In this research paper, the HIV-1 infection of CD4+ T-cells fractional mathematical model with the effect of antiviral drug therapy is handled by applying three new computational schemes to this biological model to investigate its analytical explicit wave solutions. This mathematical model is used to predict the evolution of the population dynamical systems involving virus particles. The modified Khater method, the extended simplest equation method, and sech–tanh method with a new fractional operator (Atangana–Baleanu derivative operator) is employing to find the analytical solutions in various distinct new formulas of the biological suggested model. Moreover, the stability of the obtained solutions is investigated by using the characterizes of the Hamiltonian system to show their applicability in making the antivirals that protect our human life. Some plots are explained under specific conditions of the contained constants to reveal the dynamical behavior of the evolution of the population dynamical systems involving virus particles. A comparison between our results and that obtained in previous work is also represented and discussed in detail to show the novelty for our solutions. The performance of the used methods shows power, practical, and ability to apply to other nonlinear partial differential equations.

This research paper discusses the analytical and semi-analytical solutions of the quadratic–cubic fractional nonlinear Schrodinger (NLS) equation. By applying a new fractional operator we transform the fractional formula of the model to integer-order, which allows applying the analytical and numerical methods on it. The analytical solutions are obtained by the implementation of two distinct systematic schemes and the reported solutions are used in applying the Adomian decomposition method to get the semi-analytical wave solutions of this model. These solutions are used to characterize the changes over time of a physical system in which case of quantum influence, such as wave-particle duality. The comparison between the analytical and semi-analytical solutions are given to explain the accuracy of the obtained solutions.

In this work, we implement some analytical techniques such as Exp--Function method, Tan and Tanh methods, Extended Tan and Tanh methods, and Sech method for solving the fractional nonlinear partial differential equation with a truncated M-fractional derivative, which contain exponential terms its name, General Modified fractional Degasperis-Procesi-Camassa-Holm equation with a truncated M-fractional derivative. These methods can be used as an alternative to obtain exact solutions of different types of differential equations applied in engineering mathematics. Finally, we obtain the analytical solution of the M-fractional heat equation and present a graphical analysis.