MATHEMATICS

V. F. ZHDANOVICH

SOLUTION BY THE FOURIER METHOD OF NON-SELF-ADJOINT MIXED PROBLEMS FOR HYPERBOLIC SYSTEMS IN THE PLANE

(Presented by Academician I. G. Petrovskii, 10 XII 1956)

We shall solve the mixed problem for a hyperbolic system in the narrow sense ({}^{1})

[
\frac{\partial}{\partial t}u(x,t)=A(x)\frac{\partial}{\partial x}u(x,t)+B(x)u(x,t)
\tag{1}
]

((0\le x\le l,\ 0\le t\le T<+\infty)) with boundary and initial conditions

[
\begin{aligned}
\text{a)}\quad &M\frac{\partial}{\partial t}u(0,t)+Nu(0,t)+P\frac{\partial}{\partial t}u(l,t)+Qu(l,t)=0;\
\text{b)}\quad &u(x,0)=f(x),
\end{aligned}
\tag{2}
]

where (u(x,t)) is an (n)-dimensional vector function with complex coordinates. The matrix (A(x)) is twice, and (B(x)) once, continuously differentiable for (x\in[0,l]); moreover, the eigenvalues of the matrix (A(x)) do not vanish on the segment ([0,l]). The matrices (M,N,P,Q) are complex, and

[
\operatorname{rang}\left|\begin{matrix} M & P\ N & Q\end{matrix}\right|=n+q,
\qquad
\operatorname{rang}|M,P|=q
\quad (0\le q\le n).
\tag{3}
]

Let (D_2(0,l)) be the Banach space of classes of measurable functions (f(x)) ((0\le x\le l)), equivalent with respect to the norm
[
|f(x)|_{D_2(0,l)}
=
\left(\int_0^l |f(x)|^2\,dx+|Mf(0)+Pf(l)|^2\right)^{1/2},
]
and suppose that in condition (2b) (f(x)\in D_2(0,l)).

Put in the boundary-value problem (1), (2a)
[
u(x,t)=y(x)e^{\lambda t}.
]
To find (y(x)) ((0\le x\le l)) and (\lambda) we obtain the parametric problem

[
\begin{aligned}
\text{a)}\quad &A(x)y'(x)+B(x)y(x)=\lambda y(x);\
\text{b)}\quad &(M\lambda+N)y(0)+(P\lambda+Q)y(l)=0.
\end{aligned}
\tag{4}
]

For system (4a) one can construct ({}^{2}) a fundamental matrix (Y(x,\lambda)), analytic in (\lambda) for each (x\in[0,l]) in each of the regions:
1) (\operatorname{Re}\lambda<-\gamma), 2) (|\operatorname{Re}\lambda|<\gamma), 3) (\operatorname{Re}\lambda>\gamma), where (\gamma) is a sufficiently large positive number, and having the asymptotic representation

[
Y(x,\lambda)=\left[K(x)+O\left(\frac{1}{\lambda}\right)\right]
\exp\left[\lambda\int_0^x \Lambda(\xi)\,d\xi\right]
\tag{5}
]

uniformly with respect to (x\in[0,l]) as (\lambda\to\infty). Here (\Lambda(x)=[\nu_1(x),\nu_2(x),\ldots,\nu_n(x)]) is the diagonal form of the matrix (A^{-1}(x)):
[
\Lambda(x)=K^{-1}(x)A^{-1}(x)K(x),
]
and (K(x)) may be chosen so that the functions (\nu_i(x)) ((i=1,2,\ldots,n;\ 0\le x\le l)) will be continuous and will be arranged in decreasing order—

order:
(\nu_1(x)>\nu_2(x)>\cdots>\nu_m(x)>0>\nu_{m+1}(x)>\cdots>\nu_n(x)). To find the eigenvalues of problem (4) we form the characteristic determinant
(\Delta(\lambda)=\det[(M\lambda+N)Y(0,\lambda)+(P\lambda+Q)Y(0,l)]) and expand it, using formula (5), as follows:
[
\Delta(\lambda)=\lambda^q\left[\varphi(\lambda)+\sum_{i=1}^{2^n} b_i(\lambda)e^{\alpha_i\lambda}\right],
]
where (b_i(\lambda)\to0) as (\lambda\to\infty) ((i=1,2,\ldots,2^n));
[
\varphi(\lambda)=\sum_{i=1}^{2^n} a_i e^{\alpha_i\lambda}
]
is a certain Dirichlet polynomial, and the (\alpha_i) ((i=1,2,\ldots,2^n)) are the numbers arranged in decreasing order:
[
0,\quad \int_0^l \nu_i(\xi)\,d\xi
]
[
(i=1,2,\ldots,2^n),
]
the sums of these numbers taken two at a time, three at a time, and so on up to (n); in particular,
[
\alpha_1=\int_0^l\sum_{i=1}^m\nu_i(\xi)\,d\xi,\qquad
\alpha_{2^n}=\int_0^l\sum_{i=m+1}^n\nu_i(\xi)\,d\xi .
]

Definition 1. The boundary conditions (4b) (and also (2a)) are called regular if (a_1\ne0,\ a_{2^n}\ne0).

Theorem 1. If the boundary conditions are regular, then problem (4) has a countable set of eigenvalues, and all of them are located in the strip
[
-\gamma<\operatorname{Re}\lambda<\gamma<+\infty .
]

Let ({\lambda_s}) ((s=0,\pm1,\ldots)) be the zeros of the function (\Delta(\lambda)), renumbered in increasing order of their imaginary parts; let ({k_s}) be their multiplicities; and let ({p_s}) be the multiplicities of the eigenvalues (\lambda_s); then, as is known ((^3)), (p_s\le k_s). If (p_s<k_s) for at least one (s), then in the boundary-value problem (1), (2) we shall take the unknown function (u(x,t)) to be a rectangular matrix, and set
[
u(x,t)=\mathscr{Y}{nk}(x)e^{J_k(\lambda)t}\qquad (k=1,2,\ldots),
\tag{6}
]
where (\mathscr{Y}}(x)) ((0\le x\le l)) is an unknown rectangular matrix with (n) rows and (k) columns, and (J_k(\lambda)) is a (k)-dimensional Jordan cell. To find (\mathscr{Y{nk}(x)) ((k=1,2,\ldots)) we obtain the parametric problems
[
\begin{aligned}
\text{a)}\quad & A(x)\mathscr{Y}'}(x)+B(x)\mathscr{Y{nk}(x)=\mathscr{Y}(x)J_k(\lambda);\
\text{b)}\quad & M\mathscr{Y}{nk}(0)J_k(\lambda)+N\mathscr{Y}(0)J_k(\lambda)
+P\mathscr{Y}{nk}(l)J_k(\lambda)+Q\mathscr{Y}(l)=0.
\end{aligned}
\tag{7}
]

Using the theory of eigenfunctions and associated functions ((^{3,4})), we establish the following properties of these problems: 1) all problems (7), for (k=1,2,\ldots), have common eigenvalues, namely the numbers (\lambda_s) ((s=0,\pm1,\ldots)); 2) the number of linearly independent solutions of problem (7), as (k) increases, does not decrease and for (k\ge k_s) is equal to (k_s); 3) these (k_s) solutions can be chosen so that they consist of (p_s) groups of matrices:
[
|0,\ldots,0,y_1^{(i)}(x),y_2^{(i)}(x),\ldots,y_{m_i}^{(i)}(x)|,
]
[
|0,\ldots,0,0,y_1^{(i)}(x),\ldots,y_{m_i-1}^{(i)}(x)|,\ldots,
|0,\ldots,0,0,0,\ldots,0,y_1^{(i)}(x)|
]
((i=1,2,\ldots,p_s)), where (y_j^{(i)}(x)) ((0\le x\le l;\ i=1,2,\ldots,p_s;\ j=1,2,\ldots,m_i)) are (n)-dimensional columns. From the matrices
(\mathscr{Y}{nm_i}^{(s)}(x)=|y_1^{(i)}(x),y_2^{(i)}(x),\ldots,y(x)|)}^{(i)
((i=1,2,\ldots,p_s)), by formula (6) we construct the matrices
[
U_{nm_i}^{(s)}(x,t)=\mathscr{Y}{nm_i}^{(s)}(x)\exp J(\lambda_s)t
\quad (s=0,\pm1,\ldots;\ i=1,2,\ldots,p_s),
]
and these latter, for each (s) ((s=0,\pm1,\ldots)), are combined into one
[
U_{nk_s}^{(s)}(x,t)=\mathscr{Y}{nk_s}^{(s)}(x)e^{I_s t},
\tag{8}
]
where
[
I_s=[J(\lambda_s)]}(\lambda_s),\ J_{m_2}(\lambda_s),\ \ldots,\ J_{m_{p_s}
]
is a quasidiagonal matrix;
[
\mathscr{Y}{nk_s}^{(s)}(x)=|\mathscr{Y}}^{(s)}(x),\ \mathscr{Y{nm_2}^{(s)}(x),\ \ldots,\ \mathscr{Y}(x)|.}}^{(s)
\tag{9}
]

Each column of the matrix (8) is a solution of the boundary-value problem (1), (2a), and therefore, if (f(x)\in \overline{D}_2(0,l)) is represented in the form of the sum of the series

[
f(x)=\sum_{s=-\infty}^{+\infty} y_{nk_s}^{(s)}(x)a_s,
\tag{10}
]

then the formal solution of problem (1), (2) will have the form

[
\sum_{s=-\infty}^{+\infty} y_{nk_s}^{(s)}(x)e^{\lambda_s t}a_s.
\tag{11}
]

Definition 2. The boundary conditions

[
s^{*}v(t)=-R\frac{\partial}{\partial t}v(0,t)+Sv(0,t)-V\frac{\partial}{\partial t}v(l,t)+Wv(l,t)=0
\tag{12}
]

are called adjoint to the boundary conditions (2a) if there are matrices (|H_1,H_2|) and (|G_1,G_2|) such that, for arbitrary (u(x,t)\in C^{(1)}(\Omega)), (v(x,t)\in C^{(1)}(\Omega)) ((\Omega=[0,l]\times[0,T])),

[
v^{}(x,t)A(x)u(x,t)\bigg|_{x=0}^{x=l}
=
\frac{\partial}{\partial t}[Rv(0,t)+Vv(l,t)]^{}[Mu(0,t)+Pu(l,t)]+
]

[
+[H_1v(0,t)+H_2v(l,t)]^{}su(t)+[s^{}v(t)]^{*}[G_1u(0,t)+G_2u(l,t)],
\tag{13}
]

where (su(t)) is the left-hand side of the boundary condition (2a)(^*).

The matrices (R,S,V,W) are found by comparing the coefficients in the identity (13). Without restricting the generality of the results, in the condition (2a) one may assume

[
|M,P,N,Q|=
\left|
\begin{array}{cccc}
M_{qn} & P_{qn} & N_{qn} & Q_{qn}\
0_{n-qn} & 0_{n-qn} & N_{n-qn} & Q_{n-qn}
\end{array}
\right|,
]

where (0_{n-qn}) is a matrix of zeros. Then it is not difficult to prove that

[
|R,V,S,W|=
\left|
\begin{array}{cccc}
R_{qn} & V_{qn} & S_{qn} & W_{qn}\
0_{n-qn} & 0_{n-qn} & S_{n-qn} & W_{n-qn}
\end{array}
\right|.
]

Definition 3. The boundary-value problem for the system

[
-\frac{\partial}{\partial t}v(x,t)
=
-\frac{\partial}{\partial x}\bigl[A^{}(x)v(x,t)\bigr]+B^{}(x)v(x,t)
\tag{14}
]

with the boundary conditions (12) is called adjoint to problem (1), (2a).

Carrying out separation of variables in problem (14), (12) by the formula (v(x,t)=z(x)e^{-\mu t}), we obtain the parametric problem:

[
\begin{aligned}
\text{a)}\quad &-\frac{d}{dx}\bigl[A^{}(x)z(x)\bigr]+B^{}(x)z(x)=\mu z(x);\
\text{b)}\quad &(R\mu+S)z(0)+(V\mu+W)z(l)=0.
\end{aligned}
\tag{15}
]

Theorem 2. If (\lambda_s) is a zero of the function (\Delta(\lambda)) of multiplicity (k_s) and an eigenvalue of problem (4) of multiplicity (p_s), then (\mu_s=\overline{\lambda}_s) will be a zero of the characteristic determinant (\Delta_1(\mu)) of problem (15) of the same multiplicity (k_s) and an eigenvalue of problem (15) of the same multiplicity (p_s).

Let (F_{nk}(x)) and (G_{nm}(x)) ((k,m=1,2,\ldots;\ 0\leq x\leq l)) be two rectangular matrices. Introduce the notation

[
[F_{nk}(x),G_{nm}(x)]=
]

[
=\int_0^l G_{nm}^{}(x)F_{nk}(x)\,dx
+[RG_{nm}(0)+VG_{nm}(l)]^{}[MF_{nk}(0)+PF_{nk}(l)].
]

[
\underline{\qquad}
]

(^) (v^{}u) here and below denotes the scalar product of the vectors (u) and (v).

Theorem 3. If (\mathscr{Y}{n k_s}^{(s)}(x)) ((s=0,\pm 1,\ldots)) is the system of matrices constructed from the eigenfunctions and adjoint functions of problem (4), and (Z(x)) ((r=0,\pm 1,\ldots)) is the same system for problem (15), then for (s\ne r)}^{(r)
[
[\mathscr{Y}{n k_s}^{(s)}(x), Z(x)] = 0 .}^{(r)
]

Theorem 4. If (f(x)\in D_2(0,l)) is represented in the form of the sum of the series (10), then
[
a_s=[f(x), B_s^{*-1} Z_{n k_s}^{(s)}(x)],
]
where
[
B_s=[\mathscr{Y}{n k_s}^{(s)}(x), Z(x)] \quad (s=0,\pm 1,\ldots).}^{(s)
]

To justify the scheme presented, we shall use the problems generated by (4) by the linear differential operator
[
\mathscr{L}y=A(x)y'(x)+B(x)y(x),
]
where (y(x)\in C^{(1)}(0,l)) and
[
MA(0)y'(0)+[MB(0)+N]y(0)+PA(l)y'(l)+[PB(l)+Q]y(l)=0 .
]

Theorem 5. If the boundary conditions are regular, (f(x)) ((0\le x\le l)) belongs to the domain of definition of the operator (\mathscr{L}y), (f_1(x)=\mathscr{L}f(x)) is continuous and satisfies the condition
[
N_{n-q n} f_1(0)+Q_{n-q n} f_1(l)=0,
]
and (f'_1(x)\in \mathscr{L}_2(0,l)), then the series (11), under a certain grouping of terms independent of the choice of the function (f(x)), converges uniformly on (\Omega) to a continuously differentiable function (u(x,t)) satisfying equation (1) and conditions (2).

Introduce the Banach space (M_2(\Omega)) of measurable functions (f(x,t)), ((x,t)\in\Omega), such that for each (t\in[0,T]), (f(x,t)) belongs to (D_2(0,l)) and
[
|f(x,t)|_{D_2(0,l)}