ABSTRACT
Quite often, mathematical models for applied problems arising in natural sciences lead to hyperbolic systems of partial differential equations of the first order. This is especially true of hydrodynamics and aerodynamics. One more important case of such hyperbolic systems is connected with the system of Maxwell equations capable of describing electromagnetic fields. Until now the most profound research was devoted to systems of equations with two independent variables associated with one-dimensional models which do not cover fully the diversity of problems arising time and again in theory and practice. The situation becomes much more complicated in the case of multidimensional problems for which careful analysis requires a somewhat different technique. Moreover, the scientists were confronted with rather difficult ways of setting up and treating them on the same footing. Because of these and some other reasons choosing the most complete posing of several ones that are at the disposal of the scientists is regarded as one of the basic problems in this field. On the other hand, a one-dimensional problem can serve, as a rule, as a powerful tool for
and
AU au A(x, t) ax + B(x, t) 8t + C(x, t) u =F(x, t),
ov . ov ox +}( 8t + D v = G ,
D = T-1 aT + T-1A-1B aT + T-1A-1CT ax 8t '
8VI(Z,t) 8VI(Z,t) () () O~z~L, t ~ 0,at + 8z =V2 x, t + PI X ,
8V2(Z, t) 8V2(Z, t) O~z~L, t ~ 0,(2.1.3) at - 8z =VI (x, t) + P2 (x) ,
VI(X,O) = <PI (x) , V2(X,O) =<P2(X) , O$z$L, VI (L, t) =tPI (x) , V2(0, t) = tP2(X) , t ~ 0.
