Accélération de la convergence de méthodes numériques pour résoudre des équations intégrales
| dc.contributor.author | ABDENNEBI Issam | |
| dc.date.accessioned | 2025-07-08T10:36:37Z | |
| dc.date.issued | 2025 | |
| dc.description | n this thesis, we have presented accurate spectral solution methods for Volterra and Fred holm IEs, which naturally arise from the mathematical modeling of initial and boundary value problems associated with ordinary and partial DEs. At first, we have applied the Legendre spectral Galerkin method and its iterated version to solve FIEs of the second kind and analyzed their convergence in the L 2 norm. The numerical results obtained in Subsection 4.1.3 show that the iterated Legendre spectral Galerkin solution provides a better approximation than the Legendre spectral Galerkin method. Hence, the iterated Legendre spectral-Galerkin solution improves over the Leg endre spectral-Galerkin solution. The advantage of this method is that its convergence behavior depends solely on the smoothness properties of the solution and kernel. Secondly, we have developed and analyzed an adaptive spectral collocation method for Volterra and Fredholm IEs of the second kind, where the underlying solution exhibits lo calized rapid variations, steep gradients, or a steep front. The suggested approach can enhance the efficiency of the classical algorithm by providing significant computational advantages. This is particularly achieved through the use of a mapping strategy, as shown in the numerical results presented in Subsections 4.3.4. These results clearly show that our method is better than the classical method for such problems. The main advantages of the suggested approach include: (i) Dynamically adjust resolution based on solution characteristics; (ii) Implement moving grid points for capturing the localized rapid variations in the solution of the given problem; (ii) Clustering of grid points around steep gradients or a steep front to effectively track critical features of the solution. The perspectives are: • Extending the results obtained in this thesis to nonlinear IEs with similar challenging properties. • Extending the results obtained in this thesis to higher-dimensions IEs with similar challenging properties. These issues will be addressed in future work. | |
| dc.description.abstract | 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 | |
| dc.identifier.issn | MD/35 | |
| dc.identifier.uri | https://dspace.univ-bba.dz/handle/123456789/332 | |
| dc.language.iso | en | |
| dc.publisher | university of bordj bou arreridj | |
| dc.subject | : Linear integral equations | |
| dc.subject | mappings for improved accuracy | |
| dc.subject | adaptive spectral collocation method | |
| dc.subject | spectral accuracy | |
| dc.subject | convergence analysis | |
| dc.title | Accélération de la convergence de méthodes numériques pour résoudre des équations intégrales | |
| dc.type | Thesis |