Stabilité et convergence des méthodes spectrales Application aux équations intégrales
| dc.contributor.author | Radjai, abir | |
| dc.date.accessioned | 2024-10-22T11:46:47Z | |
| dc.date.available | 2024-10-22T11:46:47Z | |
| dc.date.issued | 2024 | |
| dc.description.abstract | In 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 approach | en_US |
| dc.identifier.issn | MD/30 | |
| dc.identifier.uri | http://10.10.1.6:4000/handle/123456789/5658 | |
| dc.language.iso | fr | en_US |
| dc.publisher | Université de Bordj Bou Arreridj Faculty of Mathematics and Computer Science | en_US |
| dc.subject | Equations intégrales non lineaires, la méthode de collocation, approximation rationnelle, les polynômes de Legendre analyse de la convergence, la stabilité, | en_US |
| dc.title | Stabilité et convergence des méthodes spectrales Application aux équations intégrales | en_US |
| dc.type | Thesis | en_US |
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- Spectral methods have developed rapidly in the past two decades. They have been applied successfully to numerical simulations in many fields, such as heat conduction, fluid dynamics, quantum mechanics, and so on. Nowadays, they are some of the most powerful tools for numerical solutions of integral equations. The principal aim is to present the fundamental principles of spectral methods to the solution of integral equations, and to demonstrate the improved convergence and stability obtained with classical basis functions for certain problems. Firstly, we present an introduction to the terminology and classification of integral equations through certain characteristics and criteria. We saw the aspect of a spectral method, general principle and advantages. We also present some spectral methods namely the collocation and Galerkin method. Secondly, we recall basic consepts of numerical stability with some examples then we analyze the basic theoretical properties of spectral method (stability, convergence and consistency). We illustrate also the relation between the convergence and stability of the previous methods. Then, we close with some numerical examples they are presented to illustrate the convergence and stability of collocation method for the resolution of Fredholm integral equations on the real line. Finally, we have tried to apply the rational Legendre functions which are created by combining the classical Legendre polynomials with algebraic mapping for the numerical solution of quadratic Urysohn integral equations in the half line. We have stated the theorems on the convergence and error estimates of the method, and we have proved them for both L∞ and weighted L 2 norms. The obtained results from the numerical examples have shown that the present method is efficient and stable. This approach can be extended to solve a broad class of unbounded interval problems as quadratic Hammerstein integral equations on the half-line by the use of the rational collocation method where the idea of a future work will be.
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