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Using discrete fractional calculus, a wide variety of physiological phenomena with various time scales have been productively investigated. In order to comprehend the intricate dynamics and activity of neuronal processing, we investigate the behavior of a slow-fast FitzHugh-Rinzel (FH-R) simulation neuron that is driven by physiological considerations via the Caputo fractional difference scheme. Taking into account the discrete fractional commensurate and incommensurate mechanisms, we speculate on the numerical representations of various excitabilities and persistent activation reactions brought about by the administered stimulation. Furthermore, the outcomes concentrate on the variability of several time scales, encompassing mixed-mode oscillations and mixed-mode bursting oscillations formed by the canard occurrence. It is confirmed that the fast-analyzing component, which was isolated within this framework with the slow-fast evaluation process, is bistable, and the criterion for bistability is added as well. The architecture appears to be bistable based on this. The pertinent factors for examining time evolution, Poincaré maps, the bifurcation configuration of the system and chaos illustrations involve the inserted power stimulation using commensurate and incommensurate fractional-order values. We investigate the canards adjacent to the folded platforms using the folded node hypothesis. Additionally, we are employing mixed-mode oscillations to illustrate the homoclinic bifurcation and the resulting chaotic trajectory. Also, we determine our research results by computing the Lyapunov spectra as an expression of time in conjunction with the dominating factor ℑ to demonstrate the chaotic behavior in a particular domain. Besides that, we estimate intricacy employing the sample entropy (Sp-En) approach and C0 complexity. The emergence of chaos within the hypothesized discrete fractional FH-R system is verified using the 0−1 criterion. Ultimately, we examine the prospective implications of mixed-mode oscillations in neuroscience and draw the inference that our observed outcomes could potentially be of great relevance. As a result, the predicted intricacy decreases while applying it to non-horizontal significant models. Finally, the simulation's characteristic phases, canards and mixed model oscillations are achieved statistically with the assistance of varying fractional orders.

Nowadays, multistability assessment in discrete nonlinear fractional difference systems is an emotive topic. This paper introduces a novel discrete, nonequilibrium, memristor-based Hindmarsh–Rose neuron (HRN) with the Caputo fractional difference scheme. Furthermore, in a setting involving commensurate and incommensurate scenarios, the complex structure of the proposed discrete fractional model, which includes its multistability, concealed chaos and hyperchaotic attractor, is examined using a wide range of computational approaches, such as Lyapunov exponents, phase portraits, bifurcation illustrations, and the 0–1 evaluation emergence of synchronization. These evolving properties imply that the fractional discrete HRN has an undetected multistability. Ultimately, an elaborate investigation is performed to confirm the existence of unpredictability employing approximation entropy (ApEn) and the ℂ0 measurements. It is demonstrated that whenever the immediate synchronization indicates that it happens, the neurons’ interactions modify, recurring in decreasing fractional orders. Furthermore, minimizing the derivative order increases the incidence of explosions in the synchronization manifold, which is opposite to the behaviour of one nerve cell.

A novel nonlinear fractional order damped Pantographs equation involving input delay is formulated. We explored the controllability of the underlying nonlinear damped dynamical Pantograph equation of fractional order with input delay. The solution of the problem under consideration has initially expressed in the form of the Mittag–Leffler function in two different domains. The controllability Grammians matrices for both cases have been defined. Controllability in the linear case has been shown and applied Schaefer’s fixed point theorem as well as the Arzela-Ascoli theorem to derive necessary conditions for the controllability of the nonlinear case.

In this paper, we investigate gross domestic product model with four different fractional derivatives. We find the solutions of fractional models by utilizing Sumudu transform. We prove the efficiency of the Sumudu transform by some theoretic outcomes and applications. We demonstrate the simulations by some figures. We show the agreement of different fractional derivatives on the model.

The nanofluid technology has added significant contributions towards heat exchange increase and decrease in energy loss. This article focuses, to analyze the heat and mass transfer for two distinct hybrid nanofluids, namely (Ag-CuO/NE), (Ag-CuO/SE), under the impact of magnetic field and Rosseland radiation over a rotating cone embedded in the porous medium. The set of non-linear partial differential equations is derived with help of boundary layer approximation. A suitable and appropriate similarity set is used to obtain a dimensionless set of equations which, are then converted into ordinary differential equations. The problem is tackled numerically with the BVP-4c technique, keeping tolerance at 10−06. The influence of flow study parameters is discussed with help of graphical and tabulated data. The study shows that (Ag-CuO/NE) hybrid nanofluid has produced better heat and mass transfer rates as compared (Ag-CuO/SE) hybrid nanofluid. Therefore, the study suggests more frequent use of (Ag-CuO/NE).

Humans have observed complex behaviors presented by nature. They have observed behaviors resembling power-law, to model these problems; they have used fractional differential operators with a power-law kernel. They have observed behaviors resembling a declining process, to model these problems; they used fractional differential operators based on exponential decay kernel. They have observed crossover behaviors from fading process to power-law, to model these problems; they used fractional derivatives based on the generalized Mittag-Leffler function. However, they have also observed a passage from randomness to power-law or vice-versa, or even from deterministic to randomness, these behaviors could not be depicted using above mentioned operators. To overcome these problems, the concepts of piece-wise differential and integral calculus have been introduced. In this paper, we adopt such a concept to model the dynamical interaction of predator–prey, several scenarios are considered and simulations are provided for each case. According to the results, it seems that the idea of using piece-wise operators seems provides a better understanding of the behavior of dynamic systems.

The main objective of this paper is to retrieve the analytical solutions for a perturbed nonlinear Schrödinger’s equation. These achievements are obtained by following the basic steps of an efficient approach. The method we have used for the model is a completely novel and powerful technique that has not been used for the model so far. The basic common idea of the method is to convert the structure of the equation from a partial differential equation to an ordinary one using a new soliton wave variable. During the implementation of these methods, a variety of explicit solutions for different structures are extracted. For the sake of better understanding the dynamics of the solutions, several diagrams related to the introduced solutions are also plotted. We conclude that the employed method is an efficient tool for solving partial differential equations that paves the way to solve other practical models in the field. The software used in this article is Wolfram Mathematica.

This paper aims to establish conditions for the existence, uniqueness and Ulam–Hyers stability of solutions for a coupled system of pantograph problem with three sequential fractional derivatives. Two results on the uniqueness and existence of solutions are proved to utilize the Leray–Schauder and Banach fixed point theorems and positive contraction-type inequalities. Also, the stability in the sense of Ulam–Hyers and Ulam–Hyers–Rassias are studied. An illustrative example with graphical and numerical simulations is also proposed.

In the present article, we have utilized two efficient techniques to obtain new optical solitons for the nonlinear Schrödinger’s equation. The advantages of the employed methods are that they are straightforward to use and able extraction of different types of solutions in single frameworks. The solutions presented in this paper evoke new applications and concepts for the considered special form of Schrödinger’s equation. Numerical simulations of some of the acquired solutions are also presented in the paper. After applying some necessary changes in these techniques, they can be implemented in solving other applied complex models arising in mathematics, physics, and applied mathematics.

In this paper, we provide a generic mathematical framework for scabies transmission mechanisms. The infections involving susceptible, highly contagious people and juvenile scabiei mites are characterized by a framework of ordinary differential equations (DEs). The objective of this study is to examine the evolution of scabies disease employing a revolutionary configuration termed a fractal–fractional (FF) Atangana–Baleanu (AB) operator. Generic dynamical estimates are used to simulate the underlying pace of growth of vulnerable people, clinical outcomes, and also the eradication and propagation rates of contaminated people and immature mites. We study and comprehend our system, focusing on a variety of restrictions on its basic functionalities. The model’s outcomes are assessed for positivity and boundedness. The formula includes a fundamental reproducing factor, R0, that ensures the presence and stability of all relevant states. Furthermore, the FF-AB operator is employed in the scabies model, and its mathematical formulation is presented using a novel process. We analyze the FF framework to construct various fractal and fractional levels and conclude that the FF theory predicts the affected occurrences of scabies illness adequately. The relevance and usefulness of the recently described operator has been demonstrated through simulations of various patterns of fractal and fractional data.

Partial differential equations are among the most important mathematical tools for scientists in describing many physical and engineering problems related to real life. Accordingly, in this work, an efficient technique is proposed for getting different collections of solutions to a third-order version of nonlinear Schrödinger’s equation. Some numerical simulations related to the acquiblack solutions are also provided in this research. All results presented in this article can be consideblack as new achievements for the model. Further, it is emphasized that the used technique enables us to study other forms of nonlinear models. It is obtained that the technique is a reliable tool in handling many nonlinear partial differential equations arising in engineering, fluid mechanics, nonlinear optics, oceans, seas, and many mathematical physics. Moreover, the present results help the plasma physics researchers for investigating many nonlinear modulated structures that can generate and propagate in laboratory and space plasmas.

The main purpose of the present paper is to introduce a reliable method, for the first time, in solving differential equations with partial derivatives. The significant idea behind this method is the modification of a well-known method. The main base functions used in the solution’s structures are Jacobi elliptic functions which are known functions and have many applications in practice. In order to achieve these results, the perturbed Schrödinger equation with Kerr law nonlinearity is considered. This nonlinear equation illustrates the propagation of optical solitons in nonlinear optical fibers. Thereupon, several analytical solutions corresponding to this model are obtained through employing an improved generalized exponential rational function method. Then we present a new version of it that is presented for the first time in the literature. Using these methods, several wave solutions related to the considered model are obtained. Moreover, several numerical simulations relevant to the resulting solutions are performed in the paper. As a notable features of the proposed method, the obtained solutions are characterized by the Jacobi elliptic functions. These solutions contain an index that for some specific choices is reduced to standard functions such as trigonometric and hyperbolic. The obtained results and solutions can be used for investigating the mechanism of several nonlinear phenomena laboratory and space plasmas. All necessary calculations are performed via symbolic packages of Wolfram Mathematica.