The multi-resolution Haar wavelets collocation procedure for fractional Riccati equations

In this paper, we present a Haar wavelet collocation method (HWCM) for solving fractional Riccati equations. The primary goal of this study is to bypass the requirement of calculating the Jacobian of the nonlinear system of algebraic equations by using an iterative quasi-linearization technique. The Haar wavelet series is then utilized to approximate the first-order derivative, which is incorporated into the Caputo derivative framework to express the fractional-order derivative. This process transforms the nonlinear Riccati equation into a linear system of algebraic equations, which does not require calculating the Jacobian and can be efficiently solved using any standard linear solver. We evaluate the performance of HWCM on various forms of fractional Riccati equations, demonstrating its efficiency and accuracy. Compared to existing methods in the literature, our proposed HWCM produces more precise results, making it a valuable tool for solving fractional-order differential equations.