On Some Asymptotic Properties of Solutions to Biharmonic Equations

The author considers the application of the approximation theory methods to the principles of optimality in the decision-making theory. In finding optimal solutions, the risk function often has rather complex structure for studying its properties, which makes it necessary to approximate the risk function to another function with simple and clear characteristics. In this regard, the asymptotic properties of the solutions of biharmonic equations as approximate functions are investigated. Complete asymptotic expansions of the upper limits of deviations of the Sobolev class functions W (the set that the risk functions in decision-making optimization belong to) from operators that are solutions of biharmonic equations with certain boundary conditions are obtained. The expansions allow us to find the Kolmogorov–Nikolsky constants of arbitrarily high degree of smallness, which makes it possible to estimate the approximation error when solving optimization problems with arbitrary accuracy. It is mentioned that the biharmonic equations can be used to efficiently generate mathematical models of natural and social phenomena.