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Title: | Numerical study of boundary problems for partial differential equations |
Authors: | Zineb, Laouar |
Keywords: | Partial differential equations, spectral approximation, Galerkin method/Collocation method, finite differences scheme, Integral/integrodifferential equations, Fractional equations, Legendre/Chebyshev polynomials, Gauss types quadrature. Equations diff´erentielles partielles, Approximation spectrale, M´ethode de Galerkin/collocation, Sch´ema de diff´erences finies, Equations int´egrales/integrodiff ´erentielles, Equations fractionnaires, Polynˆomes de Legendre/Chebyshev, Quadrature de Gauss. |
Issue Date: | Jul-2024 |
Publisher: | University Center of Abdelhafid boussouf -Mila |
Abstract: | The aim of this work is to study various problems of mathematical equations using spectral methods. It develops four numerical techniques suitable for every studied problem and shows efficiency throughout different numerical illustrations. This study proposes a Legendre Galerkin method coupled with finite differences technique for the advection-diffusion equation with perturbed Robin boundary conditions. The obtained results provide two ways to confirm the efficiency: firstly, by calculating the error of approximation, and simultaneously by comparing the obtained approximate solution to the exact solution of the problem with Dirichlet boundary conditions. For the same equation, a second scheme is proposed using the spectral Galerkin method for both temporal and spatial discretizations. On the same axis, a transition to integral/integro-differential equations is introduced. A novel way of writing the basis functions as compact combinations of orthogonal polynomials using the set of initial conditions is elaborated in a Galerkin method for the integral and integro-differential equations, depending on the order of derivation. Additionally, some numerical techniques, such as using Gauss types quadrature, are also investigated for more accuracy. The last set of results pertains to the study of integro-differential equations of fractional order. Some interestingestimations are formulated to approximate the solution using orthogonal polynomials in a collocation method. All the presented techniques are supported by numerical examples that cover a vast range of cases, to demonstrate the efficiency of the proposed algorithms |
Description: | Le but de ce travail est d’´etudier quelques probl`emes d’´equations math´ematiques `a l’aide des m´ethodes spectrales. On d´eveloppe quatre techniques num´eriques adapt´ees aux ´equations ´etudi´ees et on d´emontre leur efficacit´e. On propose une m´ethode de Galerkin Legendre pour l’´equation d’advection-diffusion, avec des conditions aux limites perturb´ees de type Robin. Les r´esultats obtenus offrent deux fa¸cons de confirmer l’efficacit´e de la m´ethode: d’abord, en calculant l’erreur d’approximation, puis en comparant la solution approximative obtenue `a la solution exacte du probl`eme avec des conditions aux limites de Dirichlet. Pour la mˆeme ´equation, un second sch´ema est propos´e en utilisant une m´ethode de Galerkin pour les discr´etisations temporelles et spatiales. Sur le mˆeme axe, une transition vers les ´equations int´egrales/int´egro-diff´erentielles est introduite. Une nouvelle m´ethode d’´ecrire les fonctions de base, sous forme de combinaisons compactes de polynˆomes orthogonaux utilisant l’ensemble des conditions initiales est ´elabor´ee dans une m´ethode de Galerkin, en fonction de l’ordre de d´erivation. Certaines techniques num´eriques telles que l’utilisation de la quadrature de Gauss sont ´egalement explor´ees pour plus de pr´ecision. Les derniers r´esultats sont li´es `a l’´etude des ´equations int´egro-diff´erentielles d’ordre fractionnaire. Des estimations int´eressantes sont formul´ees pour approximer la solution de tels probl`emes en utilisant des polynˆomes orthogonaux dans une m´ethode de collocation. Toutes les techniques pr´esent´ees sont ´etay´ees par des exemples num´eriques traitant le maximum de cas possibles afin de d´emontrer l’efficacit´e des algorithmes propos´es. |
URI: | http://dspace.centre-univ-mila.dz/jspui/handle/123456789/3436 |
Appears in Collections: | Mathematics |
Files in This Item:
File | Description | Size | Format | |
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Numerical study of boundary problems for partial defferential equations.pdf | 4,19 MB | Adobe PDF | View/Open |
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