Mathématique
URI permanent de cette collectionhttps://dspacee.univ-temouchent.edu.dz/handle/123456789/747
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Item Modélisation Mathématique des Populations en Interactions(2026) LECHLECHE Safaa; BOUKHALFA FatimaThe purpose of this thesis is the mathematical study of the dynamics of interacting populations. We use ordinary differential equations as the main modeling tool to analyze the local and global stability of fundamental models of Lotka-Volterra (predator prey), competition and mutualism, then a system of three populations (rabbits, sheep, wolves). Then we extend this study to more complex systems integrating the epidemiological di- mension.Item Existence et unicité des solutions des équations différentielles semi-linéaires d’ordre fractionnaire(2026) KHELLADI Khedidja Samah; MEKHALFI KheiraThis dissertation studies the existence and uniqueness of mild solutions for semi- linear fractional differential equations with finite and infinite delay in the framework of the Riemann–Liouville fractional derivative. Banach’s Fixed Point Theorem is used to establish uniqueness, while the Leray–Schauder nonlinear alternative pro- vides existence results. The theory is illustrated by a tumor growth model and an infinite-memory viscoelastic plate model.Item Global Dynamics of a Nonlocal Delayed Viral Infection Model in Heterogeneous Environments(2026) BELHADRI Mohammed Abdelmalek; BENTOUT SoufianeThis work is devoted to the mathematical analysis of within-host viral infection dynamics, with particular emphasis on the hepatitis B virus. We first review three classical models that progressively incorporate spatial structure: the foundational ODE model of Nowak et al [23], the local diffusion model of Wang and Wang [29], and the nonlocal dispersal frame- work of Wang et al [30]. We then conduct a rigorous study of the spatially heterogeneous model proposed by Liu et al [17], which simultaneously incorporates nonlocal diffusion of free virions governed by a probability kernel J, an intracellular delay τ > 0 reflecting the latency between viral entry and viral production, and position-dependent biological parameters. The analysis is carried out in an infinite-dimensional functional framework. We establish the global well-posedness and positivity of solutions via semigroup theory and the Banach fixed point theorem, and prove that the semiflow is bounded dissipative. Asymptotic smoothness of the semiflow and the existence of a compact global attractor are obtained through a Kuratowski measure argument. Finally, the global asymptotic stability of the infection-free steady state P0 is established when R0 ≤ 1, via an explicit Lyapunov functional whose integral kernel is the positive eigenfunction of the linearised problem around P0.Item Méthodes de Monte Carlo par chaînes de Markov et estimation bayésienne(2026) BELABBES Nebia; BALASKA LamiaThis thesis focuses on Bayesian estimation and the use of Markov Chain Monte Carlo (MCMC) methods. It begins by presenting the theoretical foundations of Bayesian estima- tion, explaining the concepts of prior distribution, posterior distribution, and associated Bayesian estimators. Then, it introduces classical Monte Carlo methods as well as Markov chains, which are fundamental tools for approximating integrals and generating samples from complex distributions. Special attention is given to the Gibbs sampler, a powerful and widely used MCMC algorithm in Bayesian statistics. Finally, practical applications of these methods are presented, particularly in the context of Gaussian models. This work highlights the importance of Bayesian approaches for integrating prior information into statistical analyses and demonstrates the power of MCMC methods in solving inference problems when exact calculations are infeasible.Item Existence de solutions d’un problème aux limites avec impulsions(2026) BENAISSA RAHAL Hanene; BELLATAR ZOKHAIn this thesis, we analyzed boundary value problems related to second-order impulsive differential equations and impulsive fractional differential equations. By applying the Banach contraction principle as well as the two fixed point theorems of Leray-Schauder, we established existence and uniqueness results for solutions under suitable hypotheses. This work confirms the importance of fixed point methods in the study of impulsive differential systems.Item Spatiotemporal Dynamics and Wave Propagation in the Diffusive Kermack-McKendrick Model(2026) BAKHTI Manel Nor El Houda; BENTOUT SoufianeWe study the existence of traveling wave solutions for a diffusive Kermack–McKendrick SIR model with standard incidence. The three compartments—susceptible S, infected I, and recovered R—are all taken into account in the traveling waves analysis. We demonstrate that the minimum wave speed of traveling waves for this three-dimensional system can be obtained from the linearization around the initial disease-free equilibrium. The proof is based on Schauder’s fixed point theorem and the Arzelà-Ascoli theorem. This work offers a valuable approach for studying high-dimensional epidemic models.Item Contrôlabilité et stabilisabilité des systèmes linéaires dans les espaces de Hilbert(2026) BERREZOUK Halima; HARIRI MohammedCe mémoire est consacré à l’étude de la contrôlabilité et de la stabilisabilité des sys- tèmes linéaires dans les espaces de Hilbert. Ces notions occupent une place importante dans la théorie moderne du contrôle, en particulier dans l’étude des systèmes gouvernés par des équations différentielles ordinaires, des équations aux dérivées partielles et des modèles infini-dimensionnels. Dans un premier temps, nous rappelons les notions de base relatives aux espaces de Hilbert, aux opérateurs linéaires bornés et non bornés, aux opérateurs adjoints, aux projecteurs, aux bases hilbertiennes, aux opérateurs compacts et aux semi-groupes for- tement continus. Ces outils permettent de formuler correctement les systèmes d’évolu- tion abstraits. Dans un deuxième temps, nous étudions les systèmes linéaires infini-dimensionnels de la forme x 0 (t) = Ax(t) + Bu(t), t ≥ 0; x(0) = x0, où A est le générateur infinitésimal d’un semi-groupe fortement continu sur un es- pace de Hilbert X, B est l’opérateur de contrôle et u représente la commande. Nous présentons la notion de solution mild, qui est adaptée aux systèmes infini-dimensionnels lorsque les solutions classiques ne sont pas toujours disponibles. Ensuite, nous analysons la contrôlabilité exacte, la contrôlabilité approchée, l’opé- rateur de contrôlabilité, l’espace atteignable et les critères associés. Une attention par- ticulière est accordée à la dualité entre contrôlabilité et observabilité, qui joue un rôle central dans l’étude des équations aux dérivées partielles contrôlées. Enfin, nous abordons la stabilisabilité des systèmes linéaires dans les espaces de Hilbert. Nous présentons la stabilisation forte, la stabilisation exponentielle, la stabi- lisation par retour d’état et quelques critères spectraux. Une application numérique 5 sous MATLAB est proposée afin d’illustrer les notions étudiées à travers une approxi- mation finie-dimensionnelle d’un système gouverné par un opérateur de type Laplacien.Item Contrôle optimal et ses applications(2026) REZOUG Ahlem; Mohammed HARIRIOptimal control is a fundamental area of applied mathematics, situated at the in- tersection of dynamical systems theory, analysis, and optimization. Its objective is to determine a control that allows a system to evolve from an initial state to a desired state while minimizing a cost functional representing, depending on the context, energy expenditure, trajectory error, transfer time, or economic cost. In this thesis, we first present the mathematical foundations necessary for the study of optimal control, including dynamical systems, existence and uniqueness results for solutions, and some notions of convexity. We then formulate the general problem of optimal control by introducing state variables, admissible controls, constraints, and the cost functional. Particular attention is paid to the main theoretical methods in the field, especially Pontryagin’s maximum principle, dynamic programming, the Hamilton- Jacobi-Bellman equation, and linear quadratic control based on the Riccati equation. Finally, a numerical section using MATLAB is devoted to illustrating the theore- tical results on linear and non-linear systems. The simulations obtained demonstrate the effectiveness of the methods studied for stabilizing dynamic systems and improving their performance while limiting the control effort. This work thus highlights the relevance of optimal control both theoretically and in its numerous applications in automation, engineering, economics, and energy systems.Item Existence et stabilité des solutions du problème de Cauchy pour une équation d’ondes dans Rn avec amortissement par dérivée fractionnaire(2026) BRINI Khadidja; BENAISSA AbdelkaderWave equation with fractional derivative in generalized Caputo sense is investigated. In this thesis, we analyze the Cauchy problem in Rn with linear fractional damping (frictional). First, we establish the system’s well-posedness through semi group theory, ensuring the existence and uniqueness of solutions, named the quantitative results. In addition, we show the strong stability of our system, where qualitative results are obtained by applying the general criterion of Arendt and Batty fombined with C0-semi group approach.Item Étude et analyse d’un modèle mathématique de la pêche(2026) SIDI YAKOUB Wafaa; MAMMAR ImaneThis work focuses on the study of bioeconomic models applied to fisheries. We analyze a multi-site fishery model with variable price as well as a model integrating marine protected areas. The study is based on system reduction, equilibrium analysis and their stability.Item Étude mathématique et numérique de l’équation de poisson avec des conditions mixtes(2026) BENABI Anes; BENKHEDDA HananeThis memoir is dedicated to the study and numerical approximation of solutions to the Poisson problem, a fundamental model of elliptic partial differential equations. In a first step, we established the theoretical framework by recalling Green’s formulas and introducing Sobolev spaces, the natural functional framework of variational analysis. We then demonstrated, using the Lax-Milgram theorem, the existence and uniqueness of the weak solution of the problem. In a second step, we developed a numerical approximation using the finite element method on a conforming triangular mesh, studying two types of elements : P1 elements (affine polynomials) and P2 elements (quadratic polynomials). For each, we constructed the canonical basis of shape functions, established the discrete variational problem, and showed that it reduces to solving a linear system AX = b, where the stiffness matrix A is symmetric positive definite, hence invertible.Item Study of Fractional Boundary Value Problems(2026) LARICHI Nafisa; Dr.BEKRI ZouaouiThis work studies two nonlinear fractional boundary value problems involving the Caputo-Katugampola and Riemann-Liouville derivatives; using theorems (Banach Contraction, leray-schauder nonlinear alternative), it establishes existence, uniqueness and nontrivial existence results for the solution.Item Estimation et tests dans les processus de diffusion à dérive non régulière(2025) Djebbour, Khadîdja; BALASKA, LamiaMany natural, economic, and biological phenomena involve random fluctuations that cannot be captured by purely deterministic models. Stochastic differential equations (SDEs) provide a suitable framework by combining drift and diffusion components. This thesis focuses on a diffusion process with proportional delay and non-regular drift. Two main objectives are addressed: estimating the delay parameter via maximum likelihood in the small-diffusion regime, and constructing a simple versus simple parametric test based on the likelihood ratio. Theoretical results, inspired by the works of Ibragimov, Has’minskii, and Kutoyants, are supported by numerical simulations.Item Théorème de Girsanov et Applications(2025) Slimani, fatima zahra; Messabihi, AichaThis thesis explores Girsanov's Theorem and its applications in stochastic processes, focusing on the Ornstein-Uhlenbeck (O-U) process and Itô-Lévy processes. The study begins with foundational concepts of stochastic processes, including Brownian motion, martingales, and Itô calculus. It then rigorously presents Girsanov’s Theorem, which allows changing the probability measure to transform a stochastic process with drift into a driftless one (or vice versa). The applications demonstrate how Girsanov’s Theorem simplifies the analysis of O- U processes (used in finance and physics) and Itô-Lévy processes (incorporating jumps). Simulations in R illustrate the theoretical results, highlighting the theorem’s utility in financial modeling and risk-neutral pricing.Item Les Séries Stationnaires Appliquées(2025) BOUCHETA, Anissa; BENNAFLA, DjamilaThis thesis aims to study stationary time series, with a particular focus on their properties, identification, modelling, and forecasting. Emphasis will be placed on the application of these methods in specific domains. The project will include a practical analysis of real-world data and an implementation using R.Item Analyse d’un réseau de file d’attente sous des disciplines de service avec priorité(2025) Yekhlef, Hadjer; SAKHI, HananeIn this work, we study the stabilization of a queueing network model under service disciplines with priority; a multi-class fluid queueing system with priority consisting of N stations (N ≥ 3) and 2N classes (each station accommodates two classes). We base our stability analysis on the fluid model.Item Prévision des Séries Chronologiques à l’aide de la Régression Linéaire(2025) OURRAG, Nourhane Keltoum; BENNAFLA, DjamilaThis thesis explores the use of linear regression for time series forecasting. We analyze real-world time series data and implement linear regression techniques to predict future values. We pay special attention to data analysis, model diagnostics, and the evaluation of predictive performance. We also include practical examples and simulations using R, enabling the reader to understand the concepts and apply the presented methods.Item COMPORTEMENT ASYMPTOTIQUE D’UN MODÈLE D’HÉROÏNE AVEC LE TRAITEMENT ÂGE(2020) GASMI, Hadjira; BENTOUT, SoufianeItem ANALYSE MATHÉMATIQUE D’UN MODÈLE ÉPIDÉMIOLOGIQUE(2020) SELLAK, selima; BOUKHALFA, FatemaItem Existence de Solutions pour un Probleme Non Lineaire d’une Equation Differentielle Fractionnaire Implicite(2020) TAHRAOUI, Safaâ; Mami, Tawfiq FawziThe main objective of this paper is to study the existence of solutions for a nonlinear problem of an implicit fractional differential equation. The derivatives considered are in the sense of Caputo and of order between 0 and 1 in a Banach space with boundary conditions and in the second case with non-local ones. These results were obtained by applying the fixed point theory. Examples are included to illustrate the applicability of theoretical results.