*} ( \xi )  \equiv \  
( - 1 )  ^ {n} ( \overline{a}\; _ {0} \xi )  ^ {(n)} + ( - 1 )
+
( - 1 )  ^ {n} ( \overline{a} _ {0} \xi )  ^ {(n)} + ( - 1 )
  ^ {n-1} ( \overline{a}\; _ {1} \xi )  ^ {(n-1)} + \dots + \overline{a}\; _ {n} \xi ,
+
  ^ {n-1} ( \overline{a} _ {1} \xi )  ^ {(n-1)} + \dots + \overline{a} _ {n} \xi ,
 
$$
 
$$
  
Line 51: Line 50:
 
l _ {1}  ^ {*} + l _ {2}  ^ {*} ,\ \  
 
l _ {1}  ^ {*} + l _ {2}  ^ {*} ,\ \  
 
( \lambda l )  ^ {*}  = \  
 
( \lambda l )  ^ {*}  = \  
\overline \lambda \; l  ^ {*} ,
+
\overline \lambda l  ^ {*} ,
 
$$
 
$$
  
Line 57: Line 56:
 
The adjoint of the equation  $  l  ^ {*} ( \xi ) = 0 $
 
The adjoint of the equation  $  l  ^ {*} ( \xi ) = 0 $
 
is  $  l (y) = 0 $.  
 
is  $  l (y) = 0 $.  
For all  $  n $-
+
For all  $  n $-times continuously-differentiable functions  $  y (t) $
times continuously-differentiable functions  $  y (t) $
 
 
and  $  \xi (t) $,  
 
and  $  \xi (t) $,  
 
Lagrange's identity holds:
 
Lagrange's identity holds:
  
 
$$  
 
$$  
\overline{ {\xi l }}\; (y) - \overline{ {l  ^ {*} ( \xi ) }}\; y  =   
+
\overline{ {\xi l }} (y) - \overline{ {l  ^ {*} ( \xi ) }} y  =   
 
\frac{d}{dt}
 
\frac{d}{dt}
  
 
\left \{
 
\left \{
 
\sum _ { k=1 } ^ { n }  \sum _ { j=0 } ^ { k-1 }  
 
\sum _ { k=1 } ^ { n }  \sum _ { j=0 } ^ { k-1 }  
( - 1 )  ^ {j} ( a _ {n-k} \overline \xi \; )  ^ {(j)} y ^ {( k - j - 1 ) }
+
( - 1 )  ^ {j} ( a _ {n-k} \overline \xi )  ^ {(j)} y ^ {( k - j - 1 ) }
 
\right \} .
 
\right \} .
 
$$
 
$$
Line 76: Line 74:
 
$$  
 
$$  
 
\int\limits _ { s } ^  \tau   
 
\int\limits _ { s } ^  \tau   
[ \overline \xi \; l (y) -
+
[ \overline \xi l (y) -
\overline{ {l  ^ {*} ( \xi ) }}\;
+
\overline{ {l  ^ {*} ( \xi ) }}
 
y ]  d t =
 
y ]  d t =
 
$$
 
$$
Line 84: Line 82:
 
= \  
 
= \  
 
\left . \sum _ { k=1 } ^ { n }  \sum _ { j=0 } ^ { k-1 }  ( - 1 )  ^ {j} (
 
\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 } .
+
a _ {n-k} \overline \xi )  ^ {(j)} y ^ {( k - j - 1 ) } \right | _ {t=s} ^ {t = \tau } .
 
$$
 
$$
  
Line 95: Line 93:
 
$$  
 
$$  
 
\sum _ { k=1 } ^ { n }  \sum _ { j=0 } ^ { k-1 }  
 
\sum _ { k=1 } ^ { n }  \sum _ { j=0 } ^ { k-1 }  
( - 1 )  ^ {j} ( a _ {n-k} \overline \xi \; )  ^ {(j)}
+
( - 1 )  ^ {j} ( a _ {n-k} \overline \xi )  ^ {(j)}
 
y  ^ {(k-j-1)}
 
y  ^ {(k-j-1)}
 
  \equiv  \textrm{ const } ,\ \  
 
  \equiv  \textrm{ const } ,\ \  
Line 104: Line 102:
 
linearly independent solutions of the equation  $  l  ^ {*} ( \xi ) = 0 $
 
linearly independent solutions of the equation  $  l  ^ {*} ( \xi ) = 0 $
 
enables one to reduce the order of the equation  $  l (y) = 0 $
 
enables one to reduce the order of the equation  $  l (y) = 0 $
by  $  m $(
+
by  $  m $ (see [[#References|[1]]]–[[#References|[3]]]).
see [[#References|[1]]]–[[#References|[3]]]).
 
  
 
For a system of differential equations
 
For a system of differential equations
Line 117: Line 114:
  
 
where  $  A (t) $
 
where  $  A (t) $
is a continuous complex-valued  $  ( n \times n ) $-
+
is a continuous complex-valued  $  ( n \times n ) $-matrix, the adjoint system is given by
matrix, the adjoint system is given by
 
  
 
$$  
 
$$  
Line 132: Line 128:
  
 
$$  
 
$$  
( \overline \psi \; , L (x) ) -
+
( \overline \psi , L (x) ) -
( \overline{ {L  ^ {*} ( \psi ) }}\; , x )  =   
+
( \overline{ {L  ^ {*} ( \psi ) }} , x )  =   
 
\frac{d}{dt}
 
\frac{d}{dt}
  
( \overline \psi \; , x ) ,\ \  
+
( \overline \psi , x ) ,\ \  
 
$$
 
$$
  
 
$$  
 
$$  
\left . \int\limits _ { s } ^  \tau  [ ( \overline \psi \; , L (x) ) - (
+
\left . \int\limits _ { s } ^  \tau  [ ( \overline \psi , L (x) ) - (
\overline{ {L  ^ {*} ( \psi ) }}\; , x ) ]  d t  = \  
+
\overline{ {L  ^ {*} ( \psi ) }} , x ) ]  d t  = \  
( \overline \psi \; , x ) \right | _ {t=s} ^ {t = \tau } ;
+
( \overline \psi , x ) \right | _ {t=s} ^ {t = \tau } ;
 
$$
 
$$
  
Line 153: Line 149:
  
 
$$  
 
$$  
( \overline \psi \; (t) , x (t) )  \equiv  \textrm{ const } ,\  t \in I .
+
( \overline \psi (t) , x (t) )  \equiv  \textrm{ const } ,\  t \in I .
 
$$
 
$$
  
Line 197: Line 193:
 
$$  
 
$$  
 
\int\limits _ {t _ {0} } ^ { {t } _ {1} }
 
\int\limits _ {t _ {0} } ^ { {t } _ {1} }
[ \overline \xi \; l (y) -
+
[ \overline \xi l (y) -
\overline{ {l  ^ {*} ( \xi ) }}\;
+
\overline{ {l  ^ {*} ( \xi ) }}
 
y ]  d t  =  0
 
y ]  d t  =  0
 
$$
 
$$
Line 206: Line 202:
 
$  k = 1 \dots m $;  
 
$  k = 1 \dots m $;  
 
$  U _ {j}  ^ {*} ( \xi ) = 0 $,  
 
$  U _ {j}  ^ {*} ( \xi ) = 0 $,  
$  j = 1 \dots 2 n - m $.
+
$  j = 1, \dots, 2 n - m $.
  
 
If
 
If
Line 286: Line 282:
 
$$  
 
$$  
 
\int\limits _ {t _ {0} } ^ { {t } _ {1} } \  
 
\int\limits _ {t _ {0} } ^ { {t } _ {1} } \  
\overline \xi \; ( t ) f ( t )  dt  =  0
+
\overline \xi ( t ) f ( t )  dt  =  0
 
$$
 
$$
  
 
(see [[#References|[1]]]–[[#References|[3]]], [[#References|[7]]]).
 
(see [[#References|[1]]]–[[#References|[3]]], [[#References|[7]]]).
  
For the eigen value problem
+
For the eigenvalue problem
  
 
$$ \tag{5 }
 
$$ \tag{5 }
Line 299: Line 295:
 
$$
 
$$
  
the adjoint eigen value problem is defined as
+
the adjoint eigenvalue problem is defined as
  
 
$$ \tag{6 }
 
$$ \tag{6 }
Line 308: Line 304:
  
 
If  $  \lambda $
 
If  $  \lambda $
is an eigen value of (5), then  $  \mu = \overline \lambda \; $
+
is an eigenvalue of (5), then  $  \mu = \overline \lambda $
is an eigen value of (6). The eigen functions $  y (t) , \xi ( t ) $
+
is an eigenvalue of (6). The eigenfunctions $  y (t) , \xi ( t ) $
corresponding to eigen values $  \lambda , \mu $
+
corresponding to eigenvalues $  \lambda , \mu $
of (5), (6), respectively, are orthogonal if  $  \lambda \neq \mu $(
+
of (5), (6), respectively, are orthogonal if  $  \lambda \neq \mu $ (see [[#References|[1]]]–[[#References|[3]]]):
see [[#References|[1]]]–[[#References|[3]]]):
 
  
 
$$  
 
$$  
\int\limits _ {t _ {0} } ^ { {t } _ {1} } \overline{y}\; ( t ) \xi ( t )  dt  =  0 .
+
\int\limits _ {t _ {0} } ^ { {t } _ {1} } \overline{y} ( t ) \xi ( t )  dt  =  0 .
 
$$
 
$$
  
Line 328: Line 323:
 
is an  $  m $-
 
is an  $  m $-
 
dimensional vector functional on the space  $  C _ {n} ( \Delta ) $
 
dimensional vector functional on the space  $  C _ {n} ( \Delta ) $
of continuously-differentiable complex-valued  $  n $-
+
of continuously-differentiable complex-valued  $  n $-dimensional vector functions with  $  m < 2n $,  
dimensional vector functions with  $  m < 2n $,  
 
 
the adjoint boundary value problem is defined by
 
the adjoint boundary value problem is defined by
  
Line 339: Line 333:
  
 
(see [[#References|[1]]]). Here  $  U  ^ {*} $
 
(see [[#References|[1]]]). Here  $  U  ^ {*} $
is a  $  ( 2n - m ) $-
+
is a  $  ( 2n - m ) $-dimensional vector functional defined such that the equation
dimensional vector functional defined such that the equation
 
  
 
$$  
 
$$  

Revision as of 06:15, 10 January 2022


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 eigenvalue problem

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

the adjoint eigenvalue problem is defined as

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

If $ \lambda $ is an eigenvalue of (5), then $ \mu = \overline \lambda $ is an eigenvalue of (6). The eigenfunctions $ y (t) , \xi ( t ) $ corresponding to eigenvalues $ \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