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In this paper, Witte's conditions for the uniqueness solution of nonlinear differential equations with integer and non-integer order derivatives are investigated. We present a detailed analysis of the uniqueness solutions of four classes of nonlinear differential equations with nonlocal operators. These classes include classical and fractional ordinary differential equations in fractal calculus. For each case, theorems and lemmas and their proofs are presented in detail.

In this work, a set of nonlinear equations capable of describing the transit of the membrane potential's spiking-bursting process which is shown in experiments with a single neuron was taken into consideration. It is well known that this system, which is built on dynamical dimensionless variables, can reproduce chaos. We arrived at the chaotic number after first deriving the equilibrium point. We added different nonlocal operators to the classical model's foundation. We gave some helpful existence and uniqueness requirements for each scenario using well-known theorems like Lipchitz and linear growth. Before using the numerical solution on the model, we analyzed a general Cauchy issue for several situations, solved it numerically and then demonstrated the numerical solution's convergence. The results of numerical simulations are given.

Recently, it was proposed to use a brand-new set of nonlinear ordinary differential equations. The system aims to represent chaotic financial activities. Such a system was taken into consideration for various analyses in this paper. For each of the three axes, we first evaluated the nullcline points and then gave the formula for the associated Poincare mapping. With various differential operators, we have analyzed the existence and uniqueness of systems of solutions. We have numerically solved the model using the well-known Nystrom in the case of the classical model and the Midpoint in the case of fractional derivatives.

Academic achievement is very important, as it enables students to be well-equipped for professional and social life and shapes their future. In the case of a possible academic failure, students generally face many cognitive, social, psychological, and behavioral problems. This problem experienced by the students has been addressed with the mathematical model in this study. This mathematical model will be handled with the help of the fractional operator, and the existence, uniqueness, and positivity of the solutions to the model equation system will be examined. In addition, local and global stability analyses of the equilibrium points of the model are planned. Numerical simulations are performed with different values of fractional orders and densities of randomness. This study is very important in terms of its original and multidisciplinary approach to a subject in the field of social sciences.

A set of nonlinear ordinary differential equations has been considered in this paper. The work tries to establish some theoretical and analytical insights when the usual time-deferential operator is replaced with the Caputo fractional derivative. Using the Caratheodory principle and other additional conditions, we established that the system has a unique system of solutions. A variety of well-known approaches were used to investigate the system. The stochastic version of this system was solved using a numerical approach based on Lagrange interpolation, and numerical simulation results were produced.

Stochastic and fractional differentiation have been developed independently to depicting processes following randomness and power, a declining memory and passage from one process to another respectively. Very recently, fractional stochastic differential equations were suggested with the aim to capture processes following at the same time randomness and memory nonlocality. In this paper to further explore the applicability of this type of differential equations, a SIR model was considered and analyzed analytically and numerically. Some numerical simulations are presented for different values of fractional orders and densities of randomness.

Abstract A numerical analysis of the well-known linear partial differential equation describing the relativistic wave is presented in this work. Three different operators of fractional differentiation with power law, exponential decay law and Mittag-Leffler law are employed to extend the Klein-Gordon equation with mass parameter to the concept of fractional differentiation. The three models are solved numerically. The stability and the convergence of the numerical schemes are investigated in detail.

In this paper we investigate a possible applicability of the newly established fractional differentiation in the field of epidemiology. To do this we extend the model describing the Rubella spread by replacing the time derivative with the time fractional derivative for the inclusion of memory. Detailed analysis of existence and uniqueness of exact solution is presented using the Banach fixed point theorem. Finally some numerical simulations are showed to underpin the effectiveness of the used derivative.

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.

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.

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.

A fractional-order nonlinear dynamical model of couple has been introduced. Upper bounds are obtained for a fractional-order nonlinear dynamical model. Also different from other models, a model with the order 2α is discussed. We are expecting an acceleration in feelings; that is why we increase the order of the derivative between 1

The main objective of this study is to investigate the consumption behaviour and consumption awareness of energy drink among university students. Data were collected from 750 students in Afyon Kocatepe University by questionnaire which is improved by writers of this article. Tests were done with SPSS end of the research. Chi-square tests were done in %95 confidence interval to determine the relation of consumption and awareness of energy drink among the university students with gender, age range, school which is graduated, monthly expense, smoking and use of alcohol, the most consumed beverage types, and degree of licence. According to search results, it was found that energy drink consumption behaviour did not change with regard to the different age. On the other hand, male college graduates compared to other types of high school, it was determined that 701 TL per month and over spenders in relation to the lower income groups consumed more energy drinks . In addition, it was found that the groups that use alcohol, smokers, coffee drinkers and undergraduate students were consumed much more energy drinks than the others. When the answers measured the energy drink consumption behaviour of the students participated in the survey were considered, it was concluded that the awareness of the energy drink consumption was not high enough.