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The maximum principle for partial differential equations plays essential role in various applicatins. There is usually a natural physical interpretation of the maximum principle in those problems in differential equations that arise in physics. In such situations the maximum principle helps us apply physical intuition to mathematical models. Consequently, anyone learning about the maximum principle becomes acquainted with the classically important partial differential equations and, at the same time, discovers the reasons for their importance. The proofs required to establish the maximum principle are extremely elementary. By concentrating on those applications which can be derived from the maximum principle by elementary methods, such as characteristics methods and Stock’s theorem, Green's theorem, integrating by characteristics and others. The maximum principle enables us to obtain information about solutions of differential equations without any explicit knowledge of the solutions themselves. In particular, the maximum principle is a useful tool in the approximation of solutions, a subject of great interest to many scientists. For the cases of elliptic and parabolic partial differential equations maximum principle is well-known fact, at the same time, in the case of hyperbolic equations classical formulation of maximum principle is not valid.
This paper deals with maximum principle for second order hyperbolic equatins this lower terms. The forms that these principles take reflect the structure of properly posed problems for hyperbolic equations. Both the statements of the theorems and the methods of proof for hyperbolic operators, presented in this paper are quite different from those for elliptic and parabolic operators. In particular, the role of characteristic curves and surfaces becomes evident in the hyperbolic case. The maximum principle occurs in so many places and in such varied forms that we have found it impossible to discuss some topics which we had originally hoped to treat. For example, the maximum principle for finite difference operators is omitted entirely. The mains results of the paper are theorems on maximum principle for second order hyperbolic equations, lower terms of which contain amplitudes and first-order derivatives.
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