A NEW NUMERICAL TECHNIQUE FOR SOLVING <i>ψ</i>-FRACTIONAL RICCATI DIFFERENTIAL EQUATIONS

Abstract

This paper proposes a new numerical technique for solving a specific class of fractional differential equations, which includes the ${\psi}$-Caputo fractional derivative. The class under consideration is nonlinear ${\psi}$-fractional Riccati differential equations (${\psi}$-FRDEs). Our approach relies on the ${\psi}$-Haar wavelet (${\psi}$-HW) operational matrix, which is a novel type of operational matrix of fractional integration. We derive an explicit formula for the ${\psi}$-fractional integral of the HW. This operational matrix has been used successfully to solve nonlinear ${\psi}$-FRDEs.The Quasi-linearization technique is employed to linearize the non-linear $\bm\psi$-FRDEs. This technique reduces the problem to an algebraic equation that can be easily solved. The technique is a useful and straightforward mathematical tool for solving nonlinear ${\psi}$-FRDEs. The computational complexity of the operational matrix technique is minimal. The error analysis of the proposed method is thoroughly investigated. To justify the method's accuracy and efficiency, numerical results are given.