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A mathematical model for current-induced magnetization dynamics in ferromagnets at elevated temperatures is considered. The model is represented by a classical Landau-Lifshitz-Bloch equation containing adiabatic and non-adiabatic torques. In the case of a ferromagnet confined to a bounded domain of R 3 , we prove existence of a weak solution by using Faedo-Galerkin approach and a compactness method. This paper extends the work done by Le et al. (2016) in the absence of adiabatic and non-adiabatic terms. Furthermore, uniqueness is proved in the case of dimension one and two. Finally, the limiting behaviour in large time is studied for the non-adiabatic case.

In this paper, we analyze the dynamics of a new proposed stochastic non-autonomous SVIR model, with an emphasis on multiple stages of vaccination, due to the vaccine ineffectiveness. The parameters of the model are allowed to depend on time, to incorporate the seasonal variation. Furthermore, the vaccinated population is divided into three subpopulations, each one representing a different stage. For the proposed model, we prove the mathematical and biological well-posedness. That is, the existence of a unique global almost surely positive solution. Moreover, we establish conditions under which the disease vanishes or persists. Furthermore, based on stochastic stability theory and by constructing a suitable new Lyapunov function, we provide a condition under which the model admits a non-trivial periodic solution. The established theoretical results along with the performed numerical simulations exhibit the effect of the different stages of vaccination along with the stochastic Gaussian noise on the dynamics of the studied population.

The aim is to explore a COVID-19 SEIR model involving Atangana-Baleanu Caputo type (ABC) fractional derivatives. Existence, uniqueness, positivity, and boundedness of the solutions for the alternative model are established. Some stability results of the proposed system are also presented. Numerical simulations results obtained in this paper, according to the real data, show that the model is more suitable for the disease evolution. © 2021. All Rights Reserved.

Long-run bifurcation analysis aims to describe the asymptotic behavior of a dynamical system. One of the main objectives of mathematical epidemiology is to determine the acute threshold between an infection’s persistence and its elimination. In this study, we use a more comprehensive SVIR epidemic model with large jumps to tackle this and related challenging problems in epidemiology. The huge discontinuities arising from the complexity of the problem are modelled by four independent, tempered, α-stable quadratic Lévy processes. A new analytical method is used and for the proposed stochastic model, the critical value R0🟉 is calculated. For strictly positive value of R0🟉, the stationary and ergodic properties of the perturbed model are verified (continuation scenario). However, for a strictly negative value of R0🟉, the model predicts that the infection will vanish exponentially (disappearance scenario). The current study incorporates a large number of earlier works and provides a novel analytical method that can successfully handle numerous stochastic models. This innovative approach can successfully handle a variety of stochastic models in a wide range of applications. For the tempered α-stable processes, the Rosinski (2007) algorithm with a specific Lévy measure is implemented as a numerical application. It is concluded that both noise intensities and parameter α have a great influence on the dynamical transition of the model as well as on the shape of its associated probability density function.

We deal in this paper with a diffusive SIR epidemic model described by reaction-diffusion equations involving a fractional derivative. The existence and uniqueness of the solution are shown, next to the boundedness of the solution. Further, it has been shown that the global behavior of the solution is governed by the value of R0 , which is known in epidemiology by the basic reproduction number. Indeed, using the Lyapunov direct method it has been proved that the disease will extinct for R0 < 1 for any value of the diffusion constants. For R0 > 1, the disease will persist and the unique positive equilibrium is globally stable. Some numerical illustrations have been used to confirm our theoretical results.Subject classification: 26A33; 34A08; 92D30; 35K57.

The present paper deals with global existence of weak solutions of a time-space fractional Landau–Lifshitz–Bloch equation involving the weak Caputo derivative and a fractional Laplacian. We use Faedo–Galerkin method with some commutator estimates in order to prove global existence of weak solutions for the model. The uniqueness is also discussed in a special one dimensional case.

In this work, a fractional order SIR epidemic model is proposed. We first prove the existence, uniqueness, non-negativity and boundedness of solutions to the considered model. We also study the existence of equilibrium points. Some sufficient conditions are derived to ensure, in terms of the basic reproduction number, the global asymptotic stability of the disease free equilibrium point and endemic equilibrium point. Finally, numerical simulations are illustrated to verify the validity of our theoretical results.

Fluid viscosity is considered as constant in several boundary layer analyses, but this fluid property can change remarkably when the temperature difference exists in the boundary layer. The Prandtl number and Schmidt number can also change significantly as the fluid viscosity changes depending on temperature. Therefore, this framework is exploring the consequences of varying viscosity and varying Prandtl number on Falkner–Skan flow of Williamson nanofluid over a wedge, plate and stagnation point. The Buongiorno nanofluid model has been employed to manifest the fluid transport properties of the Williamson nanofluid. Similarity transformations are utilized to transform the governing equations into ordinary differential equation and solved numerically using Runge–Kutta (RK) Fehlberg method. Williamson fluid velocity, temperature, concentration, skin friction factor, rate of heat transfer and rate of mass transfer are investigated with emerging parameters, and the outcomes are presented graphically. Computed results manifest that the Williamson nanofluid expresses the opposite nature in velocity and temperature for higher values of Weissenberg number parameter. Positive values of variable viscosity parameter diminish the significance of variable Prandtl number and variable Schmidt number in the boundary layer. Furthermore, it is noticed that the Williamson nanofluid temperature is higher over a plate compared with wedge and stagnation point cases.

In this work, we investigate the stability of an SIR epidemic model with a generalized nonlinear incidence rate and distributed delay. The model also includes vaccination term and general treatment function, which are the two principal control measurements to reduce the disease burden. Using the Lyapunov functions, we show that the disease-free equilibrium state is globally asymptotically stable if $\mathcal{R}_{0} \leq 1 $R0≤1, where $\mathcal{R}_{0} $R0 is the basic reproduction number. On the other hand, the disease-endemic equilibrium is globally asymptotically stable when $\mathcal{R}_{0} > 1 $R0>1. For a specific type of treatment and incidence functions, our analysis shows the success of the vaccination strategy, as well as the treatment depends on the initial size of the susceptible population. Moreover, we discuss, numerically, the behavior of the basic reproduction number with respect to vaccination and treatment parameters.

To describe the main propagation of the COVID-19 and has to find the control for the rapid spread of this viral disease in real life, in current manuscript a discrete form of the SEIR model is discussed. The main aim of this is to describe the viral disease in simplest way and the basic properties that are related with the nature of curves for susceptible and infected individuals are discussed here. The elementary numerical examples are given by using the real data of India and Algeria.

In this paper, the global dynamics of an S E I epidemic model with constant immigration and general nonlinear incidence function is investigated. It is shown that there is neither a disease free equilibrium nor a basic reproduction number for this kind of models containing immigration terms. Moreover, the existence of a unique endemic equilibrium is proved. Using second Lyapunov method, we establish the global stability of the positive equilibrium. For a specific type of incidence function, some numerical simulations are presented to validate the theoretical results.

We investigate the global behaviour of a SIRI epidemic model with distributed delay and relapse. From the theory of functional differential equations with delay, we prove that the solution of the system is unique, bounded, and positive, for all time. The basic reproduction number $R_{0}$ for the model is computed. By means of the direct Lyapunov method and LaSalle invariance principle, we prove that the disease free equilibrium is globally asymptotically stable when $R_{0} < 1$. Moreover, we show that there is a unique endemic equilibrium, which is globally asymptotically stable, when $R_{0} > 1$., This is a preprint of a paper whose final and definite form is with 'Statistics Opt. Inform. Comput.', Vol. 7, No 2 (2019). See [http://www.iapress.org]. Submitted 26/July/2018; Revised and Accepted 22/Dec/2018

This paper deals with the study of a SEIRI epidemic model with relapse and general incidence function. Using the next-generation matrix method, we have looked for the expression of the basic reproduction rate R 0 and analyzed the equilibria’s existence and local stability. More precisely, we have established that the two equilibrium points are locally asymptotically stables. In addition, we have used three control functions to control the evolution of the proposed model. We have determined the optimal control by means of the Pontryagin’s maximum principle. Also, some numerical simulations are given to illustrate the theoretical results.

We consider a mathematical model describing magnetization dynamics with inertial effects. The model consists of a modified form of the Landau–Lifshitz–Gilbert equation for the evolution of the magnetization vector in a rigid ferromagnet. The modification lies in the presence of an acceleration term describing inertia. A semi-implicit finite-difference scheme for the model is proposed, and a criterion of numerical stability is given. Some numerical experiments are conducted to show the performance of the scheme.

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We study the Landau-Lifshitz-Gilbert equation in a composite ferromagnetic medium made of two different materials with highly contrasted properties. Over the so-called matrix domain, the effective field, the demagnetizing field and the bulk anisotropy field are scaled with regard to a parameter $e$ representing the size of the matrix blocks. This scaling preserves the physics of the magnetization as $e$ tends to zero. Using homogenization theory, we derive the corresponding effective model. To this aim we use the concept of two-scale convergence together with a new homogenization procedure for handling with the nonlinear terms. More precisely, an appropriate dilation operator is applied in a embedded cells network, the network being constrained by the microscopic geometry. We prove that the less magnetic part of the medium contributes through additional memory terms in the effective field.

In this paper, we prove the existence of global weak solutions for a model described by the fractional Heisenberg equation for the magnetization field and the viscoelastic integro-differential equation for the displacements. We study the three-dimensional case. The demonstration of the existence of weak solution is based on the method of Faedo-Galerkin; and to get the convergence of the nonlinear terms, we introduce the commutator structure.

The International Workshop on Applied Mathematics, Errachidia (IWAM’2017) took place in Errachidia, Morocco at the Faculty of Sciences and Technics during March 13, 2017 (https://iwam2017.sciencesconf.org/). The workshop was held with the support of FST Errachidia, the University Moulay Isma¨ıl. It is aimed at celebrating the collaborative research in applied mathematics by the mathematicians of the FST Errachidia, the Systems and Control Group of CIDMA, University of Aveiro, and with the MIA Lab, University of La Rochelle. It was attended by about 100 Mathematicians and Ph.D. and M.S. students, coming from universities from different countries, such as Algeria, Egypt, France, Portugal, and Morocco. The aim of IWAM’2017 is to bring researchers and professionals to discuss recent developments in both theoretical and applied mathematics, to create the knowledge exchange platform between mathematicians. The workshop is broad-based that covers several branches of engineering sciences, mathematics and interdisciplinary researches mainly in the fields of optimization and variational analysis, theoretical, asymptotic and numerical analysis of ordinary, partial and fractional differential equations

We consider a mathematical model describing magnetization dynamics with inertial effects. The model consists of a modified form of the Landau–Lifshitz–Gilbert equation for the evolution of the magnetization vector in a rigid ferromagnet. We discuss global existence of weak solutions and characterize the \(\omega \)-limit set for the model.

We consider a mathematical model describing polarization dynamics in ferroelectric material. The model consists of a Maxwell system for electromagnetic field coupled with a second-order time-dependent equation for the evolution of polarization. We study the long-time behaviour of weak solutions and prove that all points of the ω-limit set of any trajectories are solutions of the stationary model.

The paper deals with global existence of weak solutions to a one-dimensional mathematical model describing magnetoelastic interactions. The model is described by a fractional Landau-Lifshitz-Gilbert equation for the magnetization field coupled to an evolution equation for the displacement. We prove global existence by using Faedo-Galerkin/penalty method. Some commutator estimates are used to prove the convergence of nonlinear terms.

The paper deals with the dimensional reduction from 2D to 1D in magnetoelastic interactions. We adopt a simplified, but nontrivial model described by the Landau-Lifshitz-Gilbert equation for the magnetization field coupled to an evolution equation for the displacement. We identify the limit problem by using the so-called energy method.

This paper deals with the limiting behavior of a phase transition model in ferromagnetism. The model describes the three-dimensional evolution of both thermodynamic and electromagnetic properties of the ferromagnetic material. We are concerned with the passage from 3D to 2D in the theory of the paramagnetic-ferromagnetic transition. We identify the limit problem by using the so-called energy method.

We consider a mathematical model describing magnetisation dynamics with non-scalar damping. The model consists on a generalised Landau-Lifshitz-Gilbert equation with a general damping tensor. We apply Faedo-Galerkin/penalty method to show the existence of global weak solutions in one-dimensional case.

We consider the interlayer exchange coupling between two ferromagnetic layers mediated by a nonmagnetic spacer layer. We adopt the theoretical model which is based on a macroscopic description using the Landau--Lifshitz equations and the Hoffmann boundary conditions characterizing the interlayer exchange coupling between the interfaces of the ferromagnetic films. An asymptotic study of such a layered system for either small or large spacer thicknesses is presented. The asymptotic problem and the boundary conditions at interfaces both for the magnetization and the magnetic field are characterized.

We discuss the behaviour, when the thickness ε tends to 0, of thin ferromagnetic films with surface anisotropy energy. The model equations are given by the Landau–Lifshitz equation coupled to Maxwell equations with magnetic polarization. We consider two types of materials: flat and slender cylinders. Two scalings for the surface anisotropy coefficient are used. In the first one it is assumed that the coefficient is of order ε while in the second one we suppose that it is of order 1. We prove global existence of solutions and show that the zero-thickness limit induces new effects. For example, for slender media we get a nonlocal effect for the magnetic excitation while for flat media we obtain a one-dimensional magnetic field.

This paper deals with classical dimensional reductions 3D-2D and 3D-1D in magnetoelastic interactions. We adopt a model described by the Landau-Lifshitz-Gilbert equation for the magnetization field coupled to an evolution equation for the displacement. We identify the limit problem both for flat and slender media by using the so-called energy method.

This paper is concerned with global existence of weak solutions to a model equations of magnetization reversal by spin-polarized current in a layer introduced in [19]. The local magnetization of the ferromagnet satisfies the usual Landau-Lifshitz equation which is coupled to the nonlinear heat equation satisfied by the spin accumulation field defined in all the layer. The coupling is due to the contact interaction energy. We use an hyperbolic regularization method with penalization of the saturation constraint satisfied by the local magnetization to prove global existence result, in any finite time interval, of weak solutions with finite energy. We present other models of equations describing the magnetization switching by spin-polarized current and show that our method can be used to solve them.

We consider a mathematical model describing magnetization dynamics with vertical spin stiffness. The model consists of a modified form of the Landau-Lifshitz-Gilbert equation for the evolution of the magnetization vector in a rigid ferromagnet. The modification lies in the presence in the effective field of a nonlinear term describing vertical spin stiffness. We prove the global existence of weak solutions to the model by using the Faedo-Galerkin method and discuss the limit of the obtained solutions as the vertical spin stiffness parameter tends to zero.

This thesis is devoted to the study of some mathematical problems arising in micromagnetism. The models considered here are based on the Landau-Lifshitz-Gilbert equation (LLG) describing the evolution of the magnetization field in a ferromagnetic material. Our aim is the analysis of the behavior of the models regarding the slight variations of some physical parameters. We first recall some important notions about ferromagnetism. Then, we carry out a numerical study of a model of magnetization dynamics with inertial effects. We propose a semi-implicit finite difference scheme which intrinsically respects the properties of the continuous model. Numerical simulations are provided for emphasizing the effect of the inertia parameter. These simulations also show the performance of the scheme and confirm the order of convergence obtained theoretically. We then study a model of magnetization dynamics with a non-local damping. The sensitivity of the magnetization dynamics to the damping coefficient is studied by giving the limiting problem for small and large values of the parameter. Finally, we study the homogenization of the LLG equation in two types of structures, namely a composite material with strongly contrasted magnetic properties, and a periodically perforated material with surface anisotropy energy. The homogenized models are first obtained formally. The rigorous derivation is then performed using mainly the concepts of two-scale convergence, two-scale convergence on surfaces together with a new homogenization procedure for handling with the nonlinear terms. More precisely, an appropriate dilation operator is applied in a embedded cells network, the network being constrained by the microscopic geometry., Cette thèse est consacrée à l'étude de quelques problèmes mathématiques issus du micromagnétisme. Le but est d'analyser le comportement des modèles en fonction de différents paramètres physiques, dont les fines variations sont parfois difficilement mesurables. Nous adoptons des approches numériques, asymptotiques ou d'homogénéisation. Les modèles considérés reposent sur l'utilisation de l'équation de Landau-Lifshitz-Gilbert (LLG) décrivant l'évolution du champ d'aimantation dans un matériau ferromagnétique. Nous rappelons d'abord quelques notions importantes en ferromagnétisme. Ensuite, nous menons une étude numérique d'un modèle de la dynamique d'aimantation avec effets d'inertie. Nous proposons un schéma aux différences finies semi-implicite qui respecte de façon intrinsèque les propriétés du modèle continu. Des simulations numériques sont réalisées pour cerner l'effet du paramètre d'inertie. Ces simulations montrent aussi la performance du schéma et confirment l'ordre de convergence obtenu théoriquement. Nous étudions ensuite un modèle de la dynamique de l'aimantation avec amortissement non local. La sensibilité de la dynamique d'aimantation au paramètre d'amortissement est étudiée en donnant le problème limite pour de petites et de grandes valeurs du paramètre. Enfin, nous étudions l'homogénéisation de l'équation LLG dans deux types de matériau, à savoir les composites présentant un fort contraste des propriétés magnétiques et les matériaux périodiquement perforés avec énergie d'anisotropie de surface. Des modèles homogénéisés sont d'abord obtenus formellement puis une dérivation rigoureuse est établie en se basant principalement sur les concepts de la convergence à double échelle et de la convergence à double échelle en surface. Pour traiter les non-linéarités, nous introduisons une nouvelle méthode basée sur le couplage d'un opérateur de dilatation calibré sur les contrastes d'échelle et d'un outil de réduction de dimension, par construction de grilles emboitées adaptées à la géométrie du domaine microscopique.