Fokas Diagonalization of Piecewise Constant Coefficient Linear Differential Operators on Finite Intervals and Networks

Abstract

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included. Both homogeneous and inhomogeneous problems are solved.

Appendix: Adjoints of Multipoint and Interface Differential Operators

The definition of the Fokas transform in §7 relies on the classical (or Lagrange) adjoint of the interface differential operator. In this appendix, we undertake the construction of that adjoint, and its generalization to interface operators in which the component operators $L_{r}$ are permitted to be of different orders from one another, and are permitted to have variable coefficients. The construction follows closely that laid out by Coddington and Levinson [6, Chap. 11], and generalizes those results to and beyond what is presented in [38]. We use notation aligned with the main sections of this paper, but all necessary definitions and arguments are contained within this appendix, so that it may be read independently.

1.1 A.1 Formulation of the Problem

Let $m\in \mathbb{N}$ and, for each $r\in \{1,\ldots ,m\}$, suppose $n_{r}\in \mathbb{N}$ and define

$$ n_{\mathrm{all}}= \sum _{r=1}^{m} n_{r}. $$

For suitable coefficient functions $c_{r\;j}$, consider the formal differential operators

$$ \mathscr {L}_{r} := \left ( \frac{\,\mathrm{d}}{\,\mathrm{d}x} \right )^{n_{r}} + \sum _{j=0}^{n_{r}-2} c_{r\;j}(x) \left ( \frac{\,\mathrm{d}}{\,\mathrm{d}x} \right )^{j}, \qquad r \in \{1, \ldots ,m\}, $$

in which we have assumed $c_{r\;n_{r}}=1$ and $c_{r\;n_{r}-1}=0$, and the vector formal differential operator

$$ \mathscr {L} := (\mathscr {L}_{1},\mathscr {L}_{2},\ldots ,\mathscr {L}_{m}) \circ , $$

where ∘ represents entrywise action of operators.

We define $\Phi = \prod ^{m}_{r=1} \operatorname{AC}^{n_{r}-1}[0, 1]$, a product of function spaces $\operatorname{AC}^{n_{r}-1}[0, 1]$. The inner product on this space is defined as the sum of the ordinary unweighted sesquilinear $\mathrm{L}^{2}$ inner products on the constituent spaces,

$$ \langle \phi ,\psi \rangle := \sum ^{m}_{r=1} \int _{0}^{1} \phi _{r}(x) \overline{\psi _{r}(x)} \,\mathrm{d}x. $$

Let $N\in \mathbb{Z}$ such that $0 \leqslant N\leqslant 2n_{\mathrm{all}}$ denote the number of boundary conditions. Suppose that matrices of complex boundary coefficients $b^{r},\beta ^{r}\in \mathbb{C}^{N\times n_{r}}$, $r\in \{1, \ldots , m\}$ are chosen such that the concatenated matrix $(b^{1}:\beta ^{1}:\ldots :b^{m}:\beta ^{m})\in \mathbb{C}^{N\times 2n_{ \mathrm{all}}}$ has full rank. Define boundary forms $B_{k}:\Phi \to \mathbb{C}$ by

$$ B_{k}(\phi ) = \sum _{r=1}^{m} \sum _{j=1}^{n_{r}} \left ( b_{k\;j}^{r} \phi _{r}^{(j-1)}(0) + \beta _{k\;j}^{r}\phi _{r}^{(j-1)}(1) \right ), \qquad k\in \left \{ 1,2,\ldots ,N\right \} , $$

(A.1)

and let $\mathbf{B} = (B_{1}, B_{2}, \ldots , B_{N})$ be the vector of boundary forms. Provided that, for each $r\in \{1,\ldots ,m\}$ and $j\in \{0,1,\ldots ,n_{r}-2\}$, $c_{r\;j}\in \operatorname{AC}^{n_{r}-j-1}[0,1]$, on the space

$$ \Phi _{\mathbf{B}}=\{\phi \in \Phi : \mathbf{B}\phi =\mathbf{0}\}, $$

we define the differential operator $L:\Phi _{\mathbf{B}}\to \prod _{r=1}^{m} \mathrm{L}^{1}[0,1]$ by $L\phi =\mathscr {L}\phi$.

1.1.1 A.1.1 Adjoint Problem

For the operator $L$ defined above, we aim to construct the classical adjoint $L^{\star }:\Phi _{\mathbf{B}^{\star }} \to \prod _{r=1}^{m} \mathrm{L}^{1}[0,1]$. That is, we aim to find a formal differential operator $\mathscr {L}^{\star }$ and adjoint vector boundary form $\mathbf{B}^{\star }$ such that, for all $\phi \in \Phi _{\mathbf{B}}$ and all $\psi \in \Phi _{\mathbf{B}^{\star }}$,

$$ \langle \mathscr {L}\phi , \psi \rangle = \langle \phi , \mathscr {L}^{\star }\psi \rangle . $$

By [6, theorem 3.6.3] applied entrywise, the formal adjoint is the operator $\mathscr {L}^{\star }= (\mathscr {L}_{1}^{\star },\mathscr {L}_{2}^{\star }, \ldots ,\mathscr {L}_{m}^{\star }) \circ$, in which

$$\begin{aligned} \mathscr {L}_{r}^{\star }\psi _{r} := (-1)^{n_{r}} \frac{\,\mathrm{d}^{n_{r}}\psi _{r}}{\,\mathrm{d}x^{n_{r}}} + \sum ^{n_{r}-2}_{j=0} (-1)^{j} \frac{\,\mathrm{d}^{j}}{\,\mathrm{d}x^{j}}\left ( \overline{c_{r\;j}(x)} \psi _{r} \right ). \end{aligned}$$

(A.2)

It remains to determine an appropriate adjoint vector boundary form $\mathbf{B}^{\star }$.

Remark 34

The only application of the results of this appendix in the present work is for the case $n_{1}=n_{2}=\cdots =n_{m}=:n$, all $c_{r\;j}$ constant, and $N=n_{\mathrm{all}}=mn$, and therefore the reader may restrict themself to this case in what follows, if they desire.

1.2 A.2 Green’s Formula

Following [6, theorem 3.6.3], for $\phi ,\psi \in \Phi$, we define $[\phi \psi ]_{r}$ to be the form in

$$ (\phi _{r},\phi '_{r},\ldots ,\phi ^{n_{r}-1}_{r}) \qquad \mbox{and} \qquad (\psi _{r},\psi '_{r},\ldots ,\psi ^{n_{r}-1}_{r}) $$

given by

$$ [\phi \psi ]_{r} = \sum _{\ell =1}^{n_{r}}\sum _{ \substack{j+k=\ell -1\\j,k\geqslant 0}} (-1)^{j} \phi _{r}^{(k)} \left ( c_{r\;n_{r}-\ell }\overline{\psi _{r}} \right )^{(j)}. $$

Application of Green’s formula [6, corollary to theorem 3.6.3] yields

$$\begin{aligned} \langle \mathscr {L}\phi , \psi \rangle - \langle \phi , \mathscr {L}^{\star }\psi \rangle &= \sum _{r=1}^{m} \left ( [\phi \psi ]_{r} (1) - [ \phi \psi ]_{r} (0) \right ) \\ &= \sum _{r=1}^{m}\sum _{j,k=1}^{n_{r}} \left (F_{j\;k}^{r}(1) \phi _{r}^{(k-1)}(1) \psi _{r}^{(j-1)}(1) - F_{j\;k}^{r}(0) \phi _{r}^{(k-1)}(0) \psi _{r}^{(j-1)}(0) \right ), \end{aligned}$$

where $F^{r}(x)$ denotes an $n_{r} \times n_{r}$ matrix at the point $x \in [0,1]$. Following [6, §11.1], the entries of $F^{r}(x)$, for $j,k \in \{1, \ldots , n_{r}\}$, are given by

$$ F_{j\, k}^{r}(x) = \textstyle\begin{cases} \sum ^{n_{r}-k}_{\ell = j-1} (-1)^{\ell }\binom{\ell }{j-1} \left ( \frac{\,\mathrm{d}}{\,\mathrm{d}x}\right )^{\ell - j +1} c_{r\; \ell + k}(x) & j + k< n_{r}+ 1, \\ (-1)^{j-1} c_{r\;n_{r}}(x) = (-1)^{j-1} & j+k = n_{r} + 1, \\ 0 & j+k > n_{r} + 1. \end{cases} $$

(A.3)

Observe that since $\mathrm{det}\left (F^{r}(x)\right ) = 1$, the matrix $F^{r}$ is non-singular for all $r \in \{1, \ldots , m\}$.

Our goal is to rewrite the right side of Green’s formula as a sesquilinear form $\mathscr {S}$. First, let

$$ \vec{\phi _{r}} := \left (\phi _{r}, \phi '_{r}, \ldots , \phi _{r}^{(n_{r} - 1)} \right )^{\top }, $$

and observe that

$$\begin{aligned}{} [\phi \psi ]_{r}(x) = \sum _{k,j=1}^{n_{r}}F_{j\;k}^{r} (x) \phi _{r}^{(k-1)} (x) \psi _{r}^{(j-1)} (x) &= \sum _{j=1}^{n}\left [ \left (\sum _{k=1}^{n} F_{j\;k}^{r} \phi _{r}^{(k-1)}\right ) \psi _{r}^{(j-1)}\right ](x) \\ &= F^{r} (x) \vec{\phi _{r}}(x) \odot \vec{\psi _{r}}(x), \end{aligned}$$

where ⊙ refers to the sesquilinear dot product on $\mathbb{C}^{n_{r}}$. Green’s formula can then be rewritten as

$$\begin{aligned} \langle \mathscr {L}\phi , \psi \rangle - \langle \phi , \mathscr {L}^{\star }\psi \rangle &= \sum _{r=1}^{m} [\phi \psi ]_{r}(1) - [\phi \psi ]_{r}(0) \\ &= \sum _{r=1}^{m} F^{r} (1) \vec{\phi _{r}} (1) \odot \vec{\psi _{r}} (1) - F^{r} (0) \vec{\phi _{r}} (0) \odot \vec{\psi _{r}} (0) \\ &= \sum _{r=1}^{m} \begin{bmatrix} - F^{r}(0) & 0_{n_{r} \times n_{r}} \\ 0_{n_{r} \times n_{r}} & F^{r}(1) \\ \end{bmatrix} \begin{bmatrix} \vec{\phi _{r}}(0) \\ \vec{\phi _{r}}(1) \\ \end{bmatrix} \odot \begin{bmatrix} \vec{\psi _{r}}(0) \\ \vec{\psi _{r}}(1) \\ \end{bmatrix}. \end{aligned}$$

(A.4)

Expansion of the sum yields

$$\begin{aligned} &\langle \mathscr {L}\phi , \psi \rangle - \langle \phi , \mathscr {L}^{\star }\psi \rangle \\ &\hspace{5em}= \begin{bmatrix} - F^{1}(0) & 0_{n_{1}\times n_{1}} \\ 0_{n_{1}\times n_{1}} & F^{1}(1) \\ \end{bmatrix} \begin{bmatrix} \vec{\phi _{1}}(0) \\ \vec{\phi _{1}}(1) \\ \end{bmatrix} \odot \begin{bmatrix} \vec{\psi _{1}}(0) \\ \vec{\psi _{1}}(1) \\ \end{bmatrix} \\ &\hspace{14em} + \cdots + \begin{bmatrix} - F^{m}(0) & 0_{n_{m}\times n_{m}} \\ 0_{n_{m}\times n_{m}} & F^{m}(1) \\ \end{bmatrix} \begin{bmatrix} \vec{\phi _{m}}(0) \\ \vec{\phi _{m}}(1) \\ \end{bmatrix} \odot \begin{bmatrix} \vec{\psi _{m}}(0) \\ \vec{\psi _{m}}(1) \\ \end{bmatrix} \\ &\hspace{5em}= \underbrace{ \begin{bmatrix} - F^{1}(0) & 0 & \cdots & 0 & 0 \\ 0 & F^{1}(1) & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & -F^{m}(0) & 0 \\ 0 & 0 & \cdots & 0 & F^{m}(1) \end{bmatrix}}_{\text{$2n_{\mathrm{all}}\times 2n_{\mathrm{all}}$}} \begin{bmatrix} \vec{\phi _{1}}(0) \\ \vec{\phi _{1}}(1) \\ \vec{\phi _{2}}(0) \\ \vec{\phi _{2}}(1) \\ \vdots \\ \vec{\phi _{m}}(0) \\ \vec{\phi _{m}}(1) \end{bmatrix} \odot \begin{bmatrix} \vec{\psi _{1}}(0) \\ \vec{\psi _{1}}(1) \\ \vec{\psi _{2}}(0) \\ \vec{\psi _{2}}(1) \\ \vdots \\ \vec{\psi _{m}}(0) \\ \vec{\psi _{m}}(1) \end{bmatrix} \\ &\hspace{5em}=: S \begin{bmatrix} \vec{\phi _{1}}(0) \\ \vec{\phi _{1}}(1) \\ \vdots \\ \vec{\phi _{m}}(0) \\ \vec{\phi _{m}}(1) \end{bmatrix} \odot \begin{bmatrix} \vec{\psi _{1}}(0) \\ \vec{\psi _{1}}(1) \\ \vdots \\ \vec{\psi _{m}}(0) \\ \vec{\psi _{m}}(1) \end{bmatrix} =: \mathscr {S} \left ( \begin{bmatrix} \vec{\phi _{1}}(0) \\ \vec{\phi _{1}}(1) \\ \vdots \\ \vec{\phi _{m}}(0) \\ \vec{\phi _{m}}(1) \end{bmatrix}, \begin{bmatrix} \vec{\psi _{1}}(0) \\ \vec{\psi _{1}}(1) \\ \vdots \\ \vec{\psi _{m}}(0) \\ \vec{\psi _{m}}(1) \end{bmatrix} \right ), \end{aligned}$$

(A.5)

where the matrix $S$ is associated with the sesquilinear form $\mathscr {S}$, and $S$ is a block diagonal matrix whose diagonal blocks are $n_{r} \times n_{r}$. This may be compared with [6, equation (11.1.3)].

1.3 A.3 Interface Boundary Form Formula

We turn to characterising adjoint interface boundary conditions, by extending the boundary form formula for two point problems, as given in [6, theorem 11.2.1].

Equation (A.1) can be expressed as

$$ \mathbf{B}\phi = \sum ^{m}_{r=1} \sum ^{n_{r}-1}_{j=0} \begin{bmatrix} b_{1\;j}^{r} \\ \vdots \\ b_{N\;j}^{r}\end{bmatrix} \phi _{r}^{(j)}(0) + \begin{bmatrix} \beta _{1\;j}^{r} \\ \vdots \\ \beta _{N\;j}^{r}\end{bmatrix} \phi _{r}^{(j)}(1) = \sum ^{m}_{r=1} b^{r} \vec{\phi _{r}}(0) + \beta ^{r} \vec{\phi _{r}}(1). $$

(A.6)

Using $(b^{r}:\beta ^{r})$ to represent concatenation of matrices, we can write

$$ \mathbf{B}\phi = \sum ^{m}_{r=1} \begin{bmatrix} b^{r} : \beta ^{r} \end{bmatrix} \begin{bmatrix} \vec{\phi _{r}}(0) \\ \vec{\phi _{r}}(1) \end{bmatrix} = \underbrace{\begin{bmatrix} b^{1} : \beta ^{1} :{} \ldots {}: b^{m} : \beta ^{m} \end{bmatrix}}_{\text{$N\times 2n_{\mathrm{all}}$}} \underbrace{\begin{bmatrix} \vec{\phi _{1}}(0) \\ \vec{\phi _{1}}(1) \\ \vdots \\ \vec{\phi _{m}}(0) \\ \vec{\phi _{m}}(1) \end{bmatrix}}_{\text{$2n_{\mathrm{all}}\times 1$}}. $$

(A.7)

Thus we have two compact ways to write vectors of boundary forms, namely equations (A.6) and (A.7). Next, we extend the notion of complementary vectors of boundary forms from that in [6, §11.2] to the present setting.

Definition 35

Let $N\in \mathbb{Z}$, with $0 \leqslant N\leqslant 2n_{\mathrm{all}}$. If $\mathbf{B} = (B_{1}, \ldots , B_{N})$ is any vector of boundary forms with $\mathrm{rank}(\mathbf{B}) = N$, and $\mathbf{B}_{\mathrm{c}}= (B_{N+1}, \ldots , B_{2n_{\mathrm{all}}})$ is a vector of forms with $\mathrm{rank}(\mathbf{B}_{\mathrm{c}}) = 2n_{\mathrm{all}}-N$ such that the rank of $(B_{1}, \ldots , B_{2n_{\mathrm{all}}})$ is $2n_{\mathrm{all}}$, then $\mathbf{B}$ and $\mathbf{B}_{\mathrm{c}}$ are said to be complementary vectors of boundary forms.

Note that extending $(B_{1}, \ldots , B_{N})$ to $(B_{1}, \ldots , B_{2n_{\mathrm{all}}})$ is equivalent to embedding the matrices $b^{r}, \beta ^{r}$ in a $2n_{\mathrm{all}}\times 2n_{\mathrm{all}}$ non-singular matrix. That is

$$\begin{aligned} \begin{bmatrix} \mathbf{B}\phi \\ \mathbf{B}_{\mathrm{c}}\phi \end{bmatrix} &= \sum ^{m}_{r=1} \begin{bmatrix} b^{r} & \beta ^{r} \\ b^{r}_{\mathrm{c}}& \beta ^{r}_{\mathrm{c}} \end{bmatrix} \begin{bmatrix} \vec{\phi _{r}}(0) \\ \vec{\phi _{r}}(1) \end{bmatrix} \\ &= \underbrace{ \begin{bmatrix} b^{1} & \beta ^{1} & b^{2} & \beta ^{2} & \cdots & b^{m} & \beta ^{m} \\ b^{1}_{\mathrm{c}}& \beta ^{1}_{\mathrm{c}}& b^{2}_{\mathrm{c}}& \beta ^{2}_{\mathrm{c}}& \cdots & b^{m}_{\mathrm{c}}& \beta ^{m}_{\mathrm{c}}\end{bmatrix} }_{\text{$2n_{\mathrm{all}}\times 2n_{\mathrm{all}}$}} \underbrace{\begin{bmatrix} \vec{\phi _{1}}(0) \\ \vec{\phi _{1}}(1) \\ \vec{\phi _{2}}(0) \\ \vec{\phi _{2}}(1) \\ \vdots \\ \vec{\phi _{m}}(0) \\ \vec{\phi _{m}}(1) \end{bmatrix}}_{\text{$2n_{\mathrm{all}}\times 1$}} =: H \begin{bmatrix} \vec{\phi _{1}}(0) \\ \vec{\phi _{1}}(1) \\ \vec{\phi _{2}}(0) \\ \vec{\phi _{2}}(1) \\ \vdots \\ \vec{\phi _{m}}(0) \\ \vec{\phi _{m}}(1) \end{bmatrix}, \end{aligned}$$

(A.8)

where $\mathrm{rank}(H) = 2n_{\mathrm{all}}$ and $b^{r}_{\mathrm{c}}, \beta ^{r}_{\mathrm{c}}\in \mathbb{C}^{(2n_{ \mathrm{all}}-N)\times n_{r}}$. We will use equation (A.8) to express Green’s formula as a combination of vector boundary forms $\mathbf{B}$ and $\mathbf{B}_{\mathrm{c}}$ in Theorem 37. Its proof also requires the following lemma, whose proof is immediate from the fact that, in the sesquilinear dot product, the conjugate transpose of a matrix is the adjoint of the matrix.

Lemma 36

Let $\sigma$ be the sesquilinear form associated with a nonsingular matrix $\Sigma$;

$$ \sigma (f,g) = \Sigma f \odot g. $$

For each nonsingular matrix $F$, there exists a unique nonsingular matrix $G$ such that $\sigma (f,g) = Ff \odot Gg$ for all $f,g$. Moreover, $G=(\Sigma F^{-1})^{\dagger }$, in which ^† represents the (Hermitian) conjugate transpose.

Theorem 37

Interface boundary form formula

Given any vector of forms $\mathbf{B}$ of rank $N$, and any complementary vector of forms $\mathbf{B}_{\mathrm{c}}$, there exist unique vectors of forms $\mathbf{B}^{\star }_{\mathrm{c}}$ and $\mathbf{B}^{\star }$ of rank $N$ and $2n_{\mathrm{all}}-N$, respectively, such that

$$ \sum _{r=1}^{m} [\phi \psi ]_{r}(1) - [\phi \psi ]_{r}(0) = \mathbf{B}\phi \odot \mathbf{B}^{\star }_{\mathrm{c}}\psi + \mathbf{B}_{\mathrm{c}}\phi \odot \mathbf{B}^{\star }\psi . $$

(A.9)

Moreover, the complementary adjoint boundary coefficient matrices $b_{\mathrm{c}\;}^{r\;\star },\beta _{\mathrm{c}\;}^{r\;\star }\in \mathbb{C}^{N\times n_{r}}$ and the adjoint boundary coefficient matrices $b_{\;}^{r\;\star },\beta _{\;}^{r\;\star }\in \mathbb{C}^{(2n_{ \mathrm{all}}-N)\times n_{r}}$ are given by

$$ \begin{bmatrix} b_{\mathrm{c}\;}^{1\;\star } & \beta _{\mathrm{c}\;}^{1\;\star } & b_{ \mathrm{c}\;}^{2\;\star } & \beta _{\mathrm{c}\;}^{2\;\star } & \cdots & b_{\mathrm{c}\;}^{m\;\star } & \beta _{\mathrm{c}\;}^{m\; \star } \\ b_{\;}^{1\;\star } & \beta _{\;}^{1\;\star } & b_{\;}^{2\;\star } & \beta _{\;}^{2\;\star } & \cdots & b_{\;}^{m\;\star } & \beta _{\;}^{m \;\star } \end{bmatrix} = \left (SH^{-1}\right )^{\dagger }, $$

(A.10)

for $H$ as defined in equation (A.8) and $S$ as defined in equation (A.5).

Proof

Let $H$ be as defined by equation (A.8). By Lemma 36, there exists a unique $2n_{\mathrm{all}}\times 2n_{\mathrm{all}}$ nonsingular matrix $J$ such that

$$ \mathscr {S} \left ( \begin{bmatrix} \vec{\phi _{1}}(0) \\ \vec{\phi _{1}}(1) \\ \vdots \\ \vec{\phi _{m}}(0) \\ \vec{\phi _{m}}(1) \end{bmatrix} , \begin{bmatrix} \vec{\psi _{1}}(0) \\ \vec{\psi _{1}}(1) \\ \vdots \\ \vec{\psi _{m}}(0) \\ \vec{\psi _{m}}(1) \end{bmatrix} \right ) = H \begin{bmatrix} \vec{\phi _{1}}(0) \\ \vec{\phi _{1}}(1) \\ \vdots \\ \vec{\phi _{m}}(0) \\ \vec{\phi _{m}}(1) \end{bmatrix} \odot J \begin{bmatrix} \vec{\psi _{1}}(0) \\ \vec{\psi _{1}}(1) \\ \vdots \\ \vec{\psi _{m}}(0) \\ \vec{\psi _{m}}(1) \end{bmatrix} . $$

(A.11)

Moreover, $J=\left (SH^{-1}\right )^{\dagger }$. We define $\mathbf{B}^{\star }, \mathbf{B}^{\star }_{\mathrm{c}}$ by

$$ \begin{bmatrix} \mathbf{B}^{\star }_{\mathrm{c}}\psi \\ \mathbf{B}^{\star } \psi \end{bmatrix} = J \begin{bmatrix} \vec{\psi _{1}}(0) \\ \vec{\psi _{1}}(1) \\ \vdots \\ \vec{\psi _{m}}(0) \\ \vec{\psi _{m}}(1) \end{bmatrix} , $$

(A.12)

from which equation (A.10) follows by the same argument as that used to derive equation (A.8). Equations (A.5), (A.11), (A.8), and (A.12) yield

$$ \sum _{r=1}^{m} [\phi \psi ]_{r}(1) - [\phi \psi ]_{r}(0) = \begin{bmatrix} \mathbf{B}\phi \\ \mathbf{B}_{\mathrm{c}}\phi \end{bmatrix} \odot \begin{bmatrix} \mathbf{B}^{\star }_{\mathrm{c}}\psi \\ \mathbf{B}^{\star } \psi \end{bmatrix} = \mathbf{B}\phi \odot \mathbf{B}^{\star }_{\mathrm{c}}\psi + \mathbf{B}_{\mathrm{c}}\phi \odot \mathbf{B}^{\star }\psi . $$

By unicity of $J$, no other definition of $\mathbf{B}^{\star }_{\mathrm{c}}$ and $\mathbf{B}^{\star }$ satisfies equation (A.9). □

The interface boundary form formula theorem 37 allows us to define adjoint boundary conditions, whence we get the adjoint boundary value problem.

Definition 38

Suppose $\mathbf{B} = (B_{1}, \ldots , B_{N})$ is a vector of forms with $\mathrm{rank}(\mathbf{B}) = N$. Suppose $\mathbf{B}^{\star }$ is any vector of forms with $\mathrm{rank}(\mathbf{B}^{\star }) = 2n_{\mathrm{all}}-N$, determined as in Theorem 37. Then $\mathbf{B}^{\star }$ is an adjoint boundary form and the equation $\mathbf{B}^{\star }\psi = \mathbf{0}$ is an adjoint boundary condition to boundary condition $\mathbf{B}\phi = \mathbf{0}$ in function space $\Phi$.

Definition 39

Suppose $\mathbf{B} = (B_{1}, \ldots , B_{N})$ is a vector of forms with $\mathrm{rank}(\mathbf{B}) = N$ and $\mathbf{B}^{\star }$ is an adjoint boundary form. Then the problem of solving

$$ L\phi = 0, \qquad \phi \in \Phi _{\mathbf{B}} $$

is called a boundary value problem of rank $N$. The problem of solving

$$ L^{\star }\psi = 0, \qquad \psi \in \Phi _{\mathbf{B}^{\star }} $$

is the adjoint boundary value problem.

In Theorem 37, the adjoint boundary conditions $\mathbf{B}^{\star }$ are determined uniquely not by $\mathbf{B}$ but by the pair $(\mathbf{B},\mathbf{B}_{\mathrm{c}})$. However, the function space $\Phi _{\mathbf{B}^{\star }}$ is determined uniquely from $\mathbf{B}$. Indeed, because $\mathbf{B}_{\mathrm{c}}$ is (or, more precisely, the complementary boundary coefficient matrices $b_{\mathrm{c}}^{r}, \beta _{\mathrm{c}}^{r}$ from which it is composed are) determined by $\mathbf{B}$ uniquely up to a rank $(2n_{\mathrm{all}}-N)\times (2n_{\mathrm{all}}-N)$ perturbation, so also is $\mathbf{B}^{\star }$ (or the boundary coefficient matrices from which it is composed).

Finally, we justify that the term “adjoint boundary condition” is faithfully applied in the classical (Lagrange) sense.

Corollary 40

If $\phi \in \Phi _{\mathbf{B}}$ and $\psi \in \Phi _{\mathbf{B}^{\star }}$, then $\langle L\phi , \psi \rangle = \langle \phi , L^{\star }\psi \rangle$.

Proof

We apply the boundary form formula theorem 37 and the definitions of $\Phi _{\mathbf{B}}$ and $\Phi _{\mathbf{B}^{\star }}$ to obtain

$$ \langle L\phi , \psi \rangle - \langle \phi , L^{\star }\psi \rangle = \mathbf{B}\phi \odot \mathbf{B}^{\star }_{\mathrm{c}}\psi + \mathbf{B}_{\mathrm{c}}\phi \odot \mathbf{B}^{\star }\psi = \mathbf{0} \odot \mathbf{B}^{\star }_{\mathrm{c}}\psi + \mathbf{B}_{\mathrm{c}}\phi \odot \mathbf{0} = 0. $$

 □

1.4 A.4 Checking Adjointness

Theorem 41

Suppose that vector boundary form $\mathbf{B}$ has full rank boundary coefficient matrices $b^{r},\beta ^{r}\in \mathbb{C}^{N\times n_{r}}$, and vector boundary form $\mathbf{Z}$ has full rank boundary coefficient matrices $z^{r},\zeta ^{r}\in \mathbb{C}^{(2n_{\mathrm{all}}-N)\times n_{r}}$. The boundary condition $\mathbf{Z}$ is adjoint to $\mathbf{B}$ if and only if

$$ \sum ^{m}_{r=1} b^{r} (F^{r})^{-1}(0) z_{\;}^{r\;\dagger } = \sum ^{m}_{r=1} \beta ^{r} (F^{r})^{-1}(1) \zeta _{\;}^{r\;\dagger }, $$

(A.13)

where $F^{r}(x)$ is the $n_{r}\times n_{r}$ matrix defined by equation (A.3).

Proof of Theorem 41: “only if”

Suppose that $\mathbf{B}$ and $\mathbf{Z}$ are adjoint. By Definition 38, $\mathbf{Z}$ is determined as in Theorem 37. Thus, in determining $\mathbf{Z}$, there exist vectors of forms $\mathbf{B}_{\mathrm{c}}, \mathbf{B}_{\mathrm{c}}^{\star }$ of rank $2n_{\mathrm{all}}-N$ and $N$ respectively, such that Theorem 37 holds. As such, there exist full rank matrices $b_{\mathrm{c}}^{r}, \beta _{\mathrm{c}}^{r} \in \mathbb{C}^{(2n_{ \mathrm{all}}-N)\times n_{r}}$, and $b_{\mathrm{c}\;}^{r\;\star },\beta _{\mathrm{c}\;}^{r\;\star }\in \mathbb{C}^{N\times n_{r}}$ such that

$${\mathbf{B}}_{\mathrm{c}}\varphi =\sum _{r=1}^m{b}_{\mathrm{c}}^r{\overrightarrow{\varphi }}_r(0)+{\beta }_{\mathrm{c}}^r{\overrightarrow{\varphi }}_r(1),\mathrm{rank}\left[\begin{array}{c}{b}_{\mathrm{c}}^1:{\beta }_{\mathrm{c}}^1:\dots :b_{\mathrm{c}}^m:{\beta }_{\mathrm{c}}^m\end{array}\right]=2n_{\mathrm{all}}-N,$$

(A.14)

$${\mathbf{B}}_{\mathrm{c}}^{\star }\psi =\sum _{r=1}^m{b}_{\mathrm{c}}^{r\star }{\overrightarrow{\psi }}_r(0)+{\beta }_{\mathrm{c}}^{r\star }{\overrightarrow{\psi }}_r(1),\mathrm{rank}\left[\begin{array}{c}{b}_{\mathrm{c}}^{1\star }:{\beta }_{\mathrm{c}}^{1\star }:\dots :b_{\mathrm{c}}^{m\star }:{\beta }_{\mathrm{c}}^{m\star }\end{array}\right]=N.$$

(A.15)

Moreover, it is reasonable to denote $\mathbf{Z}$ by $\mathbf{B}^{\star }$ and $z^{r},\zeta ^{r}$ by $b_{\;}^{r\;\star },\beta _{\;}^{r\;\star }$, respectively.

By equations (A.9), (A.8), (A.15), (A.14), and (A.10), then distributing the inner products over the sums,

$$\begin{array}{c}\sum _{r=1}^m{[\varphi \psi ]}_r(1)-{[\varphi \psi ]}_r(0)=\sum _{r=1}^m\sum _{i=1}^m((b^r\overrightarrow{{\varphi }_r}(0)+{\beta }^r\overrightarrow{{\varphi }_r}(1))\odot (b_{\mathrm{c}}^{i\star }{\overrightarrow{\psi }}_i(0)+{\beta }_{\mathrm{c}}^{i\star }{\overrightarrow{\psi }}_i(1))\hfill \\ \hfill +(b_{\mathrm{c}}^r{\overrightarrow{\varphi }}_r(0)+{\beta }_{\mathrm{c}}^r{\overrightarrow{\varphi }}_r(1))\odot (b_{}^{i\star }{\overrightarrow{\psi }}_i(0)+{\beta }_{}^{i\star }{\overrightarrow{\psi }}_i(1))).\end{array}$$

By additivity of inner product and the fact that the conjugate transpose is the adjoint with respect to the sesquilinear dot product, we write the above as

$$\begin{array}{c}\sum _{r=1}^m\sum _{i=1}^m({\beta }_{\mathrm{c}}^{i\star †}{\beta }^r+{\beta }_{}^{i\star †}{\beta }_{\mathrm{c}}^r)\overrightarrow{{\varphi }_r}(1)\odot \overrightarrow{{\psi }_i}(1)+(b_{\mathrm{c}}^{i\star †}{\beta }^r+b_{}^{i\star †}{\beta }_{\mathrm{c}}^r)\overrightarrow{{\varphi }_r}(1)\odot \overrightarrow{{\psi }_i}(0)\hfill \\ \hfill +({\beta }_{\mathrm{c}}^{i\star †}{b}^r+{\beta }_{}^{i\star †}{b}_{\mathrm{c}}^r)\overrightarrow{{\varphi }_r}(0)\odot \overrightarrow{{\psi }_i}(1)+(b_{\mathrm{c}}^{i\star †}{b}^r+b_{}^{i\star †}{b}_{\mathrm{c}}^r)\overrightarrow{{\varphi }_r}(0)\odot \overrightarrow{{\psi }_i}(0).\end{array}$$

(A.16)

From Green’s formula (A.4), we have

$$ \sum ^{m}_{r=1} [\phi \psi ]_{r}(1) - [\phi \psi ]_{r}(0) = \sum _{r=1}^{m} F^{r}(1) \vec{\phi _{r}}(1) \odot \vec{\psi _{r}}(1) - F^{r}(0) \vec{\phi _{r}}(0) \odot \vec{\psi _{r}}(0). $$

(A.17)

Equating coefficients of each of the inner products in equations (A.16) and (A.17) reveals that

$$ \begin{aligned} \left (\beta _{\mathrm{c}\;}^{i\;\star \;\dagger } \beta ^{r} + \beta _{ \;}^{i\;\star \;\dagger } \beta _{\mathrm{c}}^{r}\right ) &= \bigg\{ \hspace{-0.2em} \textstyle\begin{array}{l} F^{r}(1), \\ 0, \end{array}\displaystyle & \left (b_{\mathrm{c}\;}^{i\;\star \;\dagger } b^{r} + b_{\;}^{i\; \star \;\dagger } b_{\mathrm{c}}^{r}\right ) &= \bigg\{ \hspace{-0.2em} \textstyle\begin{array}{l} -F^{r}(0), \\ 0, \end{array}\displaystyle & &\textstyle\begin{array}{l} \mbox{if }i=r, \\ \mbox{otherwise,}\end{array}\displaystyle \\ \left (b_{\mathrm{c}\;}^{i\;\star \;\dagger } \beta ^{r} + b_{\;}^{i\; \star \;\dagger } \beta _{\mathrm{c}}^{r} \right ) &= \hphantom{\bigg\{ \hspace{-0.2em}} \textstyle\begin{array}{l} 0,\end{array}\displaystyle & \left (\beta _{\mathrm{c}\;}^{i\;\star \;\dagger } b^{r} + \beta _{\;}^{i \;\star \;\dagger } b_{\mathrm{c}}^{r}\right ) &= \hphantom{\bigg\{ \hspace{-0.2em}} \textstyle\begin{array}{l} 0,\end{array}\displaystyle & &\textstyle\begin{array}{l} \mbox{for all } i,r.\end{array}\displaystyle \end{aligned} $$

(A.18)

Note that not all of the 0 matrices above are the same, as each of the above equations relates matrices in $\mathbb{C}^{n_{i}\times n_{r}}$; the nonzero cases are square matrices in $\mathbb{C}^{n_{r}\times n_{r}}$.

Interlacing the $i=r$ cases of the upper two of identities (A.18) as blocks on matrix diagonals, and filling out the matrices with appropriate sized zero blocks elsewhere, we obtain

$$\begin{array}{c}\left[\begin{array}{ccccccc}-F^1(0)& 0& 0& \cdots & 0& 0& 0\\ 0& F^1(1)& 0& \cdots & 0& 0& 0\\ 0& 0& -F^2(0)& \cdots & 0& 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & F^{m-1}(1)& 0& 0\\ 0& 0& 0& \cdots & 0& -F^m(0)& 0\\ 0& 0& 0& \cdots & 0& 0& F^m(1)\end{array}\right]\hfill \\ \hfill =\left[\begin{array}{ccccc}{b}_{\mathrm{c}}^{1\star †}{b}^1+b_{}^{1\star †}{b}_{\mathrm{c}}^1& 0& \cdots & 0& 0\\ 0& {\beta }_{\mathrm{c}}^{1\star †}{\beta }^1+{\beta }_{}^{1\star †}{\beta }_{\mathrm{c}}^1& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & b_{\mathrm{c}}^{m\star †}{b}^m+b_{}^{m\star †}{b}_{\mathrm{c}}^m& 0\\ 0& 0& \cdots & 0& {\beta }_{\mathrm{c}}^{m\star †}{\beta }^m+{\beta }_{}^{m\star †}{\beta }_{\mathrm{c}}^m\end{array}\right].\end{array}$$

(A.19)

Since the boundary matrices $F^{r}$ are each nonsingular, the block diagonal matrix on the left of equation (A.19) must also be invertible. Premultiplying on both sides by the inverse yields the following expression for the identity matrix

$$\begin{array}{c}\left[\begin{array}{ccc}-{(F^1)}^{-1}(0)(b_{\mathrm{c}}^{1\star †}{b}^1+b_{}^{1\star †}{b}_{\mathrm{c}}^1)& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & {(F^m)}^{-1}(1)({\beta }_{\mathrm{c}}^{m\star †}{\beta }^m+{\beta }_{}^{m\star †}{\beta }_{\mathrm{c}}^m)\end{array}\right]\hfill \\ \hfill =\left[\begin{array}{cc}-{(F^1)}^{-1}(0)b_{\mathrm{c}}^{1\star †}& -{(F^1)}^{-1}(0)b_{}^{1\star †}\\ {(F^1)}^{-1}(1){\beta }_{\mathrm{c}}^{1\star †}& {(F^1)}^{-1}(1){\beta }_{}^{1\star †}\\ ⋮& ⋮\\ -{(F^m)}^{-1}(0)b_{\mathrm{c}}^{m\star †}& -{(F^m)}^{-1}(0)b_{}^{m\star †}\\ {(F^m)}^{-1}(1){\beta }_{\mathrm{c}}^{m\star †}& {(F^m)}^{-1}(1){\beta }_{}^{m\star †}\end{array}\right]\left[\begin{array}{ccccc}{b}^1& {\beta }^1& \cdots & b^m& {\beta }^m\\ b_{\mathrm{c}}^1& {\beta }_{\mathrm{c}}^1& \cdots & b_{\mathrm{c}}^m& {\beta }_{\mathrm{c}}^m\end{array}\right];\end{array}$$

(A.20)

the equation is justified by using the rest of identities (A.18) to show that all blocks off the diagonal are zero blocks of appropriate dimension. Since the two matrices in the product on the right of equation (A.20) are square and full rank, they are inverse to each other, and so we have, for identity matrices $\mathrm{Id}$,

$$\begin{array}{c}\left[\begin{array}{cc}{\mathrm{Id}}_{N\times N}& 0_{N\times (2n_{\mathrm{all}}-N)}\\ 0_{(2n_{\mathrm{all}}-N)\times N}& {\mathrm{Id}}_{(2n_{\mathrm{all}}-N)\times (2n_{\mathrm{all}}-N)}\end{array}\right]\hfill \\ \hfill =\left[\begin{array}{ccccc}{b}^1& {\beta }^1& \cdots & b^m& {\beta }^m\\ b_{\mathrm{c}}^1& {\beta }_{\mathrm{c}}^1& \cdots & b_{\mathrm{c}}^m& {\beta }_{\mathrm{c}}^m\end{array}\right]\left[\begin{array}{cc}-{(F^1)}^{-1}(0)b_{\mathrm{c}}^{1\star †}& -{(F^1)}^{-1}(0)b_{}^{1\star †}\\ {(F^1)}^{-1}(1){\beta }_{\mathrm{c}}^{1\star †}& {(F^1)}^{-1}(1){\beta }_{}^{1\star †}\\ ⋮& ⋮\\ -{(F^m)}^{-1}(0)b_{\mathrm{c}}^{m\star †}& -{(F^m)}^{-1}(0)b_{}^{m\star †}\\ {(F^m)}^{-1}(1){\beta }_{\mathrm{c}}^{m\star †}& {(F^m)}^{-1}(1){\beta }_{}^{m\star †}\end{array}\right].\end{array}$$

(A.21)

The top right block (that is, the block which contains neither complementary boundary coefficient matrices nor adjoint complementary boundary coefficient matrices) yields

$$\begin{array}{c}-b^1{(F^1)}^{-1}(0)b_{}^{1\star †}+{\beta }^1{(F^1)}^{-1}(1){\beta }_{}^{1\star †}+\cdots \hfill \\ \hfill -b^m{(F^m)}^{-1}(0)b_{}^{m\star †}+{\beta }^m{(F^m)}^{-1}(1){\beta }_{}^{m\star †}=0_{N\times (2n_{\mathrm{all}}-N)},\end{array}$$

from which it follows

$$ \sum ^{m}_{r=1} b^{r} (F^{r})^{-1}(0) b_{\;}^{r\;\star \;\dagger } = \sum ^{m}_{r=1} \beta ^{r} (F^{r})^{-1}(1) \beta _{\;}^{r\;\star \; \dagger }. $$

 □

Proof of Theorem 41: “if”

Suppose that $\mathbf{Z}$ is a vector of boundary forms with boundary coefficient matrices $z^{r},\zeta ^{r}$

$$ \mathbf{Z} \psi = \sum ^{m}_{r=1} z^{r} \vec{\psi }_{r}(0) + \zeta ^{r} \vec{\psi }_{r}(1), $$

and

$$ \mathrm{rank} \begin{bmatrix} z^{1} : \zeta ^{1} : {}\ldots {} : z^{m} : \zeta ^{m} \end{bmatrix} = 2n_{\mathrm{all}}-N, $$

for which equation (A.13) holds. Then

$$ \mathrm{rank} \begin{bmatrix} z_{\;}^{1\;\dagger } \\ \zeta _{\;}^{1\;\dagger } \\ \vdots \\ z_{\;}^{m\;\dagger } \\ \zeta _{\;}^{m\;\dagger } \end{bmatrix} = 2n_{\mathrm{all}}-N $$

and equation (A.13) implies

$$ \begin{bmatrix} b^{1} : \beta ^{1} : {}\ldots {} : b^{m} : \beta ^{m} \end{bmatrix} \begin{bmatrix} -F^{1}(0)^{-1} z_{\;}^{1\;\dagger } \\ F^{1}(1)^{-1} \zeta _{\;}^{1\;\dagger } \\ \vdots \\ -F^{m}(0)^{-1} z_{\;}^{m\;\dagger } \\ F^{m}(1)^{-1} \zeta _{\;}^{m\;\dagger } \end{bmatrix} = 0_{N\times (2n_{\mathrm{all}}-N)}. $$

(A.22)

Since $F^{r}(0), F^{r}(1)$ are non-singular for each $r$, the $2n_{\mathrm{all}}-N$ columns of the matrix

$$ \mathscr {H} := \begin{bmatrix} -F^{1}(0)^{-1} z_{\;}^{1\;\dagger } \\ F^{1}(1)^{-1} \zeta _{\;}^{1\;\dagger } \\ \vdots \\ -F^{m}(0)^{-1} z_{\;}^{m\;\dagger } \\ F^{m}(1)^{-1} \zeta _{\;}^{m\;\dagger } \end{bmatrix} $$

span the solution space of the system (A.22), and $\mathrm{rank}(\mathscr {H}) = 2n_{\mathrm{all}}- N$.

Suppose that $\mathbf{B}^{\star }$ is adjoint to $\mathbf{B}$ and has boundary coefficient matrices $b_{\;}^{r\;\star },\beta _{\;}^{r\;\star }$. Following the argument in the “only if” proof, equation (A.21) holds, and both matrices on the right are full rank. Therefore, the $2n_{\mathrm{all}}-N$ columns of the rank $2n_{\mathrm{all}}- N$ matrix

$$ \mathscr{H} := \begin{bmatrix} -F^{1}(0)^{-1} b_{\;}^{1\;\star \;\dagger } \\ F^{1}(1)^{-1} \beta _{\;}^{1\;\star \;\dagger } \\ \vdots \\ -F^{m}(0)^{-1} b_{\;}^{m\;\dagger } \\ F^{m}(1)^{-1} \beta _{\;}^{m\;\star \;\dagger } \end{bmatrix} $$

also span the solution space of equation (A.22). Therefore, there is a nonsingular matrix $A$ of dimension $(2n_{\mathrm{all}}- N)$ for which $\mathscr {H}=\mathscr{H}A$. But then

$$ F^{r}(0)^{-1} z_{\;}^{r\;\dagger } = F^{r}(0)^{-1} b_{\;}^{r\;\star \; \dagger } A, \qquad \qquad F^{1}(1)^{-1} \zeta _{\;}^{r\;\dagger } = F^{r}(1)^{-1} \beta _{\;}^{r\;\star \;\dagger } A. $$

Because the inverse of $F^{r}$ is full rank,

$$ z_{\;}^{r\;\dagger } = b_{\;}^{r\;\star \;\dagger } A, \qquad \qquad \zeta _{ \;}^{r\;\dagger } = \beta _{\;}^{r\;\star \;\dagger } A. $$

That is, the boundary coefficients matrices of $\mathbf{Z}$ differ from those of $\mathbf{B}^{\star }$ only by a rank $2n_{\mathrm{all}}- N$ perturbation; $\mathbf{Z}$ is also adjoint to $\mathbf{B}$ and $\Phi _{\mathbf{Z}}=\Phi _{\mathbf{B}^{\star }}$. □