The evolution equation for the shock deformation problems of nonlinear elastic inhomogeneous mediums

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   Dynamic deformation of nonlinear elastic bodies, which is caused by the
   action of short-term intensive loads, leads to a complex
   mechanical-physical process of shock waves formation and motion.
   Inhomogeneity of the medium should be considered as an additional
   important factor in solving dynamic problems for the great length domains
   (particularly in seismology). In this paper we present results of the
   problem solution of the longitudinal shock wave in the Murnaghan medium by
   a small parameter method. The elastic moduli of the medium and its density
   have weak power type inhomogeneity in the wave direction. The joint
   integration of the weak nonlinearity and weak inhomogeneity factors leads
   to a nonlinear distortion of characteristics and the shock wave formation.
   The hypothesis of the single-wave approximation allows to provide an
   approximate solution based on the analysis of the quasi-waves evolution
   equation in the frontal area of the anterior border of the deformation
   wave. This equation fundamentally depends on the balance between the
   nonlinear and inhomogeneous properties of the medium. The general solution
   of the evolution equation is presented. Examples of particular solutions
   of various boundary value problems on the basis of this decision are
   given.