Approximating non linear higher order ODEs by a three point block algorithm.

Ahmad Fadly Nurullah Rasedee,Norizarina Ishak,Khairil Iskandar Othman,Mohammad Hasan Abdul Sathar,Siti Raihana Hamzah,Naramgari Sandeep

doi:10.1371/journal.pone.0246904

Abstract

Differential equations are commonly used to model various types of real life applications. The complexity of these models may often hinder the ability to acquire an analytical solution. To overcome this drawback, numerical methods were introduced to approximate the solutions. Initially when developing a numerical algorithm, researchers focused on the key aspect which is accuracy of the method. As numerical methods becomes more and more robust, accuracy alone is not sufficient hence begins the pursuit of efficiency which warrants the need for reducing computational cost. The current research proposes a numerical algorithm for solving initial value higher order ordinary differential equations (ODEs). The proposed algorithm is derived as a three point block multistep method, developed in an Adams type formulae (3PBCS) and will be used to solve various types of ODEs and systems of ODEs. Type of ODEs that are selected varies from linear to nonlinear, artificial and real life problems. Results will illustrate the accuracy and efficiency of the proposed three point block method. Order, stability and convergence of the method are also presented in the study.

Highlights

The multistep method was discovered by Bashforth and Adams [1] in their pursue to extend Euler’s method
The current study proposes a three-point block multistep formulation (3PBCS) in backward difference form for solving higher order ordinary differential equations (ODEs) directly
For a more just comparison, the 3PBCS method is compared against Direct Integration (DI), one point block

Summary

Introduction

The multistep method was discovered by Bashforth and Adams [1] in their pursue to extend Euler’s method. The idea which was named the Adams-Bashforth method was formulated by obtaining the approximated solution for a point by way of solution values from multiple previous steps. Milne [2, 3] established a new form of multistep method, known as the predictor-corrector formulation. The modern multistep method was widely researched by authors such as [4,5,6,7,8,9]. Krogh [6], managed to revitalize the field of numerical method for solving ordinary differential equations (ODEs) which was almost discarded as robust, with a divided difference approach. Krogh presented a comparison for two second order

Methods

Results

Conclusion