Adjoint differential equation

to an ordinary linear differential equation $l(y)=0$

The ordinary linear differential equation $l^{\ast }(\xi )=0$, where

$$\begin{array}{}\text{(1)}& l(y)\equiv a_0(t)y^{(n)}+\cdots +a_n(t)y,\end{array}$$

$$y^{(\nu )}=\frac{d^{\nu }y}{d{t}^{\nu }},y\in C^n(I),a_k\in C^{n-k}(I),$$

$$a_0(t)\ne 0,t\in I;$$

$C^m(I)$ is the space of $m$-times continuously-differentiable complex-valued functions on $I=(\alpha ,\beta )$, and

$$\begin{array}{}\text{(2)}& l^{\ast }(\xi )\equiv (-1{)}^n({\bar{a}}_0\xi {)}^{(n)}+(-1{)}^{n-1}({\bar{a}}_1\xi {)}^{(n-1)}+\cdots +{\bar{a}}_n\xi ,\end{array}$$

$$\xi \in C^n(I)$$

(the bar denotes complex conjugation). It follows at once that

$$(l_1+l_2{)}^{\ast }=l_1^{\ast }+l_2^{\ast },(\lambda l{)}^{\ast }=\bar{\lambda }{l}^{\ast },$$

for any scalar $\lambda$. The adjoint of the equation $l^{\ast }(\xi )=0$ is $l(y)=0$. For all $n$-times continuously-differentiable functions $y(t)$ and $\xi (t)$, Lagrange's identity holds:

$$\overline{\xi l}(y)-\overline{l^{\ast }(\xi )}y=\frac{d}{dt}\{\sum _{k=1}^n\sum _{j=0}^{k-1}(-1{)}^j(a_{n-k}\bar{\xi }{)}^{(j)}{y}^{(k-j-1)}\}.$$

It implies Green's formula

$$\underset{s}{\overset{\tau }{\int }}[\bar{\xi }l(y)-\overline{l^{\ast }(\xi )}y]dt=$$

$$={\sum _{k=1}^n\sum _{j=0}^{k-1}(-1{)}^j(a_{n-k}\bar{\xi }{)}^{(j)}{y}^{(k-j-1)}|}_{t=s}^{t=\tau }.$$

If $y(t)$ and $\xi (t)$ are arbitrary solutions of $l(y)=0$ and $l^{\ast }(\xi )=0$, respectively, then

$$\sum _{k=1}^n\sum _{j=0}^{k-1}(-1{)}^j(a_{n-k}\bar{\xi }{)}^{(j)}{y}^{(k-j-1)}\equiv \text{ const },t\in I.$$

A knowledge of $m(\le n)$ linearly independent solutions of the equation $l^{\ast }(\xi )=0$ 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

$$L^{\ast }(\psi )\equiv -\dot{\psi }+A^{\ast }(t)\psi =0,t\in I$$

(see [1], [4]), where $A^{\ast }(t)$ is the Hermitian adjoint of $A(t)$. The Lagrange identity and the Green formula take the form

$$(\bar{\psi },L(x))-(\overline{L^{\ast }(\psi )},x)=\frac{d}{dt}(\bar{\psi },x),$$

$${\underset{s}{\overset{\tau }{\int }}[(\bar{\psi },L(x))-(\overline{L^{\ast }(\psi )},x)]dt=(\bar{\psi },x)|}_{t=s}^{t=\tau };$$

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 $L^{\ast }(\psi )=0$, then

$$(\bar{\psi }(t),x(t))\equiv \text{ const },t\in I.$$

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 $l^{\ast }$ maps the space $C^{\ast }(I)$ adjoint to $C(I)$ into the space $C^{n\ast }(I)$ adjoint to $C^n(I)$. The restriction of $l^{\ast }$ to $C^n(I)$ is given by formula (2) (see [5]).

Adjoints are also defined for linear partial differential equations (see [6], [5]).

Let $\mathrm{\Delta }=[t_0,t_1]\subset I$, and let $U_k$ be linearly independent linear functionals on $C^n(\mathrm{\Delta })$. Then the boundary value problem adjoint to the linear boundary value problem

$$\begin{array}{}\text{(3)}& l(y)=0,t\in \mathrm{\Delta },U_k(y)=0,k=1,\dots ,m,m<2n,\end{array}$$

is defined by the equations

$$\begin{array}{}\text{(4)}& l^{\ast }(\xi )=0,U_j^{\ast }(\xi )=0,j=1,\dots ,2n-m.\end{array}$$

Here the $U_j^{\ast }$ are linear functionals on $C^n(\mathrm{\Delta })$ describing the adjoint boundary conditions, that is, they are defined in such a way that the equation (see Green formulas)

$$\underset{t_0}{\overset{t_1}{\int }}[\bar{\xi }l(y)-\overline{l^{\ast }(\xi )}y]dt=0$$

holds for any pair of functions $y,\xi \in C^n(\mathrm{\Delta })$ that satisfy the conditions $U_k(y)=0$, $k=1,\dots ,m$; $U_j^{\ast }(\xi )=0$, $j=1,\dots ,2n-m$.

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 $U_j^{\ast }(\xi )$ are linear forms in the variables

$${\xi }^{(p-1)}(t_0),{\xi }^{(p-1)}(t_1),p=1,\dots ,n.$$

Examples. For the problem

$$\ddot{y}+a(t)y=0,0\le t\le 1,$$

$$y(0)+\alpha y(1)+\beta \dot{y}(1)=0,$$

$$\dot{y}(0)+\gamma y(1)+\delta \dot{y}(1)=0,$$

with real $a(t),\alpha ,\beta ,\gamma ,\delta$, the adjoint boundary value problem has the form

$$\ddot{\xi }+a(t)\xi =0,0\le t\le 1,$$

$$\alpha \xi (0)+\gamma \dot{\xi }(0)+\xi (1)=0,$$

$$\beta \xi (0)+\delta \dot{\xi }(0)+\dot{\xi }(1)=0.$$

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

$$\underset{t_0}{\overset{t_1}{\int }}\bar{\xi }(t)f(t)dt=0$$

(see [1]–[3], [7]).

For the eigenvalue problem

$$\begin{array}{}\text{(5)}& l(y)=\lambda y,U_k(y)=0,k=1,\dots ,n,\end{array}$$

the adjoint eigenvalue problem is defined as

$$\begin{array}{}\text{(6)}& l^{\ast }(\xi )=\mu \xi ,U_j^{\ast }(\xi )=0,j=1,\dots ,n.\end{array}$$

If $\lambda$ is an eigenvalue of (5), then $\mu =\bar{\lambda }$ is an eigenvalue of (6). The eigenfunctions $y(t),\xi (t)$ corresponding to eigenvalues $\lambda ,\mu$ of (5), (6), respectively, are orthogonal if $\lambda \ne \mu$ (see [1]–[3]):

$$\underset{t_0}{\overset{t_1}{\int }}\bar{y}(t)\xi (t)dt=0.$$

For the linear boundary value problem

$$\begin{array}{}\text{(7)}& L(x)\equiv \dot{x}+A(t)x=0,U(x)=0,t\in \mathrm{\Delta },\end{array}$$

where $U$ is an $m$- dimensional vector functional on the space $C_n(\mathrm{\Delta })$ of continuously-differentiable complex-valued $n$-dimensional vector functions with $m<2n$, the adjoint boundary value problem is defined by

$$\begin{array}{}\text{(8)}& L^{\ast }(\psi )=0,U^{\ast }(\psi )=0,t\in \mathrm{\Delta }\end{array}$$

(see [1]). Here $U^{\ast }$ is a $(2n-m)$-dimensional vector functional defined such that the equation

$${(\psi (t),x(t))|}_{t=t_0}^{t=t_1}=0$$

holds for any pair of functions $x,\psi \in C_n^1(\mathrm{\Delta })$ satisfying the conditions

$$U(x)=0,U^{\ast }(\psi )=0.$$

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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