Doctora Mathématiques
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Item 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.