In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of $\alpha$-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned $\alpha$-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order $\gamma =0.99$ (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator.

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.

This research develops a model describing a relation between a predator and a prey where the prey species are divided into two classes i.e., susceptible prey species and the infected prey species. Furthermore, the susceptible prey species are also provided with a refuge. After this, the research expands the formulated model into a model with Caputo fractional derivative. A detailed analysis of the solution of Caputo fractional model for the predator–prey relation with a sickness in prey and a refuge to susceptible prey is done. Numerical solutions along with the simulations are briefed for the Caputo fractional model. Lastly, the phase portrait diagrams are also given for better understanding.

We give explicit representation of Neumann boundary value problem for Bitsadze equation on the upper half plane. We will also give solution of the inhomogeneous polyanalytic equation arising from Neumann and (n-1) Dirichlet boundary conditions on the upper half plane H.