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Abstract:
Our manuscript is related to use Caputo fractional order derivative (CFOD) to investigate results of non-linear mode in plasma. We establish results for both temporal and spatial approximate solution. For the require results, we use reduction perturbation method (RPM) to find the analytical solution of the dust acoustic shock waves. Further, using the same technique we find the solitary wave potential and compared the solutions obtained with another very useful technique known as Homotopy perturbation method (HPM). The comparison of results for both approaches are more precise and agreed with the exact solution of the problem. Finally, we present graphical representation for different fractional order for both temporal and spatial approximate solution.

Keywords:
Approximate solution, Caputo derivative, Reductive perturbation, Homotopy perturbation method

Abstract:
Mathematical models are powerful tools to study various real-world problems from different perspectives. This branch has been given much more popularity over the last several decades. Various mathematical models corresponding to different diseases have been studied so far. Keeping these details in mind, the present manuscript is devoted to present a detailed mathematical analysis of the Cutaneous Leishmaniasis disease model. Some basic properties of the model are studied including positivity, the existence of equilibrium points, and reproductive number. The existence and uniqueness of the solution for the model under consideration are also investigated. Local and global stability analyses of equilibrium points are also studied. For the required results, we use the Lyapunov function method and the third additive compound matrix technique based on the Metzler procedure. Sensitivity analysis is also investigated by using some tools from the numerical-functional analysis. A numerical analysis of the proposed model is performed by using a nonstandard finite difference scheme. Moreover, for the justification of our results, we give some graphical presentation of the model for each class in the model. Also, we present some graphical presentations related to the sensitivity analysis along with the tables for its various indices.

Keywords:
Leishmaniasis Disease model, Local and global stability, Sensitivity Analysis, Numerical analysis, Non standard finite difference method

Abstract:
In this research work, we present a mathematical analysis of a fractional sixth-order laser model of a resonant which is homogeneously extended three-level optically pumped. We use Caputo fractional order derivative in the proposed model. Our analysis includes an investigation of various chaotic behaviors under fractional order derivative and qualitative theory of the existence of the solution to the proposed model. For our required analysis of qualitative type, we use formal analysis tools. Further, numerical simulations are performed with a clustering method based on the K–Means algorithm and Adams Bashforth scheme. With the help of the aforesaid scheme, we present different chaotic behavior corresponding to various values of fractional order. Finally, we give a comparison of the CPU time of the proposed method with that of the RK4 method.

Keywords:
Qualitative theory, Chaotic behavior, K–Means algorithm, A clustering method, Adams Bashforth Scheme

Abstract:
Short memory and long memory terms are excellently explained using the concept of piecewise fractional order derivatives. In this research work, we investigate dynamical systems addressing COVID-19 under piecewise equations with fractional order derivative (FOD). Here, we study the sensitivity of the proposed model by using some tools from the nonlinear analysis. Additionally, we develop a numerical scheme to simulate the model against various fractional orders by using Matlab 2016. All the results are presented graphically.

Keywords:
Nonlinear dynamical system, Crossover behavior, Mathematical biology, Sensitivity analysis

Abstract:
This article is devoted to investigate a mathematical model consisting on susceptible, exposed, infected, quarantined, vaccinated, and recovered compartments of COVID-19. The concerned model describes the transmission mechanism of the disease dynamics with therapeutic measures of vaccination of susceptible people along with the cure of the infected population. In the said study, we use the fractal-fractional order derivative to understand the dynamics of all compartments of the proposed model in more detail. Therefore, the first model is formulated. Then, two equilibrium points disease-free (DF) and endemic are computed. Furthermore, the basic threshold number is also derived. Some sufficient conditions for global asymptotical stability are also established. By using the next-generation matrix method, local stability analysis is developed. We also attempt the sensitivity analysis of the parameters of the proposed model. Finally, for the numerical simulations, the Adams–Bashforth method is used. Using some available data, the results are displayed graphically using various fractal-fractional orders to understand the mechanism of the dynamics. In addition, we compare our numerical simulation with real data in the case of reported infected cases.

Abstract:
In the present work, we investigated the transmission dynamics of fractional order SARS-CoV-2 mathematical model with the help of Susceptible , Exposed , Infected , Quarantine , and Recovered . The aims of this work is to investigate the stability and optimal control of the concerned mathematical model for both local and global stability by third additive compound matrix approach and we also obtained threshold value by the next generation approach. The author’s visualized the desired results graphically. We also control each of the population of underlying model with control variables by optimal control strategies with Pontryagin’s maximum Principle and obtained the desired numerical results by using the homotopy perturbation method. The proposed model is locally asymptotically unstable, while stable globally asymptotically on endemic equilibrium. We also explored the results graphically in numerical section for better understanding of transmission dynamics.

Keywords:
Basic reproduction number, Stability analysis, Third additive compound matrix, Homotopy perturbation method, Next generation matrix, Fractional optimal control

Abstract:
In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such as x, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that He’s approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.