Browsing by Author "Zabidin Salleh"

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    A Differential Transformation Method for Approximating the Chaotic Chen System
    (International Journal of Mathematical Analysis, 2016) Foo Pei Ling; Zabidin Salleh
    This paper presents approximate analytical solutions for chaotic Chen system which is a three-dimensional system of ordinary differential equations using differential transform method. Comparisons between the fourth-order Runge-Kutta (RK4) methods with different time steps were done. It has been observed that the accuracy of RK4 solutions can be increased by decreasing the time step. Furthermore, the numerical results are compared to those obtained by the Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method.
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    An efficient modification of the Hestenes-Stiefel nonlinear conjugate gradient method with restart property
    (Journal of Inequalities and Applications, 2016) Zabidin Salleh; Ahmad Alhawarat
    The conjugate gradient (CG) method is one of the most popular methods to solve nonlinear unconstrained optimization problems. The Hestenes-Stiefel (HS) CG formula is considered one of the most efficient methods developed in this century. In addition, the HS coefficient is related to the conjugacy condition regardless of the line search method used. However, the HS parameter may not satisfy the global convergence properties of the CG method with the Wolfe-Powell line search if the descent condition is not satisfied. In this paper, we use the original HS CG formula with a mild condition to construct a CG method with restart using the negative gradient. The convergence and descent properties with the strong Wolfe-Powell (SWP) and weak Wolfe-Powell (WWP) line searches are established. Using this condition, we guarantee that the HS formula is non-negative, its value is restricted, and the number of restarts is not too high. Numerical computations with the SWP line search and some standard optimization problems demonstrate the robustness and efficiency of the new version of the CG parameter in comparison with the latest and classical CG formulas. An example is used to describe the benefit of using differen
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    A Note on Topological Entropy of Continuous Self-Maps
    (Journal of Mathematics and System Science, 2015-03) Zabidin Salleh
    Topological entropy can be an indicator of complicated behavior in dynamical systems. It is first introduce by Adler, Konheim and McAndrew by using open covers in 1965. After that it is still an active research by many researchers to produce more properties and applications up to nowadays. The purpose of this paper is to review and explain most important concepts and results of topological entropies of continuous self-maps for dynamical systems on compact and non-compact topological and metric spaces. We give proofs for some of its elementary properties of the topological entropy. Slight modification on Adler’s topological entropy is also presented.