to an ordinary linear differential equation 
The ordinary linear differential equation 
, where
 | (1) |
is the space of 
-times continuously-differentiable complex-valued functions on 
, and
 | (2) |
(the bar denotes complex conjugation). It follows at once that
for any scalar 
. The adjoint of the equation 
is 
. For all 
-times continuously-differentiable functions 
and 
, Lagrange's identity holds:
It implies Green's formula
If 
and 
are arbitrary solutions of 
and 
, respectively, then
A knowledge of 
linearly independent solutions of the equation 
enables one to reduce the order of the equation 
by 
(see [1]–[3]).
For a system of differential equations
where 
is a continuous complex-valued 
-matrix, the adjoint system is given by
(see [1], [4]), where 
is the Hermitian adjoint of 
. The Lagrange identity and the Green formula take the form
where 
is the standard scalar product (the sum of the products of coordinates with equal indices). If 
and 
are arbitrary solutions of the equations 
and 
, then
The concept of an adjoint differential equation is closely connected with the general concept of an adjoint operator. Thus, if 
is a linear differential operator acting on 
into 
in accordance with (1), then its adjoint differential operator 
maps the space 
adjoint to 
into the space 
adjoint to 
. The restriction of 
to 
is given by formula (2) (see [5]).
Adjoints are also defined for linear partial differential equations (see [6], [5]).
Let 
, and let 
be linearly independent linear functionals on 
. Then the boundary value problem adjoint to the linear boundary value problem
 | (3) |
is defined by the equations
 | (4) |
Here the 
are linear functionals on 
describing the adjoint boundary conditions, that is, they are defined in such a way that the equation (see Green formulas)
holds for any pair of functions 
that satisfy the conditions 
, 
; 
, 
.
If
are linear forms in the variables
then 
are linear forms in the variables
Examples. For the problem
with real 
, the adjoint boundary value problem has the form
If problem (3) has 
linearly independent solutions (in this case the rank 
of the boundary value problem is equal to 
), then problem (4) has 
linearly independent solutions (its rank is 
). When 
, problems (3) and (4) have an equal number of linearly independent solutions. Therefore, when 
, 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
has a solution if 
is orthogonal to all non-trivial solutions 
of the adjoint boundary value problem (4), i.e. if
(see [1]–[3], [7]).
For the eigen value problem
 | (5) |
the adjoint eigen value problem is defined as
 | (6) |
If 
is an eigen value of (5), then 
is an eigen value of (6). The eigen functions 
corresponding to eigen values 
of (5), (6), respectively, are orthogonal if 
(see [1]–[3]):
For the linear boundary value problem
 | (7) |
where 
is an 
-dimensional vector functional on the space 
of continuously-differentiable complex-valued 
-dimensional vector functions with 
, the adjoint boundary value problem is defined by
 | (8) |
(see [1]). Here 
is a 
-dimensional vector functional defined such that the equation
holds for any pair of functions 
satisfying the conditions
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) |
