A New Numerical Scheme for Solving Quadratic Riccati Differential Equations (QRDEs)

Abstract

Quadratic Riccati Differential Equations (QRDEs) are important in control theory, optimal stabilization, and nonlinear dynamics, thereby requiring numerical methods that are both accurate and computationally reliable. This study introduces a new numerical scheme (NNS) for solving QRDEs using an eighth-order power series basis function combined with interpolation and collocation techniques to approximate the exact solutions over a one-step integration interval. Through this procedure, the continuous differential equation is transformed into a system of nonlinear algebraic equations, which is then solved using the Gauss elimination method. A detailed analysis of the proposed scheme establishes its high order of accuracy, zero stability, consistency, convergence, and absolute stability, confirming its suitability for practical computation. The method was implemented on four different QRDE problems, and the results showed that the approximate solutions were in excellent agreement with the exact solutions throughout the integration interval. The study concludes that the proposed NNS is a robust, accurate, and broadly applicable approach for the numerical solution of QRDEs, offering a reliable contribution to computational methods for nonlinear differential equations.