66 research outputs found
Useful Results for the Qualitative Analysis of Generalized Hattaf Mixed Fractional Differential Equations with Applications to Medicine
No full textMost solutions of fractional differential equations (FDEs) that model real-world phenomena in various fields of science, industry, and engineering are complex and cannot be solved analytically. This paper mainly aims to present some useful results for studying the qualitative properties of solutions of FDEs involving the new generalized Hattaf mixed (GHM) fractional derivative, which encompasses many types of fractional operators with both singular and non-singular kernels. In addition, this study also aims to unify and generalize existing results under a broader operator. Furthermore, the obtained results are applied to some linear systems arising from medicine
A New Mixed Fractional Derivative with Applications in Computational Biology
No full textThis study develops a new definition of a fractional derivative that mixes the definitions of fractional derivatives with singular and non-singular kernels. This developed definition encompasses many types of fractional derivatives, such as the Riemann–Liouville and Caputo fractional derivatives for singular kernel types, as well as the Caputo–Fabrizio, the Atangana–Baleanu, and the generalized Hattaf fractional derivatives for non-singular kernel types. The associate fractional integral of the new mixed fractional derivative is rigorously introduced. Furthermore, a novel numerical scheme is developed to approximate the solutions of a class of fractional differential equations (FDEs) involving the mixed fractional derivative. Finally, an application in computational biology is presented
A new generalized class of fractional operators with weight and respect to another function
No full textThis paper introduces and investigates the properties of a new generalized class of fractional differential and integral operators. Such newly class covers various definitions of fractional derivatives with singular and non-singular kernels, weighted fractional derivatives with respect to another function, as well as the new mixed fractional derivative in the sense of Caputo and Riemann-Liouville. Furthermore, the newly introduced class includes all existing forms of fractional integrals, weighted fractionalintegrals and also the weighted fractional integrals with respect to another function including Riemann-Liouville, Hadamard, Katugampola and Hattaf fractional integrals. Moreover, some fundamental properties of the new generalized class of fractional differential and integral operators are rigorously derived
On the Stability and Numerical Scheme of Fractional Differential Equations with Application to Biology
No full textThe fractional differential equations involving different types of fractional derivatives are currently used in many fields of science and engineering. Therefore, the first purpose of this study is to investigate the qualitative properties including the stability, asymptotic stability, as well as Mittag–Leffler stability of solutions of fractional differential equations with the new generalized Hattaf fractional derivative, which encompasses the popular forms of fractional derivatives with non-singular kernels. These qualitative properties are obtained by constructing a suitable Lyapunov function. Furthermore, the second aim is to develop a new numerical method in order to approximate the solutions of such types of equations. The developed method recovers the classical Euler numerical scheme for ordinary differential equations. Finally, the obtained analytical and numerical results are applied to a biological nonlinear system arising from epidemiology
Three-Dimensional Cellular Automaton for Modeling the Hepatitis B Virus Infection
No full textHepatitis B is considered as the most common hepatic in the world and may lead to cirrhosis and liver cancer. It is caused by the hepatitis B virus, which attacks and can damage the liver. In this paper we investigate a new mathematical model to study the dynamic process of HBV infection on the liver. This model is based on a three dimensional cellular automaton, which is composed of four state variables. The model takes into account the heterogeneous feature and the spatial localization of the population studied. Furthemore, since the virus doesn’t remain only on the liver surface but penetrates into the organ, our model describes better the behavior of interactions between cells and hepatitis B virus in the liver than the previous works found in the literature, which have used only two cellular automata in their models
Analysis of a Fractional Reaction-Diffusion HBV Model with Cure of Infected Cells
No full textIn this paper, we propose a fractional reaction-diffusion model in order to better understand the mechanisms and dynamics of hepatitis B virus (HBV) infection in human body. The infection transmission is modeled by Hattaf–Yousfi functional response, and the fractional derivative is in the sense of Caputo. The global stability of the model equilibria is analyzed by means of Lyapunov functionals. Finally, numerical simulations are presented to support our analytical results
A Fractional Order Model for Viral Infection with Cure of Infected Cells and Humoral Immunity
No full textIn this paper, we study the dynamics of a viral infection model formulated by five fractional differential equations (FDEs) to describe the interactions between host cells, virus, and humoral immunity presented by antibodies. The infection transmission process is modeled by Hattaf-Yousfi functional response which covers several forms of incidence rate existing in the literature. We first show that the model is mathematically and biologically well-posed. By constructing suitable Lyapunov functionals, the global stability of equilibria is established and characterized by two threshold parameters. Finally, some numerical simulations are presented to illustrate our theoretical analysis
Spatiotemporal Dynamics of a Generalized Viral Infection Model with Distributed Delays and CTL Immune Response
No full textIn this paper, we propose and investigate a diffusive viral infection model with distributed delays and cytotoxic T lymphocyte (CTL) immune response. Also, both routes of infection that are virus-to-cell infection and cell-to-cell transmission are modeled by two general nonlinear incidence functions. The well-posedness of the proposed model is also proved by establishing the global existence, uniqueness, nonnegativity and boundedness of solutions. Moreover, the threshold parameters and the global asymptotic stability of equilibria are obtained. Furthermore, diffusive and delayed virus dynamics models presented in many previous studies are improved and generalized
A New Class of Generalized Fractal and Fractal-Fractional Derivatives with Non-Singular Kernels
No full textThe present paper introduces a new class of generalized differential and integral operators. This class includes and generalizes a large number of definitions of fractal-fractional derivatives and integral operators used to model the complex dynamics of many natural and physical phenomena found in diverse fields of science and engineering. Some properties of the newly introduced class are rigorously established. As applications of this new class, two illustrative examples are presented, one for a simple problem and the other for a nonlinear problem modeling the dynamical behavior of a chaotic system
A New Generalized Definition of Fractional Derivative with Non-Singular Kernel
No full textThis paper proposes a new definition of fractional derivative with non-singular kernel in the sense of Caputo which generalizes various forms existing in the literature. Furthermore, the version in the sense of Riemann–Liouville is defined. Moreover, fundamental properties of the new generalized fractional derivatives in the sense of Caputo and Riemann–Liouville are rigorously studied. Finally, an application in epidemiology as well as in virology is presented
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