Quantum mechanics

Algebraic Riccati equation

Materials science

Differential equation

Applied mathematics

Mathematical optimization

Solver

Orthonormal basis

Matrix (chemical analysis)

Algebraic equation

Fractal

Physics

Nonlinear system

Mathematics

Orthogonal matrix

Orthogonal basis

Mathematical analysis

Composite material

Fractional calculus

Riccati equation

Fractal and Fractional

Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature.

Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative