Adjoint differential equation

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

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

(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]).

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