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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.

A system of three nonlinear equations that depict chaotic behavior was considered in this work. We looked at four cases: one with a power-law property; one with a fading memory effect; one with a transition from stretched exponential to power-law; and one with a stochastic component. We have presented a detailed analysis of stability and we have suggested a number similar to the reproductive number in the case of epidemiology. At this stage, we did not provide a physical meaning for such a number. However, such a number may be useful. A numerical scheme based on the Lagrange polynomial interpolation was used to solve the respective models numerically. Numerical simulations were performed for different values of fractional orders and densities of randomness and piecewise ideas.

A work on a mathematical modeling is very popular in applied sciences. Nowadays many mathematical models have been considered and new methods have been used for approaching of these models. In this paper we are considering mathematical modeling of nuclear family model with fractional order Caputo derivative. Also the existence and uniqueness results and numerical scheme are given with Adams-Bashforth scheme via fractional order Caputo derivative.

A new concept of dynamical system of predator-prey model is presented in this paper. The model takes into account the memory of interaction between the prey and predator due to the inclusion of fractional differentiation. In addition, the model takes into account the inherent disposition of a prey or predator toward hunting or defending in time. Analysis of existence and uniqueness of the solutions is presented. A numerical method is used to generate some simulations as the fractional orders change from one to zero. A new traveling waves patterns are obtained.

In this paper, modeling the heat flow equation with fractional-fractal differentiation is considered. This problem has opened a new viewpoint for modeling the classical and the fractional differentiation. We presented the existence of positive solution of the new model using the fixed-point approach and we established the uniqueness of the positive solution. Finally, we provide an example to illustrate one of the main results.

A nonlinear system of two fractional nonlinear differential equations with Atangana-Baleanu derivative is considered in this work. General conditions under which a system solution exists and unique are presented using the fixed-point theorem method. The well-established numerical scheme is used to solve the system of equations. A numerical analysis is presented to secure the stability and convergence of the used numerical scheme.

Very recently, a new concept was introduced to capture crossover behaviors that exhibit changes in patterns. The aim was to model real-world problems exhibiting crossover from one process to another, for example, randomness to a power law. The concept was called piecewise calculus, as differential and integral operators are defined piece wisely. These behaviors have been observed in the spread of several infectious diseases, for example, tuberculosis. Therefore, in this paper, we aim at modeling the spread of tuberculosis using the concept of piecewise modeling. Several cases are considered, conditions under which the unique system solution is obtained are presented in detail. Numerical simulations are performed with different values of fractional orders and density of randomness.

Diabetes mellitus, at the forefront of the diseases of our age, is a type of disease that plays a leading role in the formation of many deadly diseases and is very common all over the world. In this work, based on the system in Shabestari et al. (Chaos Solitons Fractals 112:44–51, 2018), we give the fractional glucose–insulin regulatory system for the first time by Caputo–Fabrizio derivative. We present special solutions, stability analysis, uniqueness of the solution for this fractional system with the aid of Banach fixed point theory. Taking various values belonging to the fractional order into consideration, we present some numerical results according to the relation between glucose, insulin and $$\beta $$ -cells.

Adams-Bashforth was recognized as powerful numerical method to solve linear and non-linear ordinary differential equations. Nevertheless the method was applicable only for ordinary differential equations mostly with integer order. Atangana and Batogna have extended this method for partial differential equation with the Atangana-Baleanu fractional derivative. In this paper, to accommodate partial differential equation with Caputo-Fabrizio derivative, we suggest the corresponding method with this derivative. We applied the method to solve numerically a very interesting non-linear partial differential equation accounting for the motion of a viscous fluid. Some simulations are presented to test the efficiency of the numerical method.

Recently Hristov using the concept of a relaxation kernel with no singularity developed a new model of elastic heat diffusion equation based on the Caputo-Fabrizio fractional derivative as an extended version of Cattaneo model of heat diffusion equation. In the present article, we solve exactly the Cattaneo-Hristov model and extend it by the concept of a derivative with non-local and non-singular kernel by using the new Atangana-Baleanu derivative. The Cattaneo-Hristov model with the extended derivative is solved analytically with the Laplace transform, and numerically using the Crank-Nicholson scheme.

Recently, Atangana and Baleanu proposed a derivative with fractional order to answer some outstanding questions that were posed by many researchers within the field of fractional calculus. Their derivative has a non-singular and nonlocal kernel. In this paper, we presented further relationship of their derivatives with some integral transform operators. New results are presented. We applied this derivative to a simple nonlinear system. We show in detail the existence and uniqueness of the system solutions of the fractional system. We obtain a chaotic behavior which was not obtained by local derivative.

The model of Ebola spread within a targeted population is extended to the concept of fractional differentiation and integration with non-local and non-singular fading memory introduced by Atangana and Baleanu. It is expected that the proposed model will show better approximation than the models established before. The existence and uniqueness of solutions for the spread of Ebola disease model is given via the Picard-Lindelof method. Finally, numerical solutions for the model are given by using different parameter values.

In this work, we considered the q-differential equations of order 0 x ( 0 ) = x 0 . 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.

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.

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.

In this paper we first give a new integer order mathematical model of nuclear family. Then, we offer a fractional order mathematical model and investigate the stability of solutions. Finally, we give examples to illustrate our results.

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 < 2α ≤ 2. Stability analysis of the fractional-order nonlinear dynamical model of involving two persons is studied using the fractional Routh-Hurwitz criteria. By using stability analysis on fractional-order system, we obtain sufficient condition on the parameters for the locally asymptotic stability of equilibrium points. Finally, numerical simulations are presented to verify the obtained results.

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.

In this paper, a fractional order nonlinear dynamical model of interpersonal relationships has been introduced. The stability of equilibrium points is studied. Numerical simulations are also presented to verify the obtained results.