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We revisited the characteristic features of Yao-Cheng non-linear oscillator by employing both analytical as well as numerical approaches. Our study indicates that the solution of the non-linear oscillator shows oscillatory behaviour. Further, the phase portrait for this oscillator system reflects the closed loop signifying the stability of the system. Our results suggest the absence of damping in this physical system.

In this paper, we derive a new version of L1-Predictor–Corrector (L1-PC) method by using some previously given methods (L1-PC for single delay, PC for non-delay, and decomposition algorithm) to solve multiple delay-type fractional differential equations. The Caputo fractional derivative with singular type kernel is used to establish the results. Some important remarks related to the delay term estimation and error analysis are mentioned. In order to check the accuracy and correctness of our method, we solve a neural network system with two delay parameters. A number of graphs are given to justify the role of delays as well as the accuracy of the algorithm. The given method is fully novel and reliable to solve multiple delay type fractional order systems in Caputo sense.

In this paper, we solved a model of a well-known infectious disease called dengue fever via fractional natural decomposition and modified Predictor–Corrector (PC) methods. A study of the dengue epidemic in the Cape Verde Islands off the coast of West Africa in 2009 has been resumed here for a better understanding of the results. The results are obtained using Liouville–Caputo and new generalized Caputo-type fractional derivatives. The numerical simulations are presented for various orders of given derivatives. Existence and uniqueness analysis of the given problem are also performed in the new generalized Caputo sense. The explored results are verified using figures. The main target of this paper is to explore the different dynamics of the given dengue fever model via two types of fractional numerical algorithms.

This article considers a fractional-order neuron model under an electromagnetic field in terms of generalized Caputo fractional derivatives. The motivation for incorporating fractional derivatives in the previously proposed integer-order neuron model is that the fractional-order model impresses with efficient effects of the memory, and parameters with fractional orders can increase the model performance by amplifying a degree of freedom. The results on the uniqueness of the solution for the proposed neuron model are established using well-known theorems. The given model is numerically solved by using a generalized version of the Euler method with stability and error analysis. Several graphical simulations are performed to capture the variations in the membrane potential considering no electromagnetic field effects, various frequency brands of external forcing current, and the amplitude and frequency of the external magnetic radiation. The impacts of fractional-order cases are clearly justified.

The first reported case of coronavirus disease (COVID-19) in Brazil was confirmed on 25 February 2020 and then the number of symptomatic cases produced day by day. In this manuscript, we studied the epidemic peaks of the novel coronavirus (COVID-19) in Brazil by the successful application of Predictor–Corrector (P-C) scheme. For the proposed model of COVID-19, the numerical solutions are performed by a model framework of the recent generalized Caputo type non-classical derivative. Existence of unique solution of the given non-linear problem is presented in terms of theorems. A new analysis of epidemic peaks in Brazil with the help of parameter values cited from a real data is effectuated. Graphical simulations show the obtained results to classify the importance of the classes of projected model. We observed that the proposed fractional technique is smoothly work in the coding and very easy to implement for the model of non-linear equations. By this study we tried to exemplify the roll of newly proposed fractional derivatives in mathematical epidemiology. The main purpose of this paper is to predict the epidemic peak of COVID-19 in Brazil at different transmission rates. We have also attempted to give the stability analysis of the proposed numerical technique by the reminder of some important lemmas. At last we concluded that when the infection rate increases then the nature of the diseases changes by becoming more deathly to the population.

In this article, we propose generalized forms of three well-known fractional numerical methods namely Euler, Runge–Kutta 2-step, and Runge–Kutta 4-step, respectively. The new versions we provide of these methods are derived by utilizing a non-uniform grid which is slightly different from previous versions of these algorithms. A new generalized form of the well-known Caputo-type fractional derivative is used to derive the results. All necessary analyses related to the stability, convergence, and error bounds are also provided. The precision of all simulated results is justified by performing multiple numerical experiments, with some meaningful problems solved by implementing the code in Mathematica. Finally, we give a brief discussion on the simulated results which shows that the generalized methods are novel, effective, reliable, and very easy to implement.

In December 2019, the novel Coronavirus, also known as 2019-nCoV or SARS-CoV-2 or COVID-19, was first recognized as a deadly disease in Wuhan, China. In this paper, we analyze two different nonclassical Coronavirus models to observe the outbreaks of this disease. Caputo and Caputo–Fabrizio (C–F) fractional derivatives are considered to simulate the given epidemic models by using two separate methods. We perform all required graphical simulations with the help of real data to demonstrate the behavior of the proposed systems. We observe that the given schemes are highly effective and suitable to analyze the dynamics of Coronavirus. We find different natures of the given model classes for both Caputo and C-F derivative sense. The main contribution of this study is to propose a novel framework of modeling to show how the fractional-order solutions can describe disease dynamics much more clearly as compared to integer-order operators. The motivation to use two different fractional derivatives, Caputo (singular-type kernel) and Caputo–Fabrizio (exponential decay-type kernel) is to explore the model dynamics under different kernels. The applications of two various kernel properties on the same model make this study more effective for scientific observations.

The citrus epidemic huanglongbing (HLB), allied with an uncultured bacterial pathogen, is blusterous the industry of citrus worldwide. In this research work, we analysed a fractional huanglongbing model to study the transmission dynamics of the disease. We considered Caputo and new generalized form of Caputo type fractional derivatives to solve the proposed HLB model using two different methods. We exemplified the all necessary graphical observations by the call of real numerical data to show the nature of the given model classes. The analysis regarding to existence of the solution are mentioned with the help of theorems. We observed that the given numerical techniques are strong and worked well to show the dynamics of the model against the time variable.

Study of ecosystems has always been an interesting topic in the view of real-world dynamics. In this paper, we propose a fractional-order nonlinear mathematical model to describe the prelude of deteriorating quality of water cause of greenhouse gases on the population of aquatic animals. In the proposed system, we recall that greenhouse gases raise the temperature of water, and because of this reason, the dissolved oxygen level goes down, and also the rate of circulation of disintegrated oxygen by the aquatic animals rises, which causes a decrement in the density of aquatic species. We use a generalized form of the Caputo fractional derivative to describe the dynamics of the proposed problem. We also investigate equilibrium points of the given fractional-order model and discuss the asymptotic stability of the equilibria of the proposed autonomous model. We recall some important results to prove the existence of a unique solution of the model. For finding the numerical solution of the established fractional-order system, we apply a generalized predictor–corrector technique in the sense of proposed derivative and also justify the stability of the method. To express the novelty of the simulated results, we perform a number of graphs at various fractional-order cases. The given study is fully novel and useful for understanding the proposed real-world phenomena.

The deadly coronavirus continues to spread across the globe, and mathematical models can be used to show suspected, recovered, and deceased coronavirus patients, as well as how many people have been tested. Researchers still do not know definitively whether surviving a COVID-19 infection means you gain long-lasting immunity and, if so, for how long? In order to understand, we think that this study may lead to better guessing the spread of this pandemic in future. We develop a mathematical model to present the dynamical behavior of COVID-19 infection by incorporating isolation class. First, the formulation of model is proposed; then, positivity of the model is discussed. The local stability and global stability of proposed model are presented, which depended on the basic reproductive. For the numerical solution of the proposed model, the nonstandard finite difference (NSFD) scheme and Runge-Kutta fourth order method are used. Finally, some graphical results are presented. Our findings show that human to human contact is the potential cause of outbreaks of COVID-19. Therefore, isolation of the infected human overall can reduce the risk of future COVID-19 spread.

WOS: 000507424500001 In this paper, a numerical approach is suggested to find a semi-analytical solution for the common nonlinear boundary value problems (BVPs) of cantilever-type micro-electromechanical system (MEMS) and nano-electromechanical system (NEMS) with a set of model parameters. The nonlinear BVPs that are studied involve the states of the single and double cantilever-shaped beams under the influence of Casimir and Van der Waals force for proper distances of separation. The method is based upon an integral operator that is formed considering Green's function connected with the execution of Picard's or Mann's fixed point schemes. The numerical results for different cases of beam are presented and compared with those obtained by previous works to show the applicability, efficiency, and high accuracy of the suggested method.

An emerging treatment tool which uses replication-competent viruses to dissipate cancers without causing deficit to normal tissues, named as oncolytic virotherapy, is discussed in the article. We analysed a fractional delay dynamical model on the oncolytic virotherapy compositing viral lytic cycle and virus-specific cytotoxic T lymphocyte (CTL) response. We used a well known Caputo fractional derivative to analyse the structure of the given dynamical model. Using the literature of fixed-point theory, the given time-delay model is specified to have existence of a unique solution. We established different types of graphical simulations for the various values of R 0 and R 1 . We observed a different behaviour of the given fractional model as compare to the integer order model. The given algorithm is smooth in use and reliable to apply on different delay dynamical models.

We analyze a time-delay Caputo-type fractional mathematical model containing the infection rate of Beddington–DeAngelis functional response to study the structure of a vector-borne plant epidemic. We prove the unique global solution existence for the given delay mathematical model by using fixed point results. We use the Adams–Bashforth–Moulton P-C algorithm for solving the given dynamical model. We give a number of graphical interpretations of the proposed solution. A number of novel results are demonstrated from the given practical and theoretical observations. By using 3-D plots we observe the variations in the flatness of our plots when the fractional order varies. The role of time delay on the proposed plant disease dynamics and the effects of infection rate in the population of susceptible and infectious classes are investigated. The main motivation of this research study is examining the dynamics of the vector-borne epidemic in the sense of fractional derivatives under memory effects. This study is an example of how the fractional derivatives are useful in plant epidemiology. The application of Caputo derivative with equal dimensionality includes the memory in the model, which is the main novelty of this study.

In this research collection, we estimate the existence of the unique solution for the boundary value problem of nonlinear fractional [Formula: see text]-difference equation having the given form [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] represents the Caputo-type nonclassical [Formula: see text]-derivative of order [Formula: see text]. We use well-known principal of Banach contraction, and Leray–Schauder nonlinear alternative to vindicate the unique solution existence of the given problem. Regarding the applications, some examples are solved to justify our outcomes.

[EN] In this study, our aim is to explore the dynamics of COVID-19 or 2019-nCOV in Argentina considering the parameter values based on the real data of this virus from March 03, 2020 to March 29, 2021 which is a data range of more than one complete year. We propose a Atangana-Baleanu type fractional-order model and simulate it by using predictor-corrector (P-C) method. First we introduce the biological nature of this virus in theoretical way and then formulate a mathematical model to define its dynamics. We use a well-known effective optimization scheme based on the renowned trust-region-reflective (TRR) method to perform the model calibration. We have plotted the real cases of COVID-19 and compared our integer-order model with the simulated data along with the calculation of basic reproductive number. Concerning fractional-order simulations, first we prove the existence and uniqueness of solution and then write the solution along with the stability of the given P-C method. A number of graphs at various fractional-order values are simulated to predict the future dynamics of the virus in Argentina which is the main contribution of this paper., The third author is supported by MICINN and FEDER, Project PID2019-105011GB-I00.

The topic of coupled oscillations is rich in physical content which is both interesting and complex. In this research work we aimed to study the kinematics behaviour of an important physical system known as coupled pendulum, in which two identical pendulums were coupled together by a spring. The Lagrangian of the system was first constructed, then the equations of motion have been derived. Analytical solutions were obtained for these equations, and in addition we apply a reliable algorithm based on an adaptation of the standard differential transformation method is presented, which is the multi-step differential transformation method to obtain numerical solutions for the considered system for some selected initial conditions. The solutions of the equations of motion were obtained by the multi-step differential transform method. Of course, this method is needed if it predicts the motion of the considered system. For this reason figurative comparisons between the multi-step differential transformation method, the standard differential transformation method and the classical fourth-order Runge–Kutta method are given. The results reveal that the proposed technique is a promising tool to predict the behaviours of the considered system in long time interval. Moreover, the motion of curves represents the initial conditions that sensitive depend and how much oscillatory motion. In conclusion, the proposed technique is a reliable method to solve the considered double pendulum problem in long time interval. We believe that the system considered in this work along with the method used is interesting for physicists, mathematicians and engineers.

Several deadly epidemics that have recognized as serious problems all over the world in the last few decades. Lassa hemorrhagic fever, coronavirus, dengue fever, malaria, and HIV are well-known deadly diseases in humans. In this research, we analysed the dynamics of the canine distemper virus (CDV) and rabies epidemics in the red fox population of the northern region of Italy with the help of time-fractional models. We performed our analysis in the new generalized Caputo non-classical derivative sense with the application of the Predictor–Corrector algorithm. We used the data of northern Italy for simulations and estimated the endemic equilibrium points for both CDV and rabies models. Also, we presented the local stability of disease-free equilibrium points. Some theorems are mentioned for the purpose of existence and uniqueness analysis. Our results are perfect for giving an idea of the dynamics of the CDV and rabies epidemic in northern Italy. The dynamics of the given solutions are specified with the help of necessary graphical simulations. The projected algorithm is so effective in finding the solutions of complex dynamical systems. By this study, we give an idea of how applied mathematics is directly connected to biological studies. The major scientific aim of this study is to understand the outbreaks of CDV and rabies on the population of the red foxes by using the texture of fractional mathematical models.

In this study, a practical jointed approach in the static vibration investigation of functionally graded material (FGM) structures under internal pressure is applied by differential transform method (DTM). The FGM material consist of isotropic material that show exponential variation in the thickness. The ratio of Poisson is taken as a constant. Displacement and stress distributions due to non-homogeneous constant are intended. Stress distribution dependent on the homogeneity parameter is computed and the results obtained for cylindrical and spherical structures were compared with finite element method (FEM). The inhomogeneity parameter is empirically regulated, with a continuously varying volume fraction of the constituents. The parameters for homogeneity were randomly selected to show displacement and stress distributions.

Solutions to nonlinear boundary value problems modelling physical phenomena in engineering applications have traditionally been approximated using numerical methods. More recently, several semi-analytical methods have been developed and used extensively in diverse engineering applications. This work compares the method of variation of parameters to semi-analytical methods for solving nonlinear boundary value problems arising in engineering. The accuracy and efficiency of the method of variation of parameters are compared to those of two widely used semi-analytical methods, the Adomian decomposition method and the differential transformation method, for three practical engineering boundary value problems: (1) the deflection of a cantilevered beam with a concentrated load, (2) an adiabatic tubular chemical reactor which processes an irreversible exothermal reaction, and (3) the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylinder conduit. The accuracy and convergence of each method is investigated using the error remainder function. The method of variation of parameters is significantly more efficient than both the semi-analytical methods and traditional numerical methods while maintaining comparable accuracy. Unlike the semi-analytical methods, the efficiency of variation of parameters is independent of the nonlinearity. Variation of parameters is shown to be an attractive alternative to semi-analytical methods and traditional numerical methods for solving boundary value problems encountered in engineering applications in which solution efficiency is important.

In this research collection, we analysed two different fractional non-linear mathematical models of a well-known mosaic epidemic of plants, which is underlying by begomoviruses and is distributed to plants by whitefly. We included the role of natural microbial biostimulants which are used to increase plant performance and protects them against mosaic infection. Cause of the big expansion of the mosaic epidemic in various geographical areas, and its large privative economic and societal impacts, it is of major consequence to define dominant optimal control means of this disease. In this paper, we used Caputo (singular type kernel) and Atangana–Baleanu (Mittag-Leffler type kernel) fractional derivatives to define the structure of the proposed mosaic model. We performed some important existence and uniqueness analyses for both models by the applications of fixed point theory and the Picard–Lindelof technique. We derived the numerical solution of the Caputo fractional model by the application of the fourth-order Runge–Kutta method and the Atangana–Baleanu model by the Predictor–Corrector algorithm. A long-term discussion on the graphical interpretations of both models with different infection transmission rate and application proportion rate of MBs (microbial biostimulants) at different fractional-order values have established. We exemplified that under the case of the Mittag-Leffler kernel, the effects of different fractional-order values are much clear as compared to the singular type kernel. The main contribution of this paper is to study the dynamics of mosaic disease at different transmission rates and MBs application rates in the sense of two different kernel types.

Highlights • A compartmental mathematical model has been proposed in this paper incorporating all possible non-pharmaceutical intervention policies. • Short-term projection results have been illustrated for Bangladesh and India. • All necessary graphical simulations have been performed with the help of Caputo-Fabrizio fractional derivatives. • Optimal control problem for fractional system has been designed and the existence of unique solution has also been showed by using Picard-Lindelof technique., When the entire world is eagerly waiting for a safe, effective and widely available COVID-19 vaccine, unprecedented spikes of new cases are evident in numerous countries. To gain a deeper understanding about the future dynamics of COVID-19, a compartmental mathematical model has been proposed in this paper incorporating all possible non-pharmaceutical intervention strategies. Model parameters have been calibrated using sophisticated trust-region-reflective algorithm and short-term projection results have been illustrated for Bangladesh and India. Control reproduction numbers (Rc) have been calculated in order to get insights about the current epidemic scenario in the above-mentioned countries. Forecasting results depict that the aforesaid countries are having downward trends in daily COVID-19 cases. Nevertheless, as the pandemic is not over in any country, it is highly recommended to use efficacious face coverings and maintain strict physical distancing in public gatherings. All necessary graphical simulations have been performed with the help of Caputo-Fabrizio fractional derivatives. In addition, optimal control strategies for fractional system have been designed and the existence of unique solution has also been showed using Picard-Lindelof technique. Finally, unconditional stability of the fractional numerical technique has been proved.

[EN] The most dangerous disease of this decade novel coronavirus or COVID-19 is yet not over. The whole world is facing this threat and trying to stand together to defeat this pandemic. Many countries have defeated this virus by their strong control strategies and many are still trying to do so. To date, some countries have prepared a vaccine against this virus but not in an enough amount. In this research article, we proposed a new SEIRS dynamical model by including the vaccine rate. First we formulate the model with integer order and after that we generalize it in Atangana-Baleanu derivative sense. The high motivation to apply Atangana-Baleanu fractional derivative on our model is to explore the dynamics of the model more clearly. We provide the analysis of the existence of solution for the given fractional SEIRS model. We use the famous Predictor-Corrector algorithm to derive the solution of the model. Also, the analysis for the stability of the given algorithm is established. We simulate number of graphs to see the role of vaccine on the dynamics of the population. For practical simulations, we use the parameter values which are based on real data of Spain. The main motivation or aim of this research study is to justify the role of vaccine in this tough time of COVID-19. A clear role of vaccine at this crucial time can be realized by this study., The third author is supported by MEC, Spain, grants MTM2016-75963-P and PID2019105011GBI00

When the entire world is eagerly waiting for a safe, effective and widely available COVID-19 vaccine, un-precedented spikes of new cases are evident in numerous countries. To gain a deeper understanding about the future dynamics of COVID-19, a compartmental mathematical model has been proposed in this paper incorporating all possible non-pharmaceutical intervention policies. Model parameters have been calibrated using sophisticated trust-region-reflective algorithm and short-term projection results have been illustrated for Argentina, Bangladesh, Brazil, Colombia and India. Control reproduction numbers (ℛc) have been calculated in order to get insights about the current epidemic scenario in the above-mentioned countries. Forecasting results depict that the aforesaid countries are having downward trends in daily COVID-19 cases. However, it is highly recommended to use efficacious face coverings and maintain strict physical distancing, as the pandemic is not over in any country. Global sensitivity analysis enlightens the fact that efficacy of face coverings is the most significant parameter, which could significantly control the transmission dynamics of the novel coronavirus compared to other non-pharmaceutical measures. In addition, reduction in effective contact rate with isolated patients is also essential in bringing down the epidemic threshold (ℛc) below unity. All necessary graphical simulations have been performed with the help of Caputo-Fabrizio fractional derivatives. In addition, optimal control problem for fractional system has been designed and the existence of unique solution has also been showed by using Picard-Lindelof technique. Finally, the unconditionally stability of the given fractional numerical technique has been proved.

The first symptomatic infected individuals of coronavirus (Covid-19) was confirmed in December 2020 in the city of Wuhan, China. In India, the first reported case of Covid-19 was confirmed on 30 January 2020. Today, coronavirus has been spread out all over the world. In this manuscript, we studied the coronavirus epidemic model with a true data of India by using Predictor-Corrector scheme. For the proposed model of Covid-19, the numerical and graphical simulations are performed in a framework of the new generalised Caputo sense non-integer order derivative. We analysed the existence and uniqueness of solution of the given fractional model by the definition of Chebyshev norm, Banach space, Schauder's second fixed point theorem, Arzel's-Ascoli theorem, uniform boundedness, equicontinuity and Weissinger's fixed point theorem. A new analysis of the given model with the true data is given to analyse the dynamics of the model in fractional sense. Graphical simulations show the structure of the given classes of the non-linear model with respect to the time variable. We investigated that the mentioned method is copiously strong and smooth to implement on the systems of non-linear fractional differential equation systems. The stability results for the projected algorithm is also performed with the applications of some important lemmas. The present study gives the applicability of this new generalised version of Caputo type non-integer operator in mathematical epidemiology. We compared that the fractional order results are more credible to the integer order results.