Jumat Sulaiman

Numerical solutions for linear Fredholm integral equations of the second kind using 2-point half-sweep explicit group method

In this paper, performance of the 2-Point Half-Sweep Explicit Group (2-HSEG) iterative method with first order composite closed Newton-Cotes quadrature scheme for solving second kind linear Fredholm integral equations is investigated. The formulation and implementation of the method are described. Furthermore, numerical results of test problems are also presented to verify the performance of the method compared to 2-Point Full-Sweep Explicit Group (2-FSEG) method. From the numerical results obtained, it is noticeable that the 2-HSEG method is superior to 2-FSEG method, especially in terms of computational time.

Numerical performance of AOR methods in solving first order composite closed Newton-Cotes quadrature algebraic equations

In this paper, the application of the Accelerated Over-Relaxation (AOR) iterative method is extended to solve first order composite closed Newton-Cotes quadrature (1-CCNC) algebraic equations arising from second kind linear Fredholm integral equations. 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.

Denoising solar radiation data using coiflet wavelets

Signal denoising and smoothing plays an important role in processing the given signal either from experiment or data collection through observations. Data collection usually was mixed between true data and some error or noise. This noise might be coming from the apparatus to measure or collect the data or human error in handling the data. Normally before the data is use for further processing purposes, the unwanted noise need to be filtered out. One of the efficient methods that can be used to filter the data is wavelet transform. Due to the fact that the received solar radiation data fluctuates according to time, there exist few unwanted oscillation namely noise and it must be filtered out before the data is used for developing mathematical model. In order to apply denoising using wavelet transform (WT), the thresholding values need to be calculated. In this paper the new thresholding approach is proposed. The coiflet2 wavelet with variation diminishing 4 is utilized for our purpose. From numerical results it can be seen clearly that, the new thresholding approach give better results as compare with existing approach namely global thresholding value.

Preconditioned Jacobi-type iterative methods for solving Fredholm integral equations of the second kind

In this paper, performance analysis of the preconditioned Jacobi-type iterative methods for solving linear system arise from Fredholm integral equations of the second kind is investigated. The formulation and implementation of the proposed methods are presented. Also, numerical results are included in order to verify the performance of the methods.

Performance analysis of 2-Point explicit group iterative methods for solving four-point composite closed Newton-Cotes system

In this paper, numerical solutions of second kind Fredholm integral equation on a bounded domain are considered. We convert the integral equation to a linear system by using four-point composite closed Newton-Cotes (4-CCNC) scheme. Then, the performance of 2-Point Explicit Group (2-EG) method with complexity reduction approach for solving the generated linear system is investigated. Numerical results by solving test problems are included to verify the performance of the proposed method.

Valuing option on the maximum of two assets using improving modified Gauss-Seidel method

This paper presents the numerical solution for the option on the maximum of two assets using Improving Modified Gauss-Seidel (IMGS) iterative method. Actually, this option can be governed by two-dimensional Black-Scholes partial differential equation (PDE). The Crank-Nicolson scheme is applied to discretize the Black-Scholes PDE in order to derive a linear system. Then, the IMGS iterative method is formulated to solve the linear system. Numerical experiments involving Gauss-Seidel (GS) and Modified Gauss-Seidel (MGS) iterative methods are implemented as control methods to test the computational efficiency of the IMGS iterative method.

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.

The arithmetic mean iterative method for solving 2D Helmholtz equation

In this paper, application of the Arithmetic Mean (AM) iterative method is extended by solving second order finite difference algebraic equations. The performance of AM method in solving second order finite difference algebraic equations is comparatively studied by their application on two-dimensional Helmholtz equation. Numerical results of AM method in solving two test problems are included and compared with the standard Gauss-Seidel (GS) method. Based on the numerical results obtained, the results show that AM method is better than GS method in the sense of number of iterations and CPU time.

Complexity reduction approach for solving hyperbolic problems

Complexity reduction approach has been used to solve various science and technology problems. In this paper we will discuss the implementation of the approach to solve some hyperbolic equation such as first order hyperbolic problem and the Maxwell Equations. For solving the Maxwell equations, we implement a weighted average fourth order truncation with the complexity reduction approach. The approach shown to successfully reduce the complexity of original method. Results show to increase the speed up of its original method significantly.

Recent Development of Fast Numerical Solver for Elliptic Problem

Most elliptic solvers developed by researchers need long processing time to be solved. This is due to the complexity of the methods. The objective of this paper is to present new finite difference and finite element methods to overcome the problem. Solving scientific problems mathematically always involved partial differential equations. Two recommended common numerical methods are mesh-free solutions (Belytschko et al, 1996; Zhu 1999; Yagawa & Furukawa, 2000) and mesh-based solutions. The mesh-based solutions can be further classified as finite difference method, finite element method, boundary element method, and finite volume method. These methods have been widely used to construct approximation equations for scientific problems. The developments of numerical algorithms have been actively done by researchers. Evans and Biggins (1982) have proposed an iterative four points Explicit Group (EG) for solving elliptic problem. This method employed blocking strategy to the coefficient matrix of the linear system of equations. By implementing this strategy, four approximate equations are evaluated simultaneously. This scenario speed up the computation time of solving the problem compared to using point based algorithms. At the same time, Evans and Abdullah (1982) utilized the same concepts to solve parabolic problem. Four years later, the concept has been further extended to develop two, nine, sixteen and twenty five points EG (Yousif & Evans, 1986a). These EG schemes have been compared to one and two lines methods. As the results of comparison, the EG solve the problem efficiently compared to the lines methods. Utilizing higher order finite difference approximation, a method called Higher Order Difference Group Explicit (HODGE) was developed (Yousif & Evans, 1986b). This method have higher accuracy than the EG method. Abdullah (1991) modified the EG method by using rotated approximation scheme. The rotated scheme is actually rotate the ordinary computational molecule by 45° to the left. By rearranging the new computational molecule on the solution domain, only half of the total nodes are solved iteratively. The other half can be solved directly using the ordinary computational molecule. This method was named