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UDC 517.945.9
GENERAL MIXED PROBLEMS FOR GENERAL TWO-DIMENSIONAL HYPERBOLIC SYSTEMS
Z. O. Mel’nik
The aim of the present work is to present, from a unified point of view, a general theory of mixed problems for two-dimensional hyperbolic systems of arbitrary order, both with smooth and with discontinuous coefficients.
The mixed problem for two-dimensional hyperbolic equations and systems has been studied by various methods in a number of works. The most effective method in the study of this kind of problem has proved to be the method which may naturally be called the method of characteristics [1—3]. The essence of the method consists in the fact that, by integration along the characteristics of the system, the mixed problem is reduced to an equivalent system of Volterra integral equations, solvable by the method of successive approximations. In this process the equations and boundary conditions may contain Volterra additions. The method makes it possible to obtain both classical and various types of generalized solutions. It is equally applicable both to problems in a rectangle and to problems in domains with a curvilinear boundary.
In the present note we set forth a scheme for applying the method of characteristics to the solution of general mixed problems for general two-dimensional hyperbolic systems with smooth and discontinuous coefficients. References to many of the constructions given here may be found in [3]. In this work we are chiefly interested in the solvability conditions for the indicated mixed problems. The method of characteristics makes it possible to formulate such conditions explicitly. Moreover, as will be seen from what follows, this method makes it possible to construct a sufficiently complete theory of general linear (and weakly nonlinear) mixed problems for general two-dimensional hyperbolic systems.
§ 1. A MIXED PROBLEM FOR A HYPERBOLIC SYSTEM WITH SMOOTH COEFFICIENTS
In the rectangle \(\Pi:\ \{0<t\leq T;\ x_1<x<x_2\}\) in the \((x,t)\)-plane, consider the hyperbolic system of differential equations
\[ \sum_{i=0}^{m}\sum_{j=0}^{i} A_{ij}(x,t)\, \frac{\partial^i u}{\partial t^{\,i-j}\partial x^j} =F(x,t). \tag{1} \]
Here \(A_{ij}(x,t)\) are square \(n\times n\) matrices with elements \(a_{ijrs}(x,t)\) \((0\leq i\leq m;\ 0\leq j\leq i;\ 1\leq r,\ s\leq n)\); \(u(x,t)\) and \(F(x,t)\) are, respectively, the unknown and given columns of height \(n\), with elements \(u_i(x,t)\) and \(F_i(x,t)\) \((1\leq i\leq n)\); \(A_{m0}(x,t)\equiv E\) (\(E\) is the \(n\times n\) identity matrix).
Hyperbolicity of system (1) is understood in the sense that in the expansion
\[ \det \sum_{j=0}^{m} A_{mj}(x,t)\lambda^{m-j}\xi^j = \prod_{j=1}^{mn}(\lambda-\lambda_j(x,t)\xi) \]
all the functions \(\lambda_j(x,t)\) are real and distinct for all \((x,t)\in\overline{\Pi}\). Then, without loss of generality, one may assume that
\(\lambda_1(x,t)<\lambda_2(x,t)<\cdots<\lambda_{mn}(x,t)\). Let at each point \((x,t)\in\overline{\Pi}\), among the functions \(\lambda_j(x,t)\), \(m_1\) of them be negative and \(m_2\) positive \((0\le m_1,m_2\le mn;\ m_1+m_2=mn\), if all \(\lambda_i(x,t)\ne0\), and \(m_1+m_2=mn-1\), if one of the functions \(\lambda_i(x,t)\) is equal to zero).
Remark 1. This requirement can be weakened. In fact, what is important is only the number of positive and negative values among the functions \(\lambda_i(x_1,t)\) and \(\lambda_i(x_2,t)\) [2, 3].
For system (1) we impose the initial conditions
\[ \left.\frac{\partial^i u}{\partial t^i}\right|_{t=0} = g_i(x) \qquad (0\le i\le m-1;\quad x_1\le x\le x_2) \tag{2} \]
and the boundary conditions
\[ \sum_{i=0}^{m-1}\sum_{j=0}^{i} \alpha_{ij}^{k}(t) \frac{\partial^i u(x_k,t)}{\partial t^{\,i-j}\partial x^j} = h_k(t) \qquad (0\le t\le T;\quad k=1,2). \tag{3} \]
Here \(g_i(x)\) are given columns of height \(n\) with elements \(g_{ij}(x)\)
\((0\le i\le m-1;\ 1\le j\le n)\); \(h_k(t)\) are given columns of height \(m_k\) with elements \(h_{ki}(t)\)
\((k=1,2;\ 1\le i\le m_k)\); \(\alpha_{ij}^{k}(t)\) are given matrices of height \(m_k\) and width \(n\), with elements \(\alpha_{ijrs}^{k}(t)\)
\((k=1,2;\ 0\le i\le m-1;\ 0\le j\le i;\ 1\le r\le m_k;\ 1\le s\le n)\).
Introduce the notation
\[ \gamma_{ij}^{k}(t) = \sum_{p=0}^{m-1}\sum_{q=1}^{n} \alpha_{m-1,\,piq}^{k}(t)\, c_{pn+q,\,j+(k-1)(mn-m_k)}(x_k,t) \]
\[ (k=1,2;\quad 1\le i,j\le m_k), \]
where \(c_{ij}(x,t)\) \((1\le i,j\le mn)\) are the elements of the matrix \(C(x,t)\), which carries out the transformation
\(C^{-1}(x,t)A(x,t)C(x,t)=-\Lambda(x,t)\), and
\[ A(x,t)= \begin{pmatrix} A_{m1}(x,t) & A_{m2}(x,t) & \cdots & A_{m,m-1}(x,t) & A_{mm}(x,t)\\ -E & 0 & \cdots & 0 & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots\\ 0 & 0 & \cdots & -E & 0 \end{pmatrix}; \]
\[ \Lambda(x,t)= \begin{pmatrix} \lambda_1(x,t) & \cdots & 0\\ \cdots & \cdots & \cdots\\ 0 & \cdots & \lambda_{mn}(x,t) \end{pmatrix}. \]
The conditions
\[ \det\left\|\gamma_{ij}^{k}(t)\right\|_{i,j=1}^{m_k}\ne0 \qquad (k=1,2;\quad 0\le t\le T) \tag{4} \]
are assumed to be satisfied.
In addition, it is assumed that the initial conditions (2) and the boundary conditions (3) satisfy the natural compatibility conditions at the points \((x_1,0)\) and \((x_2,0)\):
\[ \sum_{i=0}^{m-1}\sum_{j=0}^{i} a_{ij}^{k}(0)\, \frac{d^{j} g_{i-j}(x_k)}{d x^{j}} = h_k(0) \qquad (k=1,2). \]
(To these conditions one must add a whole series of analogous conditions ensuring the continuous differentiability of the solution at all points of \(\Pi\) up to some order \(r\). Taken together, all these conditions will be called the compatibility conditions of order \(r\).)
Theorem 1. Suppose the following conditions are satisfied: 1) the matrices \(A_{mj}(x,t)\) \((1 \le j \le m)\) are uniformly continuously differentiable with respect to \(x\) and \(t\) in \(\Pi\) \(r+1\) times \((r \ge 0)\); the matrices \(A_{ij}(x,t)\) and \(F(x,t)\) are uniformly continuously differentiable with respect to \(x\) and \(t\) in \(\Pi\) \(r\) times \((0 \le i \le m-1;\ 0 \le j \le i)\); 2) the columns \(g_i(x)\) are continuously differentiable with respect to \(x\) for \(x_1 \le x \le x_2\) \(m+r-i-1\) times \((0 \le i \le m-1)\); 3) the coefficients and free terms of the boundary conditions (3) for \(0 \le t \le T\) are uniformly continuously differentiable with respect to \(t\) \(r\) times; 4) conditions (4) and the compatibility conditions of order \(r\) are satisfied. Then in \(\Pi\) there exists a unique solution of problem (1)—(3), uniformly continuously differentiable \(r\) times. This solution depends continuously on the free terms of system (1) and the boundary conditions (3), and on the initial functions (2).
Proof. Introduce new functions \(z_i(x,t)\)
\[ \left(1 \le i \le \frac{mn(m+1)}{2}\right) \]
by the substitution
\[ \frac{\partial^{m-1} u_j}{\partial t^{m-i-1}\partial x^{i}} = \sum_{s=1}^{mn} c_{in+j,s}(x,t) z_s \qquad (0 \le i \le m-1;\ 1 \le j \le n), \tag{5} \]
\[ \frac{\partial^{k} u_j}{\partial t^{k-i}\partial x^{i}} = z_{\frac{(m+k+2)(m-k-1)n}{2}+(k+1)(j-1)+i+1} \]
\[ (0 \le k \le m-2;\ 0 \le i \le k;\ 1 \le j \le n). \]
Then, as is easy to see, in view of (4) the original problem is reduced to an equivalent mixed problem for a first-order system with respect to the new unknowns \(z_i(x,t)\). Applying to the latter the methods of [1], we at once verify the validity of the theorem.
Remark 2. Without essential changes in the above reasoning, one may consider the mixed problem in nonrectangular domains, for example in the domains described in [4].
§ 2. MIXED PROBLEM FOR HYPERBOLIC SYSTEMS WITH DISCONTINUOUS COEFFICIENTS
In this section we shall consider the mixed problem for system (1) in the case when the coefficients and free terms of the system and the initial functions (2) have discontinuities of jump type. For simplicity, we assume that these quantities are discontinuous along one straight line
\[ L:\{x=x_0\}\quad (x_1<x_0<x_2), \]
parallel to the time axis. However, it will not be difficult to observe that all the subsequent calculations carry over, without essential changes, to the case of several straight lines or curved lines of discontinuity.
Let the rectangle \(\Pi\) be divided by the line \(L\) into two parts:
\(\Pi^{(1)}:\ \{0<t\leq T;\ x_1<x<x_0\}\) and
\(\Pi^{(2)}:\ \{0<t\leq T;\ x_0<x\leq x_2\}\). In what follows, by \(f^{(i)}(x,t)\) we shall denote the value of the function \(f(x,t)\) for \((x,t)\in\overline{\Pi}^{(i)}\) \((i=1,2)\). Here \(f^{(1)}(x_0,t)\) \(\bigl(f^{(2)}(x_0,t)\bigr)\) will denote the boundary value of the function \(f^{(1)}(x,t)\) \(\bigl(f^{(2)}(x,t)\bigr)\) as \(x\to x_0-0\) \((x_0+0)\).
It is assumed that now the coefficients and free terms of system (1) and the initial functions (2) satisfy conditions 1) and 2) of Theorem 1 in \(\Pi^{(1)}\) and \(\Pi^{(2)}\).
Let, at all points \((x,t)\in\overline{\Pi}^{(i)}\), among the functions \(\lambda_j^{(i)}(x,t)\) there be \(r_1^i\) negative and \(r_2^i\) positive ones \((i=1,2;\ 1\leq j\leq mn)\), and let \(R=r_1^2+r_2^1\).
Remark 3. Here Remark 1 remains in force, i.e., in fact it is only necessary to pay attention to the number of negative and positive values among the functions \(\lambda_j(x_i,t)\) \((i=0,1,2)\).
We now formulate the mixed problem for system (1) as follows: in \(\Pi_1=\Pi^{(1)}\cup\Pi^{(2)}\) it is required to find a solution of the system satisfying the initial conditions (2), the boundary conditions
\[ \sum_{i=0}^{m-1}\sum_{j=0}^{i} a_{ij}^{k}(t) \frac{\partial^i u^{(k)}(x_k,t)}{\partial t^{\,i-j}\partial x^j} = h_k(t) \qquad (k=1,2;\quad 0\leq t\leq T) \tag{6} \]
and the conjugation conditions on the line of discontinuity
\[ \sum_{s=1}^{2}\sum_{i=0}^{m-1}\sum_{j=0}^{i}(-1)^{s+1}k_{ij}^{s}(t) \frac{\partial^i u^{(s)}(x_0,t)}{\partial t^{\,i-j}\partial x^j} = h(t) \qquad (0\leq t\leq T). \tag{7} \]
Here \(a_{ij}^{k}(t)\) are prescribed matrices of width \(n\) and height \(r_k^k\), with elements
\(a_{ijpq}^{k}(t)\) \((0\leq i\leq m-1;\ 0\leq j\leq i;\ 1\leq q\leq n;\ 1\leq p\leq r_k^k;\ k=1,2)\);
\(h_k(t)\) and \(h(t)\) are prescribed columns of heights \(r_k^k\) and \(R\), respectively, with elements \(h_{ki}(t)\) and \(h_j(t)\) \((1\leq i\leq r_k^k;\ 1\leq j\leq R;\ k=1,2)\);
\(k_{ij}^{s}(t)\) are prescribed matrices of height \(R\) and width \(n\), with elements
\(k_{ijpq}^{s}(t)\) \((0\leq i\leq m-1;\ 0\leq j\leq i;\ 1\leq p\leq R;\ 1\leq q\leq n;\ s=1,2)\).
It is assumed that the coefficients and free terms of conditions (6) and (7) satisfy condition 3) of Theorem 1. Moreover, it is assumed that the initial conditions (2) and conditions (6) and (7) are naturally compatible at the points \((x_i,0)\) \((i=0,1,2)\), i.e.,
\[ \sum_{i=0}^{m-1}\sum_{j=0}^{i} a_{ij}^{k}(0) \frac{d^j g_{i-j}^{(k)}(x_k)}{dx^j} = h_k(0) \qquad (k=1,2), \]
\[ \sum_{s=1}^{2}\sum_{i=0}^{m-1}\sum_{j=0}^{i}(-1)^{s+1}k_{ij}^{s}(0) \frac{d^j g_{i-j}^{(s)}(x_s)}{dx^j} = h(0). \]
Just as in the preceding paragraph, to these conditions one must add a further set of analogous conditions which, together with those written out, would constitute compatibility conditions of some order \(r\). In view of their cumbersomeness, we shall not write out these conditions.
Introduce the notation:
\[
\Gamma_{ij}^{s}(t)=\sum_{p=0}^{m-1}\sum_{q=1}^{n}a_{m-1,piq}^{s}(t)c_{pn+q,j+(s-1)(mn-r_s)}^{(s)}(x_s,t)
\]
\[
(1\le i,j\le r_s^s;\quad s=1,2;\quad 0\le t\le T),
\]
\[
K_{ij}^{s}(t)=\sum_{p=0}^{m-1}\sum_{q=1}^{n}k_{m-1,piq}^{s}(t)c_{pn+q,j}^{(s)}(x_0,t)
\]
\[
(1\le i\le R;\quad 1\le j\le mn;\quad s=1,2;\quad 0\le t\le T).
\]
The conditions
\[
\det \left\|\Gamma_{ij}^{s}(t)\right\|_{i,j=1}^{r_s^s}\ne 0 \quad (s=1,2),
\]
\[
\left|
\begin{array}{ccccccc}
K_{1,mn-r_2^1+1}^{1}(t)&\cdots&K_{1,mn}^{1}(t)&K_{11}^{2}(t)&\cdots&K_{1,r_1^2}^{2}(t)\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\
K_{R,mn+r_2^1+1}^{1}(t)&\cdots&K_{R,mn}^{1}(t)&K_{R1}^{2}(t)&\cdots&K_{R,r_1^2}^{2}(t)
\end{array}
\right|\ne 0
\tag{8}
\]
\[
(0\le t\le T).
\]
By a solution of the problem in the case under consideration we shall mean a function defined in \(\Pi_1\) and uniformly continuously differentiable up to some order \(r\) in each \(\Pi^{(i)}\); in other words, all derivatives of order less than or equal to \(r\) are continuously extendable to each \(\overline{\Pi}^{(i)}\), although on the common boundary \(\overline{\Pi}^{(1)}\) and \(\overline{\Pi}^{(2)}\) the values of these extensions may differ. A solution in the indicated sense will be called an \(r\)-smooth generalized solution of the problem under consideration. Obviously, an \(m\)-smooth generalized solution is a classical solution of the problem.
Theorem 2. Let the coefficients and free terms of system (1), the boundary conditions (6), the conjugation conditions (7), and the initial functions (2) satisfy the smoothness conditions indicated above; let conditions (8) and the compatibility conditions of order \(r\) be fulfilled. Then in \(\Pi_1\) there exists a unique \(r\)-smooth generalized solution of system (1), satisfying the initial conditions (2), the boundary conditions (6), and the conjugation conditions (7). This solution depends continuously on the initial functions \(g_i(x)\) \((0\le i\le m-1)\) and the free terms \(F(x,t)\), \(h_k(t)\), and \(h(t)\) \((k=1,2)\).
Proof. After the substitution (5), problem (1), (2), (6), (7) passes into the equivalent mixed problem [5]:
\[
\frac{\partial z_i}{\partial t}
=\lambda_i(x,t)\frac{\partial z_i}{\partial x}
+\sum_{j=1}^{\frac{mn(m+1)}{2}} b_{ij}(x,t)z_j+f_i(x,t)
\quad (1\le i\le mn),
\tag{9}
\]
\[
\frac{\partial z_i}{\partial t}
=\sum_{j=1}^{\frac{mn(m+1)}{2}} b_{ij}(x,t)z_j
\quad \left(mn+1\le i\le \frac{mn(m+1)}{2}\right),
\]
\[
z_i(x,0)=G_i(x)
\quad \left(1\le i\le \frac{mn(m+1)}{2};\quad x_1\le x\le x_2\right),
\tag{10}
\]
\[ \sum_{j=1}^{mn}\beta_{ss}^{j}(t)\Gamma_{ij}^{s}(t)z_{j}^{(s)}(x_s,t) + \sum_{j=mn+1}^{\frac{mn(m+1)}{2}}\Gamma_{ij}^{s}(t)z_{j}^{(s)}(x_s,t) = h_{si}(t) \tag{11} \]
\[ (1\le i\le r_s^s;\quad s=1,2;\quad 0\le t\le T), \]
\[ \sum_{s=1}^{2}(-1)^{s+1} \left\{ \sum_{j=1}^{mn}\beta_{s0}^{j}(t)K_{ij}^{s}(t)z_{j}^{(s)}(x_0,t) + \sum_{j=mn+1}^{\frac{mn(m+1)}{2}}K_{ij}^{s}(t)z_{j}^{(s)}(x_0,t) \right\} = h_i(t) \quad (1\le i\le R;\quad 0\le t\le T), \tag{12} \]
where
\[ \beta_{pq}^{j}(t)= \prod_{s\ne j} \left[\lambda_{j}^{(p)}(x_q,t)-\lambda_{s}^{(p)}(x_q,t)\right]^{-1} \]
\[ (p=1,2;\quad q=0,1,2;\quad 1\le j\le mn). \]
The coefficients and the free terms of equations (9), the boundary conditions (11), the conjugation conditions (12), and the initial functions (10) are expressed in an obvious way in terms of the data of the original problem (in order to save space we do not write them out here). We note only that they possess uniformly continuous derivatives with respect to \(x\) and \(t\) up to order \(r\).
We shall regard the equalities (11) as a system of \(r_s^s\) algebraic equations with respect to the unknowns
\(z_{j+(s-1)(mn-r_s^s)}^{(s)}(x_s,t)\)
\((s=1,2;\;1\le j\le r_s^s)\); in exactly the same way, we shall regard (12) as a system of \(R\) algebraic equations with respect to the unknowns
\(z_{mn-r_2^1+1}^{(1)}(x_0,t),\ldots,z_{mn}^{(1)}(x_0,t)\),
\(z_{1}^{(2)}(x_0,t),\ldots,z_{r_1^2}^{(2)}(x_0,t)\).
In view of (8), these equations are, obviously, uniquely solvable.
Solving the indicated systems, we write the boundary conditions (11) and the conjugation conditions (12) in the form
\[ z_i^{(1)}(x_1,t)= \sum_{j=r_1^1+1}^{\frac{mn(m+1)}{2}} \omega_{ij}^{1}(t)z_j^{(1)}(x_1,t)+H_i^{1}(t) \quad (1\le i\le r_1^1), \]
\[ \tag{13} \]
\[ z_i^{(2)}(x_2,t)= \sum_{j=1}^{mn-r_2^2} \omega_{ij}^{2}(t)z_j^{(2)}(x_2,t) + \sum_{j=mn+1}^{\frac{mn(m+1)}{2}} \omega_{ij}^{2}(t)z_j^{(2)}(x_2,t)+H_i^{2}(t) \quad (mn-r_2^2+1\le i\le mn), \]
\[ z_i^{(s)}(x_0,t)= \sum_{j=1}^{mn-r_2^1} \delta_{ij}^{s}(t)z_j^{(1)}(x_0,t) + \sum_{j=mn+1}^{\frac{mn(m+1)}{2}} \delta_{ij}^{s}(t)z_j^{(1)}(x_0,t)+ \]
\[ +\sum_{j=r_1^2+1}^{\frac{mn(m+1)}{2}}\mu_{ij}^s(t)z_j^{(2)}(x_0,t)+K_i^s(t) \tag{14} \]
\[ (s=1,2;\quad (2-s)(mn-r_2^1+1)+s-1\leq i\leq (2-s)mn+(s-1)r_1^2). \]
The functions \(\omega_{ij}^s(t)\), \(\delta_{ij}^s(t)\), \(\mu_{ij}^s(t)\), \(H_i^s(t)\), and \(K_i^s(t)\) entering here are, obviously, differentiable \(r\) times with respect to \(t\) for \(0\leq t\leq T\).
Let \(x=\varphi_i^j(t,\xi,\tau)\) be the solution, in the direction of decreasing \(t\), of the characteristic equation
\[ \frac{dx}{dt}=-\lambda_i^{(j)}(x,t), \]
passing through an arbitrary point \((\xi,\tau)\in \overline{\Pi}^{(j)}\) \((j=1,2;\ 1\leq i\leq mn)\). Denote by \(\chi_i^{(j)}(\xi,\tau)\) the least value of \(t\) for such a solution [2]. Obviously, \(0\leq \chi_i^{(j)}(\xi,\tau)\leq \tau\). If \(\chi_i^{(j)}(\xi,\tau)>0\) \((j=1,2)\), then \(\varphi_i^{(1)}(\chi_i^{(1)}(\xi,\tau),\xi,\tau)\) is equal either to \(x_1\) or to \(x_0\), and \(\varphi_i^{(2)}(\chi_i^{(2)}(\xi,\tau),\xi,\tau)\) is equal either to \(x_0\) or to \(x_2\). Then the set of points \((x,t)\in\Pi^{(1)}\) is split into three parts:
\[
\Pi_{i,x_1}^{(1)}:\ \{\chi_i^{(1)}(x,t)>0;\ \varphi_i^{(1)}(\chi_i^{(1)}(x,t),x,t)=x_1\};
\]
\[
\Pi_{i,x_0}^{(1)}:\ \{\chi_i^{(1)}(x,t)>0;\ \varphi_i^{(1)}(\chi_i^{(1)}(x,t),x,t)=x_0\};
\]
\[
\Pi_{i,g}^{(1)}:\ \{\chi_i^{(1)}(x,t)=0\};
\]
correspondingly, the set of points \((x,t)\in\Pi^{(2)}\) is split into analogous three parts: \(\Pi_{i,x_0}^{(2)}\), \(\Pi_{i,x_2}^{(2)}\), and \(\Pi_{i,g}^{(2)}\). It is clear that any of the six sets introduced may turn out to be empty.
Integrating equations (3) along the characteristics issuing from an arbitrary point \((x,t)\in \overline{\Pi}^{(s)}\) \((s=1,2)\) (and for \(i\geq mn+1\) the integration is carried out with respect to \(t\) for any \(x\)), we arrive at the system of integro-functional equations:
\[ z_i^{(s)}(x,t)=v_i^{(s)}(x,t)+ \int_{\chi_i^{(s)}(x,t)}^t \sum_{j=1}^{\frac{mn(m+1)}{2}} b_{ij}^{(s)}(\varphi_i^{(s)}(\tau,x,t),\tau)\times \]
\[ \times z_j^{(s)}(\varphi_i^{(s)}(\tau,x,t),\tau)\,d\tau +\tilde f_i^{(s)}(x,t) \quad (1\leq i\leq mn), \tag{15} \]
\[ z_i^{(s)}(x,t)=G_i^{(s)}(x)+ \int_0^t \sum_{j=1}^{\frac{mn(m+1)}{2}} b_{ij}^{(s)}(x,\tau)\times \]
\[ \times z_j^{(s)}(x,\tau)\,d\tau \quad \left(mn+1\leq i\leq \frac{mn(m+1)}{2}\right), \]
where
\[ v_i^{(s)}(x,t)=G_i^{(s)}\bigl(\varphi_i^{(s)}(0,x,t)\bigr), \quad (x,t)\in\Pi_{i,g}^{(s)}, \]
\[ v_i^{(s)}(x,t)=z_i^{(s)}\bigl(x_k,\chi_i^{(s)}(x,t)\bigr), \quad (x,t)\in\Pi_{i,x_k}^{(s)}, \tag{16} \]
\[ \tilde f_i^{(s)}(x,t)= \int_{\chi_i^{(s)}(x,t)}^t f_i^{(s)}(\varphi_i^{(s)}(\tau,x,t),\tau)\,d\tau; \quad (s=1,2;\ k=0,s;\ 1\leq i\leq mn). \]
The system of integro-functional equations (15), (16) is immediately reduced to a system of integral equations. Indeed, by virtue of, for example—
for example, from (13) the values \(z_i^{(1)}\bigl(x_1,\chi_i^{(1)}(x,t)\bigr)\) \((1 \leqslant i \leqslant r_1^1)\) are expressed in terms of the known \(H_i^1\) and in terms of linear combinations of the unknowns
\[
z_j^{(1)}\bigl(x_1,\chi_i^{(1)}(x,t)\bigr)
\qquad
\left(r_1^1+1 \leqslant j \leqslant \frac{mn(m+1)}{2}\right).
\]
But these latter, by virtue of (15), (16), are expressed in terms of the known quantities \((G_i\) and \(f_i)\) and in terms of the values of certain \(z_j^{(1)}\) on the line \(x=x_0\) for values of \(t\) smaller than \(\chi_i^{(1)}(x,t)\). Continuing this process further, using first (14), then (15) and (13), and so on, we shall ultimately reach the value \(t=0\) and thus express
\[
z_i^{(1)}\bigl(x_1,\chi_i^{(1)}(x,t)\bigr)
\]
in terms of known quantities and integrals of the unknowns \(z_i^{(s)}\). We proceed in exactly the same way with
\[
z_i^{(s)}\bigl(x_0,\chi_i^{(s)}(x,t)\bigr)
\quad\text{and}\quad
z_i^{(2)}\bigl(x_2,\chi_i^{(2)}(x,t)\bigr).
\]
Thus, since \(t\) is finite and \(\chi_i^{(j)}(x,t)<t\), conditions (13), (14) make it possible to reduce the system of functional-integral equations (15), (16) to a system of Volterra integral equations. Solving this system and the systems obtained from it by differentiating \(r\) times with respect to \(x\) and \(t\), we immediately verify the validity of Theorem 2.
The indicated process of reducing the system (15), (16) to a system of integral equations becomes obvious if one considers the physical essence of the problem under consideration. Physically, this problem describes, for example, the oscillations of a certain system consisting of two heterogeneous pieces under the action of initial disturbances at \(t=0\) and boundary disturbances at \(x=x_1\) and \(x=x_2\), with a prescribed state (14) on the interface. It is clear that before the wave reaches the point \((x,t)\) for sufficiently large \(t\), it will be reflected several times from the walls \(x=x_1\), \(x=x_0\), and \(x=x_2\). But for every finite \(t\) the number of such reflections will be finite.
§ 3. Some generalizations
-
One could consider a more general mixed problem: the system (1) and the boundary conditions (3) (or conditions (6) and (7) in the case of problems with discontinuous coefficients) may contain Volterra additions involving the desired function \(u\) and all its partial derivatives with respect to \(x\) and \(t\) up to order \(m-1\) inclusive. These may be, for example, integrals from \(0\) to \(t\) of linear combinations of the indicated derivatives with variable coefficients depending on \(x\), \(t\), and the variable of integration. In this case the form of conditions (4) and (8) does not change. In addition, the system (1) may be quasilinear with nonlinearities in terms containing derivatives of \(u\) up to order \(m-1\) inclusive. Accordingly, conditions (3), (6), (7) may also contain nonlinear additions. In solving the indicated problem one must use the results of [2].
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The method indicated above makes it possible to consider problems in which the boundary conditions (and conjugation conditions) contain derivatives of the desired function with respect to \(x\) and \(t\) of any order greater than or equal to \(m\). By elimination (using the system (1)) and integration with respect to \(t\), such a problem can always be reduced to the one considered in this work. Moreover, the boundary conditions need not have the form (3)—they may relate the values of the desired function and its derivatives at different values of \(x\) [3] (in particular, these conditions may have the same form as in [6]).
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Finally, let us indicate possible generalizations in the very formulation of the problem. As is clear from the preceding, the number of boundary conditions on the sides \(x=x_i\) is determined by the number of positive and negative-
among the functions \(\lambda_i(x_i,t)\). It follows from this that, if the functions \(\lambda_j(x,t)\) are allowed to change sign inside the domains under consideration, then the number of boundary conditions on the sides \(x=x_1\) and \(x=x_2\) may be equal to any number between \(0\) and \(mn\); the corresponding number for the side \(x=x_0\) lies between \(0\) and \(2mn\) (moreover, the number of conditions on each of the sides may be completely independent of the number of conditions on the other sides). Thus, in the formulation, for example, of a problem with discontinuous coefficients, the following two extreme cases are possible: a) \(r_1^1=r_1^2=r_2^1=r_2^2=R=0\). In this case the boundary conditions and the matching conditions disappear; the initial conditions determine a unique solution in \(P\); b) \(r_1^1=r_1^2=r_2^1=r_2^2=mn\); \(R=2mn\). Naturally, all cases intermediate between a) and b) are possible. The methods of the present paper, as is not difficult to see, make it possible to study these exceptional cases.
References
- Abolina V. E., Myshkis A. D. Scientific Notes of the Latvian University, 20, issue 3, 1958, pp. 87–104.
- Abolina V. E., Myshkis A. D. Mathematics Collection, 50 (92), 1960, pp. 423–442.
- Myshkis A. D. On the maximal domain of solvability of a mixed problem for an almost linear hyperbolic system with two independent variables. Materials for the Soviet-American Symposium on Partial Differential Equations. Novosibirsk, 1963.
- Thomée V. Math. Scand., 5, 1, 93–113, 1957.
- Petrovskii I. G. Lectures on Partial Differential Equations. Moscow, 1961.
- Vaganov A. I. Doklady AN SSSR, 155, 6, 1247–1249, 1964.
Received by the editors
May 26, 1965
Lvov State University