Kamel Ariffin Mohd Atan

Partitioning ordinary differential equations using Runge-Kutta methods

Two techniques for detecting stiffness when using Runge-Kutta type of methods are discussed and compared, and a partitioning strategy for first-order system of equations into stiff and nonstiff subsystems is proposed. A few problems are solved using three-stage semi-implicit Runge-Kutta method. Newton iteration is used for the stiff part and simple iteration for the nonstiff. Finally, numerical results based on different criteria to detect stiffness are compared.

Improvement to scalar multiplication on Koblitz curves by using pseudo τ-adic non-adjacent form

Pseudo τ-adic non-adjacent form (pseudoTNAF) for elliptic scalar multiplication on Koblitz Curve was developed by Faridah et al. since 2012. This is analog to binary method and alternative to τ-adic non-adjacent form (TNAF) and reduced τ-adic non-adjacent form (RTNAF) methods that was produced by Solinas at the year 1997 and 2000 respectively. The objective of this paper is to improve the scalar multiplication algorithm with pseudoTNAF that was published earlier. Consequently, to prove that the density of the pseudoTNAF Hamming weights (HW) is less four percents than the HW of both TNAF and RTNAF.

A Nonclassical Radau Collocation Method for Nonlinear Initial-Value Problems with Applications to Lane-Emden Type Equations

We propose a numerical method for solving nonlinear initial-value problems of Lane-Emden type. The method is based upon nonclassical Gauss-Radau collocation points, and weighted interpolation. Nonclassical orthogonal polynomials, nonclassical Radau points and weighted interpolation are introduced on arbitrary intervals. Then they are utilized to reduce the computation of nonlinear initial-value problems to a system of nonlinear algebraic equations. We also present the comparison of this work with some well-known results and show that the present solution is very accurate.

The Atom Bond Connectivity Index of Some Trees and Bicyclic Graphs

The atom bond connectivity (ABC) index is one of the recently most investigated degree-based molecular structure descriptors that have applications in chemistry. For a graph G, the ABC index is defined as \( ABC(G) = \sum\nolimits_{uv \in E(G)} {\sqrt {d_{v} + d_{u} - 2/d_{v} \cdot d_{u} } } \), where d u denotes the degree of a vertex u in G. In this paper, we obtain the general formula for ABC index of some special, chemical trees, and bicyclic graphs.

Eccentric Connectivity Index of Certain Classes of Cycloalkenes

Let G be a simple connected molecular graph. The eccentric connectivity index \( \upxi(G) \) is defined as \( \xi (G) = \mathop \sum \nolimits_{v\epsilon V(G)} { \text{deg} } (v)\,\text{ec}(v) \), where \( { \text{deg} }(v) \) denotes the degree of vertex v and \( {\text{ec}}(v) \) is the largest distance between v and any other vertex \( u\,\epsilon\, G. \) In this paper, we establish the general formulas for the eccentric connectivity index of molecular graphs of cycloalkenes.

On atom bond connectivity index of some molecular graphs

The atom-bond connectivity (ABC) index is one of the newly most studied degree based molecular structure descriptors, which have chemical applications. For a graph G, the ABC index can be defined as ABC(G)=Σuv∈E(G)dv+du−2/dv.du, where du, the degree of the vertex u is the number of edges with u as an end vertex denotes the degree of a vertex u in G. In this paper, we establish the general formulas for the atom bond connectivity index of molecular graphs of alkenes and cycloalkenes.

An explicit estimation of p-adic sizes of common zeros of partial derivative polynomials associated with a complete quartic form

In this paper we obtain an estimation of p-adic sizes of common zeros of partial derivative polynomials associated with a complete quartic polynomial by applying the Newton polyhedron technique. Such estimates are obtained by examining indicator diagrams associated with the Newton polyhedra of partial derivatives polynomials considered and applying new conditions to improve the results of earlier researchers.

An estimation of p-adic sizes of common zeros of partial derivative polynomials associated with a quartic form

In this paper we apply Newton polyhedron technique in estimating the p-adic sizes of common zeros of partial derivative polynomial associated with a quartic polynomial. It is found that the p-adic sizes of a common zeros can be determined explicitly in terms of the p-adic orders of coefficients of dominant terms of polynomial.

New construction of wavelets base on floor function

In this paper, the properties of the floor function has been used to find a function which is one on the interval [0,1) and is zero elsewhere. The suitable dilation and translation parameters lead us to get similar function corresponding to the interval [a,b). These functions and their combinations enable us to represent the stepwise functions as a function of floor function. We have applied this method on Haar wavelet, Sine-Cosine wavelet, Block-Pulse functions and Hybrid Fourier Block-Pulse functions to get the new representations of these functions.