Purpose. To unify the classical theory of the Cauchy problem for parabolic systems of partial differential equations; development of the theory of spaces of fundamental and generalized functions as a medium for studying boundary value problems for differential functional equations; a general description of the correctness and uniqueness classes of the Cauchy problem for wide classes of equations of different types; study of the properties of classical solutions of such equations and systems.
Application area. Education, basic scientific research.
Advantages. Expansion and unification of the classes of Petrovsky, Eidelman, Shilov, Zhitomirsky systems of partial differential equations. Effective methods for constructing and studying the Green’s function for parabolic and hyperbolic partial differential equations and equations with fractional differentiation operators, in particular, the fractal heat equation and the telegraph equation with fractional derivatives. General statements about the stability of stochastic dynamic systems of random structure with Markov switchings and external disturbances and the conditions for the existence of optimal control for stabilizing systems to stochastically stable ones.
Description. The development involves the expansion and generalization of known classes of systems of partial differential equations and evolutionary equations of various types. Creation of new methods for constructing and studying the Green’s function of boundary value problems for such equations and description of correctness and unity classes. Studying the properties of their classical solutions with limit values from wide classes of generalized functions. Study of stability in various probabilistic senses of general stochastic systems of differential functional equations and synthesis of their optimal control.