Ilknur Koca

A method for solving differential equations of q-fractional order

In this work, we considered the q-differential equations of order 0 < q < 1 with initial condition x(0)=x0. The derivative with fractional order considered here is the Caputo time since it allows the use of initial conditions. An analytical method to obtain exact solution of this class of fractional differential equations is provided. Some illustrative examples are provided to underpin and illustrate the efficiency of the used method.

Analysis of a new model of H1N1 spread: Model obtained via Mittag-Leffler function

In the recent decades, many physical problems were modelled using the concept of power law within the scope of fractional differentiations. When checking the literature, one will see that there exist many formulas of power law, which were built for specific problems. However, the main kernel used in the concept of fractional differentiation is based on the power law function x−λ It is quick important to note all physical problems, for instance, in epidemiology. Therefore, a more general concept of differentiation that takes into account the more generalized power law is proposed. In this article, the concept of derivative based on the Mittag-Leffler function is used to model the H1N1. Some analyses are done including the stability using the fixed-point theorem.

A new nonlinear triadic model of predator–prey based on derivative with non-local and non-singular kernel

The model of predator–prey has been used by many researchers to predict the animal growth population in many countries in the world. However, the system of equations used in these models assumes a density of prey and predator, respectively, but when looking at the real-world situation, most of the time we have some prey that act at the same time like a predator, for instance, hyena, and also some predators that act as a prey, for instance, lions. In this research, we proposed a new model of triadic prey–prey–predator. The new model was constructed using the new fractional differentiation based on the generalized Mittag-Leffler function due to the non-locality of the dynamical system of the three species. We presented the existence of a positive set of the solutions for the new model. The uniqueness of the positive set of the solutions was presented in detail. The new model was solved numerically using the Crank–Nicolson numerical scheme.

Novel analysis of the fractional Zika model using the Adams type predictor-corrector rule for non-singular and non-local fractional operators

A mathematical system of equations using the concept of fractional differentiation with non-local and non-singular kernel has been analysed in this work. The developed mathematical model is designed to portray the spread of Zika virus within a given population. We presented the equilibrium point and also the reproductive number. The model was solving analytically using the Adams type predictor-corrector rule for Atangana-Baleanu fractional integral. The existence and uniqueness exact solution was presented under some conditions. The numerical replications were also presented. c ©2017 All rights reserved.

On Uncertain-Fractional Modeling: The Future Way of Modeling Real-World Problems

It has been a long time a challenge for many researchers to give a real interpretation of derivatives with fractional order. Some researchers said, fractional derivative is the shadow on the wall. This interpretation was wrong since the shadow of any object does not provide the real properties of the real object, for instance a black man has the same shadow with a white man. Using the definition and applications of a convolution, we gave new interpretation of derivative with fractional order. We gave specific interpretation for Caputo and Caputo–Fabrizio types as the fractional order changes. It was long believed that, the derivative with fractional order portray the effect of memory, this was only proved to be true in theory of elasticity and nowhere else. In this chapter, we introduced a new operator called uncertain derivative capable or portraying the memory effect in almost all situation. In order to include into mathematical formulation, the real rate of change and also the effect of memory, we introduced a new way of modeling real-world problem called uncertain-fractional modeling (UFM) and applied it to advection dispersion model. Numerical simulations of the new model show that real-world observation. This method will be the future way of modeling real-world problem efficiently.

New nonlinear model of population growth

The model of population growth is revised in this paper. A new model is proposed based on the concept of fractional differentiation that uses the generalized Mittag-Leffler function as kernel of differentiation. The new model includes the choice of sexuality. The existence of unique solution is investigated and numerical solution is provided.