Application of the Central-Difference with Half-Sweep Gauss-Seidel Method for Solving First Order Linear Fredholm Integro-Differential Equations

The objective of this paper is to analyse the application of the Half-Sweep Gauss-Seidel (HSGS) method by using the Half-sweep approximation equation based on central difference (CD) and repeated trapezoidal (RT) formulas to solve linear fredholm integro-differential equations of first order. The formulation and implementation of the Full-Sweep Gauss-Seidel (FSGS) and Half- Sweep Gauss-Seidel (HSGS) methods are also presented. The HSGS method has been shown to rapid compared to the FSGS methods. Some numerical tests were illustrated to show that the HSGS method is superior to the FSGS method

Computational solution of first order linear fredholm integro-differential equations by quarter sweep successive over relaxation method

In this paper the effectiveness of the Quarter-Sweep Successive Over Relaxation (QSSOR) iterative method has been examined corresponding to finite difference-composite trapezoidal discretization schemes in solving first order linear Fredholm integro-differential equations. The mathematical formulations of the standard or Full-Sweep Successive Over Relaxation (FSSOR) methods also presented. Analysis of computational complexities and calculation of percentages reduction in number of iterations and execution time are also given to demonstrate that the QSSOR is superior compared to the standard Successive Over Relaxation method. Several numerical experiments have been shown to support the statements

Quarter-Sweep Iteration Concept on Conjugate Gradient Normal Residual Method via Second Order Quadrature-Finite Difference Schemes for Solving Fredholm Integro-Differential

In this paper, we have examined the effectiveness of the quarter-sweep iteration concept on conjugate gradient normal residual (CGNR) iterative method by using composite Simpson’s (CS) and finite difference (FD) discretization schemes in solving Fredholm integro-differential equations. For comparison purposes, Gauss- Seidel (GS) and the standard or full- and half-sweep CGNR methods namely FSCGNR and HSCGNR are also presented. To validate the efficacy of the proposed method, several analyses were carried out such as computational complexity and percentage reduction on the proposed and existing methods

QSMSOR Iterative Method for the Solution of 2D Homogeneous Helmholtz Equations

In this paper, we consider the numerical solutions of homogeneous Helmholtz equations of the second order. The Quarter-Sweep Modified Successive Over-Relaxation (QSMSOR) iterative method is applied to solve linear systems generated form discretization of the second order homogeneous Helmholtz equations using quarter sweep finite difference (FD) scheme. The formulation and implementation of the method are also discussed. In addition, numerical results by solving several test problems are included and compared with the conventional iterative methods

Efficient iterative approximation for nonlinear porous medium equation with drainage model

The porous medium equation with drainage was applied to model various phenomena in physics and biology fields. The exact solutions of the model are limited, and an efficient numerical approach is demanded to study the model. The main contribution of this research is to develop an iterative approximation that can efficiently solve the porous medium equation with the drainage model. This paper presented a modified successive over-relaxation iterative method derived from using a quarter-sweep implicit finite-difference approximation. Several models of the porous medium equation with drainage are selected to validate the efficiency of the proposed method. The experiment used different sizes of the system of equations to analyse the efficiency of the proposed method. The proposed method is compared against several existing methods, and the results support the superiority of the proposed method in the number of iterations and computer execution time

Quarter-sweep successive over-relaxation approximation to the solution of porous medium equations

This paper investigated the use of a successive over-relaxation parameter in a quarter-sweep finite difference approximation scheme. The performance of the developed quarter-sweep successive over-relaxation method is examined by considering a nonlinear partial differential equation, namely the porous medium equation. The main contribution of this paper is to present the stability, convergence and efficiency of the proposed method. Several initial-boundary value problems of the porous medium equation are solved to illustrate the efficiency of the proposed method. The numerical results showed that the quarter-sweep successive over-relaxation method is more efficient in reducing iterations and computational time than the standard and the existing numerical methods. In addition, the accuracy of the quarter-sweep successive over-relaxation method is comparable to the tested numerical methods

Numerical performance of half-sweep SOR method for solving second order composite closed Newton-Cotes system

In this paper, application of the Half-Sweep Successive Over-Relaxation (HSSOR) iterative method is extended by solving second order composite closed Newton-Cotes quadrature (2-CCNC) system. The performance of HSSOR method in solving 2-CCNC system is comparatively studied by their application on linear Fredholm integral equations of the second kind. The derivation and implementation of the method are discussed. In addition, numerical results by solving two test problems are included and compared with the standard Gauss-Seidel (GS) and Successive Over-Relaxation (SOR) methods. Numerical results demonstrate that HSSOR method is an efficient method among the tested methods

Complexity Reduction Approach for Solving Second Kind of Fredholm Integral Equations

Initially, the concept of the complexity reduction approach was applied to solve symmetry algebraic systems that were generated from the discretization of the partial differential equations. Consequently, in this paper, the effectiveness of a complexity reduction approach based on half- and quarter-sweep iteration concepts for solving linear Fredholm integral equations of the second kind is investigated. Half- and quarter-sweep iterative methods are applied to solve dense linear systems generated from the discretization of the second kind of linear Fredholm integral equations using a repeated modified trapezoidal (RMT) scheme. The formulation and implementation of the proposed methods are presented. In addition, computational complexity analysis and numerical results of test examples are also included to verify the performance of the proposed methods