Elayaraja Aruchunan (2012) Modification of conjugate gradient iterative method for solving linear Fredholm integro-differential equations. Masters thesis, Universiti Malaysia Sabah.
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Abstract
Integro-Differential Equation (IDE) is an important branch of modern mathematics and arises frequently in many engineering and finance problems. In this study, first, second and fourth order linear Fredholm Integro-differential equations are considered and solved by using numerical techniques. The main objective of this research is to analyse the effectiveness of Conjugate Gradient (CG) iterative methods in solving systems of linear equations. CG methods implemented in this research are from the families of Krylov subspace methods. In this study, these methods can be categorized to Full-sweep Conjugate Gradient (FSCG) method and its modification namely Half-Sweep Conjugate Gradient (HSCG) and Quarter-Sweep Conjugate Gradient (QSCG) methods. The formulation and implementation of these three methods were elaborated in solving the linear systems generated from the corresponding approximation equations. In term of discretization schemes, finite difference scheme was adopted to discretize differential term and closed Newton- Cotes quadrature schemes were used for integral term to discretize the proposed problems. In the line to derive an approximation equation, the second-order central difference scheme is imposed to differential terms and the repeated trapezoidal, repeated Simpson’s 1/3 and repeated Simpson’s 3/8 schemes for integral term. Based on the numerical tests, the results showed that the CG iterative methods are more superior in terms of computational time and number of iterations as compared to the Full-Sweep Gauss-Seidel. This is because of the concept of the half-sweep and quarter-sweep complexity reduction that are applied to the CG and GS can reduce computational complexity approximately 50% and 75% respectively. In the meantime, the accuracy of approximate solutions for the proposed iterative methods are nearly similar compared to the full-sweep case.
| Item Type: | Thesis (Masters) |
|---|---|
| Keyword: | Integro-differential equations, Fredholm integro-differential equations, Conjugate gradient, Krylov subspace methods, Finite difference scheme, Newton-cotes quadrature, Numerical methods, Computational complexity, Full-sweep gauss-seidel, Iterative methods |
| Subjects: | Q Science > QA Mathematics > QA1-939 Mathematics > QA299.6-433 Analysis |
| Department: | SCHOOL > School of Science and Technology |
| Depositing User: | DG MASNIAH AHMAD - |
| Date Deposited: | 08 Apr 2025 16:07 |
| Last Modified: | 08 Apr 2025 16:07 |
| URI: | https://eprints.ums.edu.my/id/eprint/43402 |
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