Difference between revisions of "Adjoint differential equation"
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$ 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} | + | ( - 1 ) ^ {n} ( \overline{a} _ {0} \xi ) ^ {(n)} + ( - 1 ) |
− | ^ {n-1} ( \overline{a} | + | ^ {n-1} ( \overline{a} _ {1} \xi ) ^ {(n-1)} + \dots + \overline{a} _ {n} \xi , |
$$ | $$ | ||
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l _ {1} ^ {*} + l _ {2} ^ {*} ,\ \ | l _ {1} ^ {*} + l _ {2} ^ {*} ,\ \ | ||
( \lambda l ) ^ {*} = \ | ( \lambda l ) ^ {*} = \ | ||
− | \overline \lambda | + | \overline \lambda l ^ {*} , |
$$ | $$ | ||
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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 }} | + | \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 | + | ( - 1 ) ^ {j} ( a _ {n-k} \overline \xi ) ^ {(j)} y ^ {( k - j - 1 ) } |
\right \} . | \right \} . | ||
$$ | $$ | ||
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$$ | $$ | ||
\int\limits _ { s } ^ \tau | \int\limits _ { s } ^ \tau | ||
− | [ \overline \xi | + | [ \overline \xi l (y) - |
− | \overline{ {l ^ {*} ( \xi ) }} | + | \overline{ {l ^ {*} ( \xi ) }} |
y ] d t = | y ] d t = | ||
$$ | $$ | ||
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= \ | = \ | ||
\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 | + | a _ {n-k} \overline \xi ) ^ {(j)} y ^ {( k - j - 1 ) } \right | _ {t=s} ^ {t = \tau } . |
$$ | $$ | ||
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$$ | $$ | ||
\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 | + | ( - 1 ) ^ {j} ( a _ {n-k} \overline \xi ) ^ {(j)} |
y ^ {(k-j-1)} | y ^ {(k-j-1)} | ||
\equiv \textrm{ const } ,\ \ | \equiv \textrm{ const } ,\ \ | ||
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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 | ||
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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 | ||
$$ | $$ | ||
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$$ | $$ | ||
− | ( \overline \psi | + | ( \overline \psi , L (x) ) - |
− | ( \overline{ {L ^ {*} ( \psi ) }} | + | ( \overline{ {L ^ {*} ( \psi ) }} , x ) = |
\frac{d}{dt} | \frac{d}{dt} | ||
− | ( \overline \psi | + | ( \overline \psi , x ) ,\ \ |
$$ | $$ | ||
$$ | $$ | ||
− | \left . \int\limits _ { s } ^ \tau [ ( \overline \psi | + | \left . \int\limits _ { s } ^ \tau [ ( \overline \psi , L (x) ) - ( |
− | \overline{ {L ^ {*} ( \psi ) }} | + | \overline{ {L ^ {*} ( \psi ) }} , x ) ] d t = \ |
− | ( \overline \psi | + | ( \overline \psi , x ) \right | _ {t=s} ^ {t = \tau } ; |
$$ | $$ | ||
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$$ | $$ | ||
− | ( \overline \psi | + | ( \overline \psi (t) , x (t) ) \equiv \textrm{ const } ,\ t \in I . |
$$ | $$ | ||
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t \in \Delta ,\ \ | t \in \Delta ,\ \ | ||
U _ {k} (y) = 0 ,\ \ | U _ {k} (y) = 0 ,\ \ | ||
− | k = 1 \dots m ,\ \ | + | k = 1, \dots, m ,\ \ |
m < 2 n , | m < 2 n , | ||
$$ | $$ | ||
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l ^ {*} ( \xi ) = 0 ,\ \ | l ^ {*} ( \xi ) = 0 ,\ \ | ||
U _ {j} ^ {*} ( \xi ) = 0 ,\ \ | U _ {j} ^ {*} ( \xi ) = 0 ,\ \ | ||
− | j = 1 \dots 2 n - m . | + | j = 1, \dots, 2 n - m . |
$$ | $$ | ||
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$$ | $$ | ||
\int\limits _ {t _ {0} } ^ { {t } _ {1} } | \int\limits _ {t _ {0} } ^ { {t } _ {1} } | ||
− | [ \overline \xi | + | [ \overline \xi l (y) - |
− | \overline{ {l ^ {*} ( \xi ) }} | + | \overline{ {l ^ {*} ( \xi ) }} |
y ] d t = 0 | y ] d t = 0 | ||
$$ | $$ | ||
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holds for any pair of functions $ y , \xi \in C ^ {n} ( \Delta ) $ | holds for any pair of functions $ y , \xi \in C ^ {n} ( \Delta ) $ | ||
that satisfy the conditions $ U _ {k} (y) = 0 $, | that satisfy the conditions $ U _ {k} (y) = 0 $, | ||
− | $ 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 | ||
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y ^ {(p-1)} ( t _ {0} ) ,\ \ | y ^ {(p-1)} ( t _ {0} ) ,\ \ | ||
y ^ {(p-1)} ( t _ {1} ) ,\ \ | y ^ {(p-1)} ( t _ {1} ) ,\ \ | ||
− | p = 1 \dots n , | + | p = 1, \dots, n , |
$$ | $$ | ||
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\xi ^ {(p-1)} ( t _ {0} ) ,\ \ | \xi ^ {(p-1)} ( t _ {0} ) ,\ \ | ||
\xi ^ {(p-1)} ( t _ {1} ) ,\ \ | \xi ^ {(p-1)} ( t _ {1} ) ,\ \ | ||
− | p = 1 \dots n . | + | p = 1, \dots, n . |
$$ | $$ | ||
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$$ | $$ | ||
− | \ | + | \ddot{y} + a (t) y = 0 ,\ \ |
0 \leq t \leq 1 , | 0 \leq t \leq 1 , | ||
$$ | $$ | ||
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$$ | $$ | ||
− | \ | + | \ddot \xi + a (t) \xi = 0 ,\ \ |
0 \leq t \leq 1 , | 0 \leq t \leq 1 , | ||
$$ | $$ | ||
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l (y) = f (t) ,\ \ | l (y) = f (t) ,\ \ | ||
U _ {k} ( y ) = 0 ,\ \ | U _ {k} ( y ) = 0 ,\ \ | ||
− | k = 1 \dots n , | + | k = 1, \dots, n , |
$$ | $$ | ||
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$$ | $$ | ||
\int\limits _ {t _ {0} } ^ { {t } _ {1} } \ | \int\limits _ {t _ {0} } ^ { {t } _ {1} } \ | ||
− | \overline \xi | + | \overline \xi ( t ) f ( t ) dt = 0 |
$$ | $$ | ||
(see [[#References|[1]]]–[[#References|[3]]], [[#References|[7]]]). | (see [[#References|[1]]]–[[#References|[3]]], [[#References|[7]]]). | ||
− | For the | + | For the eigenvalue problem |
$$ \tag{5 } | $$ \tag{5 } | ||
l ( y ) = \lambda y ,\ \ | l ( y ) = \lambda y ,\ \ | ||
U _ {k} (y) = 0 ,\ \ | U _ {k} (y) = 0 ,\ \ | ||
− | k = 1 \dots n , | + | k = 1, \dots, n , |
$$ | $$ | ||
− | the adjoint | + | the adjoint eigenvalue problem is defined as |
$$ \tag{6 } | $$ \tag{6 } | ||
l ^ {*} ( \xi ) = \mu \xi ,\ \ | l ^ {*} ( \xi ) = \mu \xi ,\ \ | ||
U _ {j} ^ {*} ( \xi ) = 0 ,\ \ | U _ {j} ^ {*} ( \xi ) = 0 ,\ \ | ||
− | j = 1 \dots n . | + | j = 1, \dots, n . |
$$ | $$ | ||
If $ \lambda $ | If $ \lambda $ | ||
− | is an | + | is an eigenvalue of (5), then $ \mu = \overline \lambda $ |
− | is an | + | is an eigenvalue of (6). The eigenfunctions $ y (t) , \xi ( t ) $ |
− | corresponding to | + | 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} | + | \int\limits _ {t _ {0} } ^ { {t } _ {1} } \overline{y} ( t ) \xi ( t ) dt = 0 . |
$$ | $$ | ||
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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 | ||
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(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 | ||
$$ | $$ |
Latest revision as of 01:30, 5 March 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
$$ \ddot{y} + a (t) y = 0 ,\ \ 0 \leq t \leq 1 , $$
$$ y (0) + \alpha y (1) + \beta \dot{y} (1) = 0 , $$
$$ \dot{y} (0) + \gamma y (1) + \delta \dot{y} (1) = 0 , $$
with real $ a (t) , \alpha , \beta , \gamma , \delta $, the adjoint boundary value problem has the form
$$ \ddot \xi + a (t) \xi = 0 ,\ \ 0 \leq t \leq 1 , $$
$$ \alpha \xi (0) + \gamma \dot \xi (0) + \xi (1) = 0 , $$
$$ \beta \xi (0) + \delta \dot \xi (0) + \dot \xi (1) = 0 . $$
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
$$ \int\limits _ {t _ {0} } ^ { {t } _ {1} } \ \overline \xi ( t ) f ( t ) dt = 0 $$
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) |
Adjoint differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_differential_equation&oldid=45036