Abstract: A dynamical system of partial differential equations which is 3D with respect to spatial coordinates and contains as a particular case both: Navier-Stokes equations and the nonlinear systems of PDEs of the elasticity theory is proposed.
In the second part using above uniform expansion there are created and justified new 2D with respect to spatial coordinates nonlinear dynamical mathematical models von Karman-Mindlin-Reissner (KMR) type systems of partial differential equations for anisotropic porous, piezo, viscous elastic prismatic shells. Truesdell-Ciarlet unsolved problem (open even in case of isotropic elastic plates) about physical soundness respect to von Karman system is decided. There is found also new dynamical summand (is Airy stress function) to another equation of von Karman type systems . Thus, the corresponding systems in this case contains Rayleigh-Lamb wave processes not only in the vertical, but also in the horizontal direction. For comlpleteness we also introduce 2D Kirchhoff-Mindlin-Reissner type models for elastic plates of variable thickness.
Then if KMR type systems are 1D one respect to spatial coordinates at first part for numerical solution of corresponding initial-boundary value problems we consider the 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 smooth functions and Sard type best coefficients characterizing remainder proximate members on less smooth class of admissible solutions.
Corresponding 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. These processes are new even for ordinary differential equations.
In case if KMR type systems are 2D one respect to spatial coordinates at first part for numerical solution of some corresponding initial-boundary value problems we use Gauss-Hermete processes with discrete-variational and differentiate-parameteric methods.

Brief Biography of the Speaker:
Born in Tbilisi(Tb),1937,education: 1954/59 - student Tb. State Univ.(TSU),1959/62 - Post grad. stud. Razmadze Institute of Mathematics,1981-doctorant Moscow Phys.-Tech.Inst. Ph.D(Candidate of Sci.)-1964, Dr.Hab. (Full professor) Solid Mechanics-1984(Lomonosov Moscow SU),1987(Razmadze Inst.),work experience:1962-1973-Senior Reaseachers Razmadze and Vekua Institutes,1973- Head of Dept., Full Prof. Mathematics Vekua Inst. Appl. Math./TSU, First Premium Diploma Works TSU (1959); ; Honor order(2003); Premium Ilia Vekua Georgian NAS (1993), Premium of COBASE (1999, Invited Professor-Explorer Univ. Delaware), Premiums ISF (G.Soros Foundation): 1993-Short-Term, 1995-96-Long-Term-KZB 200 (Leader of Group), Academician Georgian Acad. Engineering Sci., Members Editorial Boards JAFA(Memphis), 4 regional journals in Georgia,1998-2003:Editor-in-Chief Proceed. TSU (Appl.Math.&Comp.Sci), Invited Lecturer: 1983-1989- Math. Summer Schools Pushchino Biophys.Ins.AS FSU, 1979-1996-ISIMM Symposiums, and Participant-more than 100 Inter.Math.Conferences, 175- research paper including review articles, 4- monographs, and 3-manuals. 2- books(ed.)