Ebola Virus Disease (EVD) remains a major global health concern due to its high fatality rate and rapid transmission, particularly in resource limited regions. In this study, we investigate the dynamics of EVD transmission using a fractional-order SIRD model incorporating the Atangana–Baleanu–Caputo (ABC) derivative to capture memory-dependent behaviors inherent in infectious processes. The defined model based on fractional calculus provide a more accurate description of temporal dynamics by integrating nonlocal and nonsingular kernels. That model is biologically more realistic than classical integer-order formulations. We use the Homotopy Perturbation Method (HPM) to provide a reliable and efficient means to handle the nonlinear structure. Theoretical analysis of the model includes existence and uniqueness of the solution using fixed point theory. Furthermore, the basic reproduction number is derived through the next-generation matrix approach to assess threshold behavior and local stability of the disease free equilibrium. We use graphical simulations in Mathematica, based on varying fractional orders and key epidemiological parameters, offer insights into the impact of memory effects on disease progression. The suggested model architecture allows better forecasting for managing the Ebola outbreak and exhibits increased analytical integrity.

In this paper, we introduced a new fractional derivative operator based on Lonezo Hartely function, which is called G-function. With the help of the operator, we solved a fractional diffusion equations. Some applications related to the operator is also discussed as form of corollaries.

The paper is concerned with the SIZR mathematical model for an outbreak of zombie infection with time-dependent infection rate. This class of the SIZR model involves equations that relate the susceptible S(t), the infected I(t), the zombie Z(t), and removed population R(t). The well poseness of the model is presented. The proposed model is then outstretched to the fractional order mathematical model with three different derivative operators i.e., Caputo, Caputo-Fabrizio, and Atangana-Baleanu fractional derivative operator. The conditions under which the model has a unique solution are established for different derivative operators. Using the numerical scheme which was proposed by Atangana and Toufik the numerical solutions are presented for the different fractional derivative operators.

In the current research work, we are going to discuss the Modified Bergman's Minimal blood glucose-insulin model.We are restructuring the old model by including one more component known as diet, which is also very essential for blood glucose model. Many years ago the minimal model was define which explain the relation between blood glucose and insulin level, and after that a lots of work and research has been done on it. We define the fractional model to show the strength of Caputo-Fabrizio fractional derivative and also examined the existence and uniqueness of the result along with its numerical and graphical representation.

In this paper, we will solve Generalized Korteweg-De Vries equation with Local fractional derivative bylocal fractional homotopy analysis method (LFHAM). The approximate solutions of non-differentiabletype with the generalized functions defined on Cantor sets are analyzed. Further, some examples andspecial cases of main results are also discussed.