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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.

Here, we use the modified Kudryashov’s algorithm and addendum to Kudryashov’s technique to obtain new optical solitons of the Ginsburg–Landau fractional complex of non-linearity (CGLE) laws in Kerr, which demonstrate various phenomena in physics such as non-linear waves, the transformation of the secondary phase, superconductivity, surfaces, liquid crystals, and field-theory strings. Dark, bright, singular solitons and combo bright-singular solitons are retrieved. The modified Kudryashov’s algorithm provides dark, singular solitons, and combo bright-singular solitons, although the addendum to Kudryashov’s technique ensures bright and singular solitons. These solutions are practiced for three optional definitions of derivative viz. conformable derivative, beta derivative, and M-truncated. Also shown are graphic representations of the solutions obtained.

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.

Up to day, the role of ultraviolet rays in producing cancers like melanoma has been considered extensively. However, quantum effects on the evolutions of melanoma has been discussed minorly. In this research, we propose a quantum model for the replication of DNAs within skin cells. We show that ultraviolet rays prevent of imaging information within genes of parent cells on genes of child cells and make some genetic errors in produced DNAs during replications. We compare the model with experimental data and obtain the growth rate of tumour on the skin.

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.

In this article, we investigated a deterministic model of pneumonia-meningitis coinfection. Employing the Atangana–Baleanu fractional derivative operator in the Caputo framework, we analyze a seven-component approach based on ordinary differential equations (DEs). Furthermore, the invariant domain, disease-free as well as endemic equilibria, and the validity of the model’s potential results are all investigated. According to controller design evaluation and modelling, the modulation technique devised is effective in diminishing the proportion of incidences in various compartments. A fundamental reproducing value is generated by exploiting the next generation matrix to assess the properties of the equilibrium. The system’s reliability is further evaluated. Sensitivity analysis is used to classify the impact of each component on the spread or prevention of illness. Using simulation studies, the impacts of providing therapy have been determined. Additionally, modelling the appropriate configuration demonstrated that lowering the fractional order from 1 necessitates a rapid initiation of the specified control technique at the largest intensity achievable and retaining it for the bulk of the pandemic’s duration.

New definitions always have more capabilities in describing phenomena than the more classic and well-known concepts because they usually appear to fill the gaps in existing concepts. In this paper, we employ a new stochastic fractional system that models interactions in a mutualism–parasitism food web. The main idea of piece-wise differential and integral operators is to separate the time under study into separate time domains in which the model behaves different behaviors. The idea was recently proposed by Atangana et al., and is gradually being used to solve a variety of application models. In order to demonstrate the valuable capabilities of this approach, many numerical simulations are provided in this paper. These results are mainly obtained from studying the role of the parameters in the model on its overall behavior. The results confirm that the piece-wise approach to the problem is a wise idea with more effectiveness rather than considering the whole situation is studied by a single operator. It is recommended to use these modified operators in modeling other applied problems with similar features.

Infectious diseases are the major source of increasing death rate during recent years. Among the infectious diseases, dengue fever is disseminated in more than hundred countries, especially in tropical and subtropical regions. In this work, a novel fractional model is presented which describes the dynamics of dengue including human and vector (mosquitoes) population. The mathematical model of dengue transmission is important to understand the dynamical behavior of the disease. A compartmental model of the dengue transmission is composed of five compartments representing the human and mosquito dynamics. The model is solved by using Adams-Bashforth-Moulton technique. The simulated results are compared with four years (2011–2014) real time data of dengue cases in Lahore, Pakistan. The fractional model gives a better approximation for α = 0.8 as compared to α = 1. The reasonable range of fractional order is between α = 0.7 and α = 1. The stability analysis of the equilibria is presented. Moreover, the positivity of the solution is proved. Furthermore, the parametric study using the outbreak dengue data for Lahore is presented. On the basis of results, we conclude that the fractional order model is more suitable than integer order equations to evaluate the transmission of dengue disease. The present study plays an important role in studying the factors to remove the communicable diseases like dengue.

This paper investigates the analytical solutions of the well-known nonlinear Schrödinger (NLS) equation with the higher-order through three members of Kudryashov methods (the original Kudryashov method, modified Kudryashov method, and generalized Kudryashov method). The considered model is also known as the sub-10-fs-pulse propagation model used to describe these measurements’ implications for creating even shorter pulses. We also discuss the problem of validating these measurements. Previous measurements of such short pulses using techniques. This paper’s aim exceeds the idea of just finding the traveling wave solution of the considered model. Still, it researches to compare the used schemes’ accuracy by applying the quintic-B-Spline scheme and the convergence between three methods. Many distinct and novel solutions have been obtained and sketched, along with different techniques to show more details of the model’s dynamical behavior. Finally, the matching between analytical and numerical schemes has been shown through some tables and figures.

This manuscript handles the nonlinear Klein–Fock–Gordon (KFG) equation by applying two recent computational schemes (generalized exponential function (GEF) and generalized Riccati expansion (GRE) methods) to construct abundant novel wave solutions The considered model is the generalized form of the well-known nonlinear Schrödinger equation which is considered a quantized version of the relativistic energy-momentum relation. The accuracy of the employed analytical schemes by showing the matching between computational and approximate solutions and calculating the absolute value of error between these solutions. This matching is investigated by employing the variational iteration (VI) method to show the precision of the used schemes with the previously published solutions. The physical characterization of the evaluated solutions has explained through some distinct sketches in 2D, 3D, contour, polar, and spherical plots. The originality and novelty of our investigation have been checked by comparing our solution’s accuracy with previous other solutions’ accuracy.

DNA could emit some waves and absorb or repel drugs. Thus, DNA waves have the main role in drug’s effects on cells. Recently, some scientists have shown that the probability for survival in patients who taken nivolumab plus ipilimumab is more than in patients who obtained ipilimumab alone. We show that the reason for these events is that some drug molecules which have more charges could be absorbed more by charges on DNA within cells. We propose a physical model for calculating the attractive forces between drugs and DNA within cells. In this model, a drug is formed from charged particles and by its motions along blood vessels, some electrical currents are emerged. From other side, DNA within cells are also formed from charged particles. Charges of DNA within cells interact with charges of drugs and consequently; DNA within cells vibrate. Consequently, a cellular current is emerged which interacts with drug currents and two electrical and magnetical forces are emerged. We show that by increasing number of charged atoms within drugs and DNA, the attractive forces between cells and drugs increase. Thus, a system which is formed from two drugs like nivolumab plus ipilimumab includes more charges respect to one drug like ipilimumab alone. Consequently, this combination is more absorbed by charges on DNA within cells and is more effective. This could be generalized to any other drug that can be used in curing cancers.

In this paper, we will propose a novel system for remote detecting COVID-19 patients based on artificial intelligence technology and internet of things (IoT) in order to stop the virus spreading at an early stage. In this work, we will focus on connecting several sensors to work together as a system that can discover people infected with the Coronavirus remotely, this will reduce the spread of the disease. The proposed system consists of several devices called smart medical sensors such as: pulse, thermal monitoring, and blood sensors. The system is working sequentially starting by pulse sensor and end by blood sensor including an algorithm to manage the data given from sensors. The pulse sensor is devoted to acquire a high quality data using a smartphone equipped by a mobile dermatoscope with 20× magnification. The processing is used RGB color system to perform moving window to segment regions of interest (ROIs) as inputs of the heart rate estimation algorithm. The heart rate (HR) estimation is then given by computing the dominant frequency by identifying the most prominent peak of the discrete Fourier transform (DFT) technique. The thermal monitoring is used for fever detection using a smart camera that can provide an optimum solution for fever detection. The infrared sensor can quickly measure surface temperature without making any contact with a person’s skin. A blood sensor is used to measure percentages of white, red blood (WBCs, RBCs) volume and platelets non-invasively using the bioimpedance analysis and independent component analysis (ICA). The proposed sensor consists of two electrodes which can be used to send the current to the earlobe and measure the produced voltage. A mathematical model was modified to describe the impedance of earlobe in different frequencies (i.e., low, medium, and high). The COMSOL model is used to simulate blood electrical properties and frequencies to measure WBCs, RBCs and Platelets volume. These devices are collected to work automatically without user interaction for remote checking the coronavirus patients. The proposed system is experimented by six examples to prove its applicability and efficiency.

We introduce entangled pair coherent states of a noncommutative harmonic oscillator operators associated with a q-deformed oscillator algebra. The definition of two-mode q-deformed operator and the decomposition of its eigenstates in terms of the q-deformed Fock states are obtained. The general solution of the recurrence relation of the first nondeformed quadrature is obtained. The asymptotic behavior of the general solution is studied, a complete expansion is derived, and also representations of the general solution in terms of the Gauss hypergeometric function are given. Also, we prove that the general solution is related to Meixner-Pollaczek orthogonal polynomial and calculate the weight function. The photons number distribution, the second order correlation function and the optical tomogram of the entangled pair coherent states are discussed. The recurrence relation for the first q-deformed quadrature is inferred. We evaluate the eigenstate of the q-deformed quadrature operator and give a numerical study for the optical tomogram of the q-deformed of the entangled pair coherent states. Finally, we show the change of the parameters in the entangled pair coherent states that connected with physical quantities and explain how they affect the format of optimal tomography.

It is shown that the following property (1) Eα,β(a(s+t)αβ)=Eα,β(asαβ)Eα,β(atαβ),s,t≥0,a∈R,α,β>0 is true only when α=β=1, and a=0,β=1 or β=2. Moreover, a new equality on Eα, β(atαβ) is developed, whose limit state as α↑1 and β > α is just the above property (1) and if β=1, then the result is the same as in [16]. Also, it is proved that this equality is the characteristic of the function tβ−1Eα,β(atα). Finally, we showed that all results in [16] are special cases of our results when β=1.