Doctora Mathématiques

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Accélération de la convergence de méthodes numériques pour résoudre des équations intégrales
(university of bordj bou arreridj, 2025) ABDENNEBI Issam
The 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 freedom