To the Development and Perspective of Some Linear and Non-linear Problems of Continuum Mechanics

For nonlinear continuum mechanics the unified formal representation is given, especially for poro-piezoelastic, inhomogeneous, anisotropic, electro-magnetic field responsive media. With this representation, systems of generalized Navier–Stokes and Euler equations and essentially non-linear problems of solid elastic body mechanics are obtained. Based on our earlier works, we continue research on mechanics problems of thin walled structures. Regarding this, incompleteness of the well-known representation of von Kármán system is noted, and taking into account the remark of Clifford Truesdell, corresponding refined models are built for bending and extension-compression processes in dynamic case too. The main part presents the study of the possibility of constructing solutions to multidimensional problems of both refined theories and solid bodies based on the approach of Morse and Feshbach. For BVPs we showed that the solution of a number of problems can be studied as a result of the investigation of the system of DEs, main part of which is the Laplace vector operator. We showed that generalized numerical methods are effective for constructing an approximate solution of these systems. For two independent variables, based on complex analysis, an operator DE with an essentially non-linear main part is studied. An important place is given to numerical realization of ODEs, which not only have an independent value, but also represents a way of constructing an approximation of the solution of high order of accuracy using a continuous analogue of the alternating direction method.