Discrete matrix Riccati equations with superposition formulas

TL;DR

This paper discusses discretizations of matrix Riccati equations that maintain their superposition formulas, applicable to various derivative types including q-derivatives and standard discrete derivatives.

Contribution

It introduces a general discretization method for matrix Riccati equations that preserves their superposition properties, extending to different derivative frameworks.

Findings

Discretizations preserve superposition formulas.

Applicable to q-derivatives and standard derivatives.

Provides a unified approach for discretizing Riccati equations.

Abstract

An ordinary differential equation is said to have a superposition formula if its general solution can be expressed as a function of a finite number of particular solution. Nonlinear ODE's with superposition formulas include matrix Riccati equations. Here we shall describe discretizations of Riccati equations that preserve the superposition formulas. The approach is general enough to include $q$-derivatives and standard discrete derivatives.