Doctora Mathématiques
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Item Accélération de la convergence de méthodes numériques pour résoudre des équations intégrales(university of bordj bou arreridj, 2025) ABDENNEBI IssamThe research work presented in this thesis focuses on improving the convergence speed of numerical methods for solving integral equations. These equations often introduce a very complex behavior, posing significant challenges to traditional numerical techniques, par ticularly in terms of convergence and accuracy. To address these challenges, we have de veloped and analyzed an adaptive spectral collocation method for Fredholm and Volterra integral equations of the second kind, which can achieve fast convergence and high ac curacy despite the fact that its solution exhibits localized rapid variations, steep gradi ents, or a steep front. Adaptivity is implemented using a suitable family of one-to-one mappings to generate a new equation with smoother behavior that can be approximated more accurately. The proposed method can achieve exponential accuracy by adjusting a parameter-dependent mapping in the modal approximation according to the given data. Finally, several numerical examples are given to show that the proposed method is prefer able to its classical method and some other existing approaches with a relatively smaller number of degrees of freedomItem Stabilité et convergence des méthodes spectrales Application aux équations intégrales(Université de Bordj Bou Arreridj Faculty of Mathematics and Computer Science, 2024) Radjai, abirIn recent years, there has been a growing interest in the formulation of many problems in terms of integral equations, and this has fostered a parallel rapid growth of the literature on their numerical solution. In this sense, our focus will be on spectral methods for solving integral equations. One of the purposes of this research is to provide the mathematical foundations of spectral methods and to analyze their basic theoretical properties (stability, accuracy, computational complexity, and convergence). Furthermore, we have applied the spectral collocation method to find numerical solutions to quadratic Urysohn integral equations. This method reduces the nonlinear integral equation to a system of nonlinear algebraic equations and that algebraic system has been solved by the iterative method. We have derived an error analysis for the current method, which proves that it has exponential convergence order. Finally, several numerical examples are given to show the effectiveness and stability of our approachItem Etude qualitative de quelques EDPs en temps avec amortissement(UNIVERSITY BBA, 2024) LAKEHAL, IBRAHIMThis thesis is devoted to the study of two problems related to the theory of control of PDE. In a first and second time, we study two nonlinear Euler-Bernoulli beams with a neutral type delay and viscoelastic, using controls acting on the free boundaries. By using the method of Faedo-Galerkin, we prove the existence and uniqueness of the solution for each problem. After that using the energy method and constructing an appropriate Lyapunov function, under certain conditions on the neutral delay term kernel and the viscoelastic term, we show that although, the destructive nature of delay in general, which is a very general degrading energy problem.Item Limit cycles of continuous and discontinuous piecewise differential systems separated by straight line and formed by two arbitrary quadratic centers(UNIVERSITY BBA, 2024-06-12) Imane Benabdallah, BenabdallahOur thesisisdevotedtosolvingasignificantandchallengingissueinthequalitativetheory of differentialsystemscalledthesixteenthHilbertproblem.Moreprecisely,weusethefirst integralstodeterminethemaximumnumberoflimitcyclesofsomefamiliesofdiscontinuous piecewise nonlineardifferentialsystemsseparatedbyastraightlineItem Phase planes and bifurcations in planar linear-quadratic differential systems with a pseudo-focus(UNIVERSITY BBA, 2024-06-12) Barkat, MeriemOur thesis isdividedinthreeparts,thefirstpartconsistsinsolvingthesecondpartofthe extended 16thHilbertproblemforaclassofdiscontinuouspiecewisedifferentialsystems.The second partfocusesonfindingthemaximumnumberoflimitcyclesofsmallamplitudewhichis called thecyclictyproblem,andthethirdpartwewereabletofindtheglobalphaseportraitsand the bifurcationsetsforsomespecificfamiliesofdiscontinuouspiecewisequadraticdifferential systems, characterizedbyhavingapseudo-centreattheoriginItem Limit cycles of discontinuous piecewise differential systems separated by a non–regular line and formed by an arbitrary linear center and an arbitrary quadratic center(UNIVERSITY BBA, 2024-06-11) Baymout, LouizaThis thesisconsistsoftwoimportantparts,thefirstoneisdevotedtothe study oftheupperboundonthenumberoflimitcyclesthatcanbecreatedfrom three differentnon-linearfamiliesofdiscontinuouspiecewisedifferentialsystems separated byaregularline. The secondpartfocusesonthestudyoftheexistenceandthemaximumnumber of limitcyclesofaclassofnon-lineardiscontinuouspiecewisedifferentialsystems but inthiscaseweuseanirregularlineastheseparationcurveinsteadofregular line.Item Numerical treatment of stochastic differential equations: Diffusion and jump-diffusion processes with applications(UNIVERSITY BBA, 2024-06-13) Boukhelkhal, IkramIn thisdissertation,wedealwiththeproblemofsimulatingstochasticdifferentialequations driven byBrownianmotionorthegeneralL´evy processes.First,weestablishthebasic theory ofstochasticcalculusandintroducetheIt ˆo-Taylorexpansionforstochasticdifferen- tial equations(SDEs).Inaddition,wepresentvariousnumericalschemesderivedfrom the It ˆo-Taylorexpansion.ThesemethodsareusedtosolvethestochasticLorenzequa- tion, thestochasticDuffingequation,andtheMertonmodelequation.Inaddition,spec- tral techniquesareadaptedforthenumericalsolutionofnonlinearstochasticdifferential equations. Further,generalizedLagrangeinterpolationfunctionsareproposedforsolving various typesofSDEs,offeringsignificantperformanceimprovements.Item Compactly supported radial basis functions(Université de Bordj Bou Arreridj Faculty of Mathematics and Computer Science, 2023) TAKOUK, DalilaThis thesis deals with the applications of compactly supported radial basis functions for high dimensional reconstruction of surfaces (images) based on irregular samples. These methods without mesh (meshfree) based on the introduction of radial basis functions, contrary to traditional methods, namely finite element (FEM) and finite difference (FDM) methods. We try to introduce the concept of this technique through several applications Cette thèse traite les applications des fonctions de base radiale, à support compact (CSRBF), pour la reconstruction bidimensionnelle de surfaces (images) à partir d’échantillons irréguliers. Ces méthodes sans maillage (meshefree) qui reposent sur l’introduction des fonctions de base radiale, contrairement, aux méthodes traditionnelles, a savoir la méthode des éléments finis (FEM) et la méthode des différences finies (FD). Nous essayons d’introduire le concept de cette méthode à travers plusieurs applicationsتناول ذهطروحةتطبيقاتوظائفساس الشعا المدعومة شل مضغوطلإعادة بناءسطح ذات عاد العالية اسنادا إ عينات غ منتظمة. ذه الطرق بدون شبكة سند إ إدخال وظائف ساس الشعا، ع عكس الطرق التقليدية، كطرقة الفروق ادودة وطرقة العناصر ادودة. نحاول تقديم مفوم ذه الطرقة من خلال عدة تطبيقات. .Item Periodic orbits of differential systems via averaging theory(Université de Bordj Bou Arreridj Faculty of Mathematics and Computer Science, 2022) Loubna, DAMENEفي رسالتنا قمنا بدراسة الديناميكية النوعية لبعض الفئات من األنظمة التفاضمية المستوية و غير الخطية. بالتحديد و كخطوة أولي تمكنا من حل الجزء الثاني من المسألة السادس عشرة لهيمبرت لألنظمة الخطية المفصولة بمنحنيات جبرية مكعبة غير قابمة لالختزال. حيث أننا درسنا أوال األنظمة المركزية الخطية. - بعدها مزجنا بين األنظمة الخطية لهميمتون و األنظمة المركزية. كما تمكنا أيضا من إيجاد جميع الحمول الهندسية في قرص بوانكاريه لألنظمة التفاضمية لما يسمى بأنظمة كوكمز من الدرجة الثامنة. Notre thèse est divisée en deux parties, la première partie consiste à résoudre la deuxième partie du seizième problème de Hilbert de trois classes différentiels discontinues linéaires par morceaux en utilisant les intégrales premières. La deuxième partie s'articule sur le problème de cyclicité pour une classe de système différentiel de Kukles, où on a utilisé la méthode de la moyenne jusqu'à l'ordre sept pour obtenir le nombre maximal de cycle limite de ce système. Our thesis is divided in two parts, the first part consists in solving the second part of the sixteenth Hilbert problem of three piecewise linear discontinuous differential classes by using the first integrals. The second part focuses on the cyclicity problem for a class of Kukles differential system, where we used the method of averaging up to order seven to obtain the maximum number of limit cycles of this system.Item Mapped spectral methods and rational approximations(Université de Bordj Bou Arreridj Faculty of Mathematics and Computer Science, 2023) REMILI, WALIDL’objectif principal de cette thèse est d’approximer les solutions de certains problèmes mathématiques sous la forme d’équations intégrales ou d’équations différentielles sur des domaines non bornés. Une stratégie courante et efficace pour traiter les domaines illimités consiste à utiliser un mappage approprié qui transforme un domaine infini. Dans cette thèse, nous introduisons un nouveau système orthogonal de fonctions de Jacobi mappées qui sont les images des polynômes de Jacobi classiques sous l’application inverse. Les méthodes spectrales de Jacobi modifiées sont proposées pour les équations différentielles du second ordre et intégrales non linéaires sur le domaine semi-infini. The main aim of this thesis is to approach the solutions of some mathematical problems in the form of integral equations or differential equations on unbounded domains. A common and effective strategy in dealing with unbounded domains is to use a suitable mapping that transforms an infinite domain. In this thesis, we introduce a new orthogonal system of mapped Jacobi functions which is the images of classical Jacobi polynomials under the inverse mapping. The modified Jacobi spectral methods are proposed for second-order differential and nonlinear integral equations on the semi-infinite domain.