An Optimized Computational Method for Solving Quadratic Riccati Differential Equations

Résumé

This study presents the derivation, analysis and implementation of an Optimized Computational Method (OCM) for the numerical solution of Quadratic Riccati Differential Equations (QRDEs). An OCM was developed using a power series polynomial as the basis function through interpolation and collocation techniques with an optimized mesh point introduced within a one-step interval. The optimization procedure was employed to determine the new mesh point, thereby enhancing the approximation capability of the resulting numerical scheme. Fundamental properties of the method, including order, consistency, zero-stability, convergence and absolute stability were investigated. The analysis revealed that an OCM possesses a uniform order of seven, confirming its high accuracy and small truncation error. Furthermore, the method was shown to be consistent, zero-stable and convergent according to the Dahlquist convergence criterion. Stability analysis using the boundary locus technique established that the method is A_α​-stable. To validate the performance of an OCM, the OCM was applied to three test problems involving QRDEs and the results were compared with existing methods reported in the literature. The numerical solutions obtained by an OCM showed excellent agreement with the exact solutions throughout the integration interval. Comparative analysis demonstrated that an OCM consistently outperformed the existing methods in terms of accuracy and computational reliability. The results confirm that an OCM is an efficient, stable and highly accurate numerical technique for solving QRDEs arising in scientific and engineering applications.