A five-dimensional hyperchaotic system is a dynamical system with five state variables that exhibits chaotic behavior in multiple directions. In this work, we incorporated a 5D hyperchaotic system with constant- and variable-order Caputo and the Caputo–Fabrizio fractional derivatives. These fractional 5D hyperchaotic systems are solved numerically. Through simulations, the chaotic behavior of these fractional-order hyperchaotic systems is analyzed and a comparison between constant- and variable-order fractional hyperchaotic systems is presented.

The aim of this article is to introduce analytical and approximate techniques to obtain the solution of time-fractional Navier–Stokes equations. This proposed technique consists is coupling the homotopy perturbation method (HPM) and Laplace transform (LT). The time-fractional derivative used is the Caputo–Hadamard fractional derivative (CHFD). The effectiveness of this method is demonstrated and validated through two test problems. The results show that the proposed method is robust, efficient, and easy to implement for both linear and nonlinear problems in science and engineering. Additionally, its computational efficiency requires less computation compared to other schemes.

This study introduces an innovative framework for generalized Hermite-Frobenius-Genocchi polynomials in two variables, parameterized by a single variable. The focus is on providing a comprehensive characterization of these polynomials through various mathematical tools, including generating functions, series expansions, and summation identities that uncover their essential properties. The work extends to the derivation of recurrence relations, the investigation of shift operators, and the formulation of multiple types of differential equations. In particular, the study delves into integro-differential and partial differential equations, employing a factorization technique to develop different forms and solutions. This multifaceted approach not only enhances our understanding of these polynomials, but also lays the groundwork for their further exploration in diverse areas of mathematical research.

This research paper introduces the generalized Burgers equation, a mathematical model defined using the general fractional derivative, the most recent operator in fractional calculus. The general fractional derivative can be reduced into three well-known operators, providing a more tractable form of the equation. We apply the homotopy perturbation method (HPM), a powerful analytical technique, to obtain the solution of the generalized Burgers equation. The results are illustrated using a practical example, and we present an analysis of the three reduced operators. In addition, a graphical analysis is provided to visualize the behavior of the solution. This study sheds light on the application of the homotopy perturbation method and the general fractional derivative in solving the generalized Burgers equation, contributing to the field of nonlinear differential equations.

In the present study, we aimed to derive analytical solutions of the homotopy analysis method (HAM) for the time-fractional Navier–Stokes equations in cylindrical coordinates in the form of a rapidly convergent series. In this work, we explore the time-fractional Navier–Stokes equations by replacing the standard time derivative with the Katugampola fractional derivative, expressed in the Caputo form. The homotopy analysis method is then employed to obtain an analytical solution for this time-fractional problem. The convergence of the proposed method to the solution is demonstrated. To validate the method’s accuracy and effectiveness, two examples of time-fractional Navier–Stokes equations modeling fluid flow in a pipe are presented. A comparison with existing results from previous studies is also provided. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations applied in engineering mathematics.