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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)
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