On the Identification of Solutions to Riccati Equation and the other Polynomial Systems of Differential Equations

Abstract

The authors previously proposed a general method for finding particular solutions for overdetermined PDE systems, where the number of equations is greater than the number of unknown functions. The essence of the method is to reduce the PDE to systems of PDE of a lower dimension, in particular, to ODEs by overdetermining them by additional constraint equations. Reduction of some PDE systems generates overdetermined systems of polynomial ODEs, which are studied in this paper. A method for transforming polynomial ODE systems to linear ODE systems is proposed. The result is interesting from a theoretical point of view if these systems of polynomial ODEs are with constant coefficients. The solution of such nonlinear systems using our method can be represented as a sum of a very large but finite number of oscillations. The amplitudes of these oscillations depend on the initial data nonlinearly. The Navier–Stokes equations and unified PDE systems obtained by the authors earlier can be transformed to such systems. The Riccati equation is also investigated. New special cases are indicated when it is possible to find its solution. Numerical estimates of the complexity of this method for practical implementation are presented.