Solving Nonstiff Higher-Order Ordinary Differential Equations Using 2-Point Block Method Directly

Hazizah Mohd Ijam,Soheil Salahshour,Ahmad Fadly Nurullah Rasedee,Mohamed Suleiman,Norazak Senu,Ali Ahmadian

doi:10.1155/2014/867095

Hazizah Mohd Ijam, Soheil Salahshour + Show 4 more

Open Access

https://doi.org/10.1155/2014/867095

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Abstract

We describe the development of a 2-point block backward difference method (2PBBD) for solving system of nonstiff higher-order ordinary differential equations (ODEs) directly. The method computes the approximate solutions at two points simultaneously within an equidistant block. The integration coefficients that are used in the method are obtained only once at the start of the integration. Numerical results are presented to compare the performances of the method developed with 1-point backward difference method (1PBD) and 2-point block divided difference method (2PBDD). The result indicated that, for finer step sizes, this method performs better than the other two methods, that is, 1PBD and 2PBDD.

Highlights

We formulate the corrector in terms of the predictor. Both points yn+1 and yn+2 can be written as yn(d+−1t)
The derivation for up to third-order explicit integration coefficients for the first point yn+1 has been given by Suleiman et al [8]
The derivation for up to third-order implicit integration coefficients for the first point yn+1 has been given by Suleiman et al [8]

Summary

Introduction

According to Omar [5], both implicit and explicit block Adams methods in their divided difference form are developed for the solution of higherorder ODEs. Majid [6] has derived a code based on the variable step size and order of fully implicit block method to solve nonstiff higher-order ODEs directly. Ibrahim [7] has developed a new block backward differentiation formula method of variable step size for solving first- and secondorder ODEs directly. 2. The Formulation of the Predict-EvaluateCorrect-Evaluate (PECE) Multistep Block Method in Its Backward Difference Form (MSBBD) for Nonstiff Higher-Order ODEs. The code developed will be using the PECE mode with constant stepsize. We formulate the corrector in terms of the predictor Both points yn+1 and yn+2 can be written as yn(d+−1t) n+1.

Derivation for Higher-Order Explicit Integration Coefficients

Derivation for Higher-Order Implicit Integration Coefficients

The Relationship between the Explicit and Implicit Coefficients

Method

Numerical Result