Accélération de la convergence de méthodes numériques pour résoudre des équations intégrales
Date
2025
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
university of bordj bou arreridj
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
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.
Keywords
: Linear integral equations, mappings for improved accuracy, adaptive spectral collocation method, spectral accuracy, convergence analysis