Solution for nonlinear Duffing oscillator using variable order variable stepsize block method

Real life phenomena found in various fields such as engineering, physics, biology and communication theory can be modeled as nonlinear higher order ordinary differential equations, particularly the Duffing oscillator. Analytical solutions for these differential equations can be time consuming whereas, conventional numerical solutions may lack accuracy. This research propose a block multistep method integrated with a variable order step size (VOS) algorithm for solving these Duffing oscillators directly. The proposed VOS Block method provides an alternative numerical solution by reducing computational cost (time) but without loss of accuracy. Numerical simulations are compared with known exact solutions for proof of accuracy and against current numerical methods for proof of efficiency (steps taken)

Variable step direct block multistep method for general second order ODEs

Direct block multistep method is developed for the numerical solution of second order ordinary differential equations (ODEs). This method was designed for computing the solution at four points simultaneously using variable step size. The development of this method based on numerical integration and using interpolation approach which are similar to the Adams method. In order to gain an efficient and reliable numerical approximation, this developed block method is implemented in the predictor corrector mode using simple iteration technique. This method has also been proven as a convergence method under suitable conditions of stability and consistency. Several tested problems are taken into account in the numerical experiments and were compared with the existing method. The results obtained showed that this developed block method managed to produce good results

Numerical algorithm of block method for general second order ODEs using variable step size

This paper outlines an alternative algorithm for solving general second order ordinary differential equations (ODEs). Normally, the numerical method was designed for solving higher order ODEs by converting it into an n-dimensional first order equations with implementation of constant step length. Nevertheless, this involved a lot of computational complexity which led to consumption a lot of time. Consequently, a direct block multistep method with utilization of variable step size strategy is proposed. This method was developed for computing the solution at four points simultaneously and the derivation based on numerical integration as well as using interpolation approach. The convergence of the proposed method is justified under suitable conditions of stability and consistency. Five numerical examples are considered and some comparisons are made with the existing methods for demonstrating the validity and reliability of the proposed algorithm

Variable Step Variable Order Two Point Block Fully Implicit Method for Solving Ordinary Differential Equations

The aim of this paper is to investigate the performance of the developed two point block methods of order 5, 7 and 9 for solving first order Ordinary Differential Equations (ODEs) using variable step size and order. The code will combine three proposed block methods i.e the 2-point one block fully implicit block method of order 5, the 2-point two block fully implicit block method of order 7 and the 2-point three block fully implicit block method of order 9. These methods will estimate the numerical solution at two equally spaced points simultaneously within a block. The existence multistep method involves the computations of the divided differences and integration coefficients when using the variable step size or variable step size and order. The block method developed will be presented in the simple form of the Adams Moulton type. The performances of the code will be compared in terms of maximum error, total number of steps and execution times with the existence non block method and 2-point block method of variable step size and order code

Linear multistep methods for integrating reversible differential equations

This paper studies multistep methods for the integration of reversible dynamical systems, with particular emphasis on the planar Kepler problem. It has previously been shown by Cano & Sanz-Serna that reversible linear multisteps for first-order differential equations are generally unstable. Here, we report on a subset of these methods -- the zero-growth methods -- that evade these instabilities. We provide an algorithm for identifying these rare methods. We find and study all zero-growth, reversible multisteps with six or fewer steps. This select group includes two well-known second-order multisteps (the trapezoidal and explicit midpoint methods), as well as three new fourth-order multisteps -- one of which is explicit. Variable timesteps can be readily implemented without spoiling the reversibility. Tests on Keplerian orbits show that these new reversible multisteps work well on orbits with low or moderate eccentricity, although at least 100 steps/radian are required for stability.Comment: 31 pages, 9 figures, in press at The Astronomical Journa

A One Step Method for the Solution of General Second Order Ordinary Differential Equations

In this paper, an implicit one step method for the numerical solution of second order initial value problems of ordinary differential equations has been developed by collocation and interpolation technique. The introduction of an o step point guaranteed the zero stability and consistency of the method. The implicit method developed was implemented as a block which gave simultaneous solutions, as well as their rst derivatives, at both o step and the step point. A comparison of our method to the predictor-corrector method after solving some sample problems reveals that our method performs better

Optimization of Nordsieck's Method for the Numerical Integration of Ordinary Differential Equations

Stability and accuracy of Nordsieck's integration method can be improved by choosing the zero-positions of the extraneous roots of the characteristic equation in a suitable way. Optimum zero-positions have been found by minimizing the lower bound of the interval of absolute stability and the coefficient of the truncation error. Various properties of the improved methods have been analysed, such as the behaviour of the high-order terms, the equivalence with multistep methods and the damping of perturbations

Multistep integration formulas for the numerical integration of the satellite problem

The use of two Class 2/fixed mesh/fixed order/multistep integration packages of the PECE type for the numerical integration of the second order, nonlinear, ordinary differential equation of the satellite orbit problem. These two methods are referred to as the general and the second sum formulations. The derivation of the basic equations which characterize each formulation and the role of the basic equations in the PECE algorithm are discussed. Possible starting procedures are examined which may be used to supply the initial set of values required by the fixed mesh/multistep integrators. The results of the general and second sum integrators are compared to the results of various fixed step and variable step integrators