Abstract
The main subjects of the present paper are the Goursat and Darboux boundary-value problems for hyperbolic systems with two independent variables. We show that obtained by T.V. Chekmarev in terms of successive approximations formulas for solution of the Goursat problem can be built also by the Riemann method, work out an analog of the Riemann–Hadamard method for the system, and introduce its Riemann–Hadamard matrix. We solve the Darboux problem in terms of the introduced matrix.
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1 INTRODUCTION
System of equations of the first order
was studied by a number of authors (see, for instance, [1–6] and references in these papers). It is of interest, in particular, in connection with differential equations of mixed type. The most number of publications concerns the case \(n=2\).
The author prposed [7] a version of the ariemann method for systems of differential equations both with simple and multiple characteristics.
The Darboux problem for hyperbolic equations and systems is of great interest. This problem for equations of second order with two independent variables is considered by a number of authors. We mention here the works [8, p. 228–233], [9–14]. There exist also investigations of the Darboux problem for hyperbolic equations of the third order [15, 16].
2 THE GOURSAT PROBLEM
We consider the following system of equations
in rectangle \(D=\{x_{0}<x<x_{1}\), \(y_{0}<y<y_{1}\}\). Here \(a_{11}\), \(a_{12}\), \(a_{21}\), \(a_{22}\), \(f_{1}\), \(f_{2}\in C(\overline{D})\). A solution \(u\) is called regular in \(D\) if \(u_{1},\)\(u_{2},\)\(u_{1x},\)\(u_{2y}\in C(D)\).
The Goursat problem: Find regular solution of system (2) in domain \(D\) which is continuously extendable on the boundary of domain \(D\) and satisfies boundary conditions
The Goursat problem has unique solution [].
A linear transformation of the desired functions reduces (2) to the case
In what follows we conider these identities as fulfilled.
Chekmarev [] by means of the successive approximations method obtained solution of the Goursat problem (2), (3) in the form
where \(L\), \(K\), \(M\), \(N\) are determined by coefficients \(a_{12}\), \(a_{21}\) as uniformly convergent series, and satiffy relations
There exists other approach to solving of described aboveproblems. We can apply to system (2) the Riemann method. Rewrite (2) in vector-matrix form
Introduce Riemann matrix \(\mathbf{R}=\mathop{\textrm{colon}}\nolimits(\mathbf{R}_{1},\mathbf{R}_{2})\), where vectors \(\mathbf{R}_{i}(x,y,\xi,\eta)=(r_{i1},r_{i2})\), \(i=\overline{1,2}\), are solutions of systems
\(\delta_{ij}\) is Kronecker symbol. The solutions of systems (9) exist and are unique in the class of continuous functions. We differentiate (9), and see that system (9) is equivalent to two Goursat problems
Clearly, function \(R\) with respect to the first pair of variables satisfies the conjugated to (2) system
We integrate identity
over rectangle \(D_{1}=\{x_{0}<x<\xi\), \(y_{0}<y<\eta\}\), \((\xi,\eta)\in D\), and obtain
We use the properties of the Riemann matrix, differentiate (12) with respect to variable \(\eta\) and (13) with respect to \(\xi\), and obtain solutions of the Goursat problem in the following form:
The formulas (5), (6), (7), (9), (14), (15) imply relations
3 THE DARBOUX PROBLEM
We consider the Darboux problem:Find regular solution of system (2) in domain \(D_{0}=\{(x,y):0<y<x<T\}\), which is continuable on the boundary of domain \(D\) and satisfies boundary conditions
Solution of the Darboux problem exists, and it is unique [3, p. 26–29].
We fix a point \(P(\xi,\eta)\in D_{0}\), and consider corresponding quadrangle \(D_{p}=\{(x,y):y<x<\xi,0<y<\eta\}\subset D_{0}\) with vertices at points \(O(0,0)\), \(P_{1}(\eta,\eta)\), \(P(\xi,\eta)\), \(P_{3}(\xi,0)\), rectangle \(D_{1}=\{(x,y):\eta<x<\xi,0<y<\eta\}\) with vertices at points \(P\), \(P_{1}\), \(P_{2}(\eta,0)\), \(P_{3}\), and triangle \(D_{2}=\{(x,y):y<x<\eta,0<y<\eta\}\) with vertices at points \(O\), \(P_{1}\), \(P_{2}\) (see Fig. 1).
Define Riemann–Hadamard matrix for the Darboux problem \(H(x,y,\xi,\eta)\). Let
where \(R\) is the defined above Riemann matrix, and matrix \(\mathbf{V}=\mathop{\textrm{colon}}\nolimits(\mathbf{V}_{1},\mathbf{V}_{2})\), with vectors \(\mathbf{V}_{i}(x,y,\xi,\eta)=(\bar{r}_{i1},\bar{r}_{i2})\), \(i=\overline{1,2}\), needs to be specified in domain \(D_{2}\).
We require that \(V_{1}\), \(V_{2}\) be solutions the Darboux problems in domain \(D_{2}\) for systems
with conditions relatively
for vector \(V_{1}\), and
for vector \(V_{2}\).
We integrate identity (11) over rectangle \(D_{1}\):
Integration of identity (11) over triangle \(D_{2}\) gives
We add (21) with (23) and (22) with (24), and obtain
It follows from (9), (19), (20) that \(r_{11}(\xi,y,\xi,\eta)=1\), \(r_{12}(x,\eta,\xi,\eta)=0\), \(r_{21}(\xi,y,\xi,\eta)=0\), \(r_{22}(x,\eta,\xi,\eta)=1\), \(\bar{r}_{12}(y,y,\xi,\eta)=0\), \(\bar{r}_{22}(y,y,\xi,\eta)=0\), and
Consequently, (25) and (26) are representable as
where \(F_{1}(\xi,\eta)\), \(F_{2}(\xi,\eta)\) are completely determined by the Riemann–Hadamard matrix \(H\) and boundary data of the Darboux problem.
We differentiate the first equality (27) in \(\eta\), and second in \(\xi\), and obtain explicit solution of the Darboux problem in terms of the Riemann–Hadamard matrix.
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Mironova, L.B. Boundary-value Problems with Data on Characteristics for Hyperbolic Systems of Equations. Lobachevskii J Math 41, 400–406 (2020). https://doi.org/10.1134/S1995080220030130
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DOI: https://doi.org/10.1134/S1995080220030130