Constructing Analytical Solutions of the Fractional Riccati Differential Equations Using Laplace Residual Power Series Method

Abstract

In this article, a hybrid numerical technique combining the Laplace transform and residual power series method is used to construct a series solution of the nonlinear fractional Riccati differential equation in the sense of Caputo fractional derivative. The proposed method is implemented to construct analytical series solutions of the target equation. The method is tested for eminent examples and the obtained results demonstrate the accuracy and efficiency of this technique by comparing it with other numerical methods.

Tikhonov-Tuned Sliding Neural Network Decoupling Control for an Inverted Pendulum

“…The optimal solution can be derived through algebraic or differential methods in optimal control theory. One common approach is to solve the algebraic Riccati equation [26]. By solving the following Riccati equation, we can obtain the optimal controller gain matrix K:…”

“…Intelligent control systems for the inverted pendulum often rely on combinations of adaptive control and robust control. The coupling nature of the inverted pendulum system, where the motion of the rod influences the pendulum's position and angle, makes controlling and stabilizing the system more challenging [16][17][18][19][20][21][22][23][24][25][26][27].…”

A multi-step formable transform decomposition method for solving fractional order Riccati equation

“…The method provides the solution in terms of convergent series with easily computable comp onents. In the past years, several academics have concentrated their efforts on the numerical solution of ordinary differential equations of fractional order and various numerical techniques, including the Fourier transform method ( Kemle& Beyer, 1997), Homotopy perturbation method (Wang, 2007;Odibat & Momani, 2008;Mtawal & Alkaleeli, 2020), Homotopy analysis method (Canget al, 2009;Zurigatet al, 2010;Freihat, et al, 2014), Residual power series method (Ali et al, 2017), Alternative variational iteration method (Mtawalet al, 2020), Triple Shehu transform method (Alkaleeliet al, 2021;Kapooret al, 2022), the Laplace residual power series method (Burqan, et al, 2022),. Recently, a formable transformation decomposition method (FTDM) was applied by (Al-ZouBi&Zurigat, 2014), it combines the formable integral transform (Saadeh, 2021) and the decomposition method (Momani& Al-Khaled, 2005;Shawagfeh, 2002;Khanet al, 2013;Mahdyet al, 2015).…”