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Phase-amplitude method for numerically exact solution of the differential equations of the two-center Coulomb problem Abstract. In this paper, we consider a two-point implicit block method for solving delay differential equations. For greater efficiency, the block method is implemented in variable stepsize technique. The most optimal stepsize is taken while achieving the desired accuracy. The implicit method is solved using predictor-corrector scheme where the corrector is iterated until convergence. Grid point formulae are derived using a predictor and a corrector of order five. The formulae produce two new values in a single integration step. Delay solutions are approximated using Hermite interpolation of order five. The advantage of using Hermite interpolation is that it requires less support points than the existing interpolation technique in order to achieve the overall accuracy. Numerical results indicate that the two-point block method with Hermite interpolation technique is efficient and reliable in solving a wide range of delay differential equations.

Delay differential equations (DDEs) appear naturally in modeling many real life phenomena. DDEs differ from ordinary differential equations since the derivative of the unknown function contains the expression of the unknown function at earlier and present states as well. DDEs that cannot be solved analytically are solved numerically. In this work, we solve DDEs using predictor-corrector multistep method where the corrector is iterated until convergence. The predictor uses the Adams-Bashforth four-step explicit method and the corrector uses Adams-Moulton three-step implicit method. Two types of interpolation polynomials which are Lagrange and Hermite interpolations are used to approximate the delay solutions. The accuracy of the adapted Adams-Bashforth-Moulton methods using these two polynomials is compared.

This paper considers the numerical solution of delay differential equations. The predictor-corrector scheme based on generalized multistep methods are implemented in variable order variable stepsize techniques. The formulae are represented in divided difference form where the integration coefficients are computed by a simple recurrence relation. This representation produces simpler calculation as compared with the modified divided difference form, but no sacrifice is made in efficiency and accuracy of the method. Numerical results prove that the method is reliable, efficient and accurate. The P-and Q-stability regions for a fixed stepsize of the predictor-corrector scheme are illustrated for various orders.

Conjugate Gradient (CG) methods have an important role in solving largescale unconstrained optimization problems. Nowadays, the Three-Term CG method hasbecome a research trend of the CG methods. However, the existing Three-Term CGmethods could only be used with the inexact line search. When the exact line searchis applied, this Three-Term CG method will be reduced to the standard CG method.Hence in this paper, a new Three-Term CG method that could be used with the exactline search is proposed. This new Three-Term CG method satisfies the descent conditionusing the exact line search. Performance profile based on numerical results show thatthis proposed method outperforms the well-known classical CG method and some relatedhybrid methods. In addition, the proposed method is also robust in term of number ofiterations and CPU time.