Numerical solution of Cauchy problems for evolutionary equations by Gauss&Hermite processes

In the first part there are created and justified new 2D with respect to spatial coordinates nonlinear dynamical mathematical models von Kármán-Mindlin-Reissner(KMR) type systems of partial differential equations for anisotropic porous, piezo, viscous elastic prismatic shells. Truesdell-Ciarlet unsolved( even in case of isotropic elastic plates) problem about physical soundness respect to von Kármán system is decided. There is find also new dynamical summand ( is Airy stress function) in the another equation of von Kármán type systems too. Thus the corresponding systems in this case contains Rayleigh-Lamb wave processes not only in the vertical, but also in the horizontal direction. These dynamical systems represent evolutionary equations for which the methods of Harmonic Analyses are nonapplicable. In this connection for Cauchy problem suggests new schemes having arbitrary order of accuracy and based on Gauss-Hermite processes. This processes are new even for ordinary differential equations.

In the second part if KMR type systems are 1D one respect to spatial coordinates for numerical solution of corresponding initial-boundary value problems there are constructing finite-element method using new class of B-type splain-functions. The exactness of such schemes depends from differential properties of unknown solutions: it has an arbitrary order of accuracy respect to a mesh width in case of sufficiently smoothness functions and Sard type best coefficients characterizing remainder proximate members on less smoothing class of admissible solutions.