This paper presents a direct two-point block one-step method for solving linear Dirichlet boundary value problems (BVPs) directly. The block method is formulated using Lagrange interpolating polynomial. Mathematical problems which involve higher order ordinary differential equations (ODEs) were likely to be reduced into the system of first order equations in order to solve it. However, this method will solve the second order linear Dirichlet BVPs directly without reducing it to the system of first order equations. The direct solution of the linear Dirichlet BVPs will be calculated at the two-points simultaneously using constant step size. This method will be used together with the linear shooting technique to construct the numerical solution. The implementation is based on the predictor and corrector formulas in the PE(CE) r mode. Numerical results are given to show the efficiency and performance of this method compared to the existing methods.

A new compression algorithm used to ensure a modified Baptista symmetric cryptosystem which is based on a chaotic dynamical system to be applicable is proposed. The Baptista symmetric cryptosystem able to produce various ciphers responding to the same message input. This modified Baptista type cryptosystem suffers from message expansion that goes against the conventional methodology of a symmetric cryptosystem. A new lossless data compression algorithm based on theideas from the Huffman coding for data transmission is proposed.This new compression mechanism does not face the problem of mapping elements from a domain which is much larger than its range.Our new algorithm circumvent this problem via a pre-defined codeword list.  The purposed algorithm has fast encoding and decoding mechanism and proven analytically to be a lossless data compression technique.

This paper presents 4-point 1-step block method (4LBVP) to solve the linear 2 nd order boundary value problem (BVP) with Dirichlet boundary condition. 4LBVP will solve the linear 2 nd order BVP with Dirichlet boundary condition directly without the need to reduce it first into the system of 1 st order equations. 4LBVP will produce several numerical solutions simultaneously in one step. The formulation of the 4LBVP will be based on the Lagrange interpolating polynomial. 4LBVP will be used together with the linear shooting technique to produce the numerical solutions. Comparison with other methods will be given to show the advantages of this method.

A direct two-point block one-step method for solving linear Neumann boundary value problems (LNBVP) is considered. This method will solve the second order LNBVP directly without reducing it to the system of first order equations. The direct solution of LNBVP will be calculated at two points simultaneously using constant step size. This method will be used together with the linear shooting technique to construct the numerical solution. The implementation is based on the predictor and corrector formulas in the PE(CE)m mode. Numerical results are given to show the performance of this method compared to the existing methods.