Least Square Homotopy Perturbation Method for Ordinary Differential Equations

Mubashir Qayyum,Imbsat Oscar,Gaetano Luciano

doi:10.1155/2021/7059194

Abstract

In this study, a new modification of the homotopy perturbation method (HPM) is introduced for various order boundary value problems (BVPs). In this modification, HPM is hybrid with least square optimizer and named as the least square homotopy perturbation method (LSHPM). The proposed scheme is tested against various linear and nonlinear BVPs (second to seventh order DEs). Validity of the obtained solutions is confirmed by finding absolute errors. To analyze the efficiency of the proposed scheme, tested problems have also been solved through HPM and results are compared with LSHPM. Furthermore, obtained results are also compared with other numerical schemes available in literature. Analysis reveals that LSHPM is a consistent and effective scheme which can be used for more complex BVPs in science and engineering.

Highlights

Most of the phenomena in mathematical physics, biological mathematics, and applied mathematics are modeled in the form of differential equations
A new modification of homotopy perturbation method (HPM) has been introduced by hybriding HPM with LS optimizer. e pro(71) posed scheme has been tested agaist various order linear and nonlinear boundary value problems (BVPs). e validity of least square homotopy perturbation method (LSHPM) has been checked by comparing exact and approximate solutions
For testing the efficiency LSHPM, problems are solved with HPM and results are compared with LSHPM. is can be observed in Tables 1–12. ese tables signify the efficiency of LSHPM in terms of high accuracy with less computational cost. e convergence of LSHPM can be observed in Figures 1–12. ese figures show that LSHPM is more consistent as compared other mentioned schemes

Summary

Introduction

Most of the phenomena in mathematical physics, biological mathematics, and applied mathematics are modeled in the form of differential equations. Solutions to such differential equations are needed to analyze and predict changes in a physical system. He introduced the homotopy perturbation method (HPM) for highly nonlinear equations [5]. Various modifications of HPM have been proposed by different researchers to tackle more complex problems. Bota and Caruntu applied HPM with the least square method to fluid flow problems in [15]. Ji et al further applied Le-He extension of HPM to nonlinear packaging system in [17]. HPM is combined with LS optimizer along with some refine inital guesses to obtain fast convergent semianalytical solutions of linear and nonlinear BVPs. is scheme takes few iterations (cycles) to achieve accurate solution, and it has less computational cost with improved accuracy

Basic Idea of LSHPM

Application of LSHPM

Results and Discussion

Conclusion