Doctorat

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    Etude sur quelques proprietes qualitatives pour les solutions de differentes classes d'equations differentielles abstraites fractionnaires
    (Université M'Hamed Bougara Boumerdes : Faculté des Sciences, 2023) Bourchi, Soumia; Adjabi, Yassine(Directeur de thèse)
    Cette thèse est consacrée à l’étude de quelques propriétés qualitatives de solution pour trois classes d’équations diférentielles abstraites généralisées qui peuvent être soumises à di?érentes conditions initiales locales ( respect. non locales ) dans l’espace de Banach, par le biais des semi-groupes en utilisant le générateur in?nitésimal des opérateurs linéaires bornés ( respect. non bornés) d’un semi-groupe. Pour cela, premièrement, nous examinons dans le cadre du type fractionnaire généralisé ( Caputo, > 0) d’ordre 2 ( 0; 1 ) . Deuxièmement, nous étudions aussi dans le cadre du type '-Caputo pondéré d’ordre 2 ( 0; 1 ) (respect. 2 ( 1; 2 ) ) et ' est une fonction di?érentiable strictement croissante avec la non-linéarité est une fonction dépend des opérateurs intégraux généralisés de Volterra et Fredholm. On nous prouve les opérateurs de solutions, ensuite, nous dérivons une solution douce en termes de semi-groupes pour chaque problème fractionnaire abstrait correspondant. L’approche utilisée est de transformer le problème abstrait de Cauchy en un opérateur intégral équivalent a?n que le problème d’existence de solutions se ramène à la recherche de points ?xes à un opérateur intégral. On intéresse à l’existence, l’unicité, la dépendance aux paramètres et la stabilitéde type Ulam- Hyers de la solution douce. Les résultats sont obtenus en employant la théorie du calcul fractionnaire, la théorie des semi-groupes, les théorèmes du point ?xe, la technique itérative monotone, la méthode des solutions supérieures et inférieures, le concept de la mesure de non-compacité de Kuratowski, l’inégalité de Gronwell, transformée de Laplace et les fonctions spéciales.
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    Numerical modelling of hyperbolic partial differential equations for two-phase flows
    (Université M'Hamed Bougara Boumerdes : Faculté des Sciences, 2022) Ouffa, Souheyla; Seba, Djamila(Directeur de thèse)
    This thesis aims to develop numerical tools for hyperbolic partial differential equations which can be applied to several two-phase flow problems in applied sciences research fields such as energy, environment and oil industry. Although there are enormous demands of engineering twophase flow simulations, there is no established mathematical and computational tools which can simulate a wide variety of two-phase flow problems. Here we present a numerical method to get a solution for a system of equations for two phase flow. This solution is valid for a special case of initial condition called the Riemann problem. The system consists of three hyperbolic conservation laws including gas mass balance, liquid mass balance and total momentum balance. This thesis focuses on the extension of the Weighted Average Flux (WAF) scheme for the numerical simulation of two-phase gas-liquid flow by imposing velocity equilibrium and without mechanical equilibrium of the transient drift-flux model. The model becomes a hyperbolic system of conservation laws with realistic closure relations where both phases are strongly coupled during their motion. Exploiting this, the drift-flux model discretization, simulation and investigation become very fast, simple and robust. The efficiency of the WAF scheme as being a second order in space and time without data reconstruction have been demonstrated in the published literature for compressible single-phase flows. However, the scheme is rarely applied to compressible two-phase flows. Based on a recent and complete exact Riemann solver for the drift-flux model, the model is numerically solved by the WAF scheme. The numerical algorithm accuracy and ability are validated through different published test cases. It is shown that the proposed scheme can be effectively employed to simulate two-phase flows involving discontinuities such as shocks and interfaces. The proposed WAF scheme is also compared with other numerical methods. Simulation results show appropriate agreement of WAF scheme even with the exact solution. Comparisons of the presented simulations demonstrate that the behaviour of WAF scheme is encouraging, more accurate and fast than other numerical methods
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    On the existence of positive solutions to some boundary value problems
    (Université M'Hamed Bougara Boumerdes : Faculté des Sciences, 2022) Attia, Chahira; Mechrouk, Salima(Directeur de thèse)
    The purpose of this thesis is to study the existence and multiplicity of positive solutions of two classes differential equations for singular boundary value problems associated with -Laplacian operator posed on bounded and unbounded intervals of the positive real line. First one, we provide sufficient conditions for existence and multiplicity of positive unbounded solutions for a class of singular second-order boundary value problem posed on the half-line. Second one, we present some new existence results for a nonlinear third-order three point boundary value problem, this results are obtained under some additional assumptions on the nonlinearity. The proofs are based on Krasnosel’skii’s fixed point theorem on cone expansion and compression in a Banach space with arguments of fixed point theory. As for compactness, we have used Ascoli-Arzelà theorem on bounded intervals as well as Corduneanu’s criterion on unbounded intervals. In addition, some illustrative examples are provided to validate our obtained theoretical results
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    Some star selection principles and applications
    (Université M'hamad Bougara : Faculté des Sciences, 2021) Lakehal, Rachid; Seba, Djamila(Directeur de thèse)
    In this thesis, we study some covering properties in topological and bi-topological spaces, using open and closed covers, neighborhoods and Menger, Rothberger, Hurewicz and Lindelof covering properties. New types of star selection principles are introduced in both topological and bi-topological spaces. Some selection principles properties are established. Moreover, a set of relations between the newly de?ned notions and classical selection principles are proved. Similarities and di?erences among each other are also discussed
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    On the application of a numerical method to the resolution of fractional order differential equations
    (Université M'hamad Bougara : Faculté des Sciences, 2021) Attia, Nourhane; Seba, Djamila(Directeur de thèse)
    Fractional differential equations (FDEs) are becoming increasingly popular as a modeling tool to describe a wide range of natural phenomena in physics, chemistry, biology, and so on. These FDEs help scientists to understand, analyze, and make predictions about the modeled system in one case– when their solutions are available. But most FDEs do not have exact solutions and even if there are exact solutions, they can not be evaluated exactly. Thus, one has to rely on numerical methods to obtain their approximate solutions. The purpose of this thesis is to present an efficient computational method for finding numerical solutions of some important fractional differential equations that do not have exact solutions: the fractional biological-population model, fractional cancer tumor models, time-fractional advection-diffusion equation, and time-fractional Swift-Hohenberg equation. Those models are solved by using the reproducing kernel Hilbert space method (RKHSM). The main advantages of this method that encouraged us to use it are its flexibility and simplicity. The convergence analysis and error estimations associated with the RKHSM are discussed to confirm the applicability theoretically. The impact of the fractional derivative on each model is discussed. We also illustrate the profiles of several representative numerical solutions of these models. By testing some examples for each model, we demonstrated the potentiality, validity, and effectiveness of the RKHSM. The computational results are compared with other available results in which these comparisons indicate the superiority and accuracy of the RKHSM in solving complex problems
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    Nonlinear p-laplacian boundary value problems in the frame of conformable fractional derivatives
    (Université M'hamed Bougara : Faculté des sciences, 2021) Bouloudene, Mokhtar; Heminna, Amar(Directeur de thèse)
    L' ecoulement turbulent dans un milieu poreux est un probl eme m ecanique fondamental. Pour etudier ce type de probl eme, en 1983, L. S. Leibenson a introduit l' equation p-Laplacienne. Comme cons equence du d eveloppement intensif du d eriv e fractionnaire, une g en eralisation naturelle de l' equation di erentielle p-Laplacienne est de remplacer le d eriv e ordinaire par un d eriv e fractionnaire pour donner l' equation p-laplacienne fractionnaire, qui peut ^etre consid er ee comme un cas particulier de la g en eralisation du p - Equation di erentielle Laplacienne. L'objectif de cette th ese est de d evelopper un probl eme de valeurs aux limites p-laplaciennes par un calcul fractionnaire. En utilisant la th eorie du degr e de co ncidence, l'existence d'au moins une solution pour un type de p-Laplacien avec d eriv ee conformable au sens de Caputo, les equations di erentielles non lin eaires a r esonance a deux ou trois points peuvent ^etre soumises a di erentes conditions aux limites. De plus, nous etudions une autre classe de probl eme singulier de quatre valeurs limites de l'op erateur p-Laplacien avec d eriv ee conformable par la m ethode des solutions sup erieure et inf erieure associ ee au th eor eme du point xe dans des ensembles partiellement ordonn es, une condition n ecessaire et su - isante pour l'existence d'au moins une solution positive est etablie. Aussi, nous etudions la d ependance de la solution par rapport a l'ordre de l' equation di erentielle conformable. Notre travail g en eralise certains r esultats ant erieurs de la litt erature.
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    Contribution to the study of the controllability of semilinear differential equations of fractional order
    (Université M'hamad Bougara : Faculté des Sciences, 2021) Aimene, Djihad; Seba, Djamila(Directeur de thèse)
    In view of the increased interest that the Kalman concept has received since the 1960s, this thesis is dedicated to contributing to the development of research on the controllability of a class of semilinear differential systems of fractional order with the impulsive condition in abstract spaces, by applying the techniques of fixed point theory and concepts of semigroup theory and define an admissible set of necessary and sufficient conditions to prove the existence of mild solutions and are renewed in each time with what suits the studied system. During this doctoral thesis, we rely on the use of three different definitions of fractional derivatives and discuss their fundamental properties, as well as highlighting the most important basic mathematical principles and necessary theoretical concepts to reach the desired results. This thesis permeates a valuable set of practical examples to prove the proposed approach and further clarify the given purpose