Approximate Solutions for Higher Order Linear and Nonlinear Boundary Value Problems

Abstract

Purpose: The specific objective of this study is to examine the higher order nonlinear BVPs (12th and 13th orders) which perform efficient role in the modeling of physical problems of science and engineering. Design/methodology/approach: An innovative modification of the homotopy perturbation (HP) technique by coupling it with the Laplace transform (LT) has been expended to solve linear and nonlinear higher order boundary-value problems (BVPs). A homotopy is constructed for the given problems (BVPs) by HP technique and solved it by temporal Laplace method. Then Laplace inversion procedure has been used for retrieving the initial dominion solution. Motivation: The motivation of this paper is to introduce an improved and fast converging technique to solve complex higher order nonlinear boundary value problems. Findings: The main finding in this paper is to analyze the higher order nonlinear ordinary differential equations with more accurate approximate solutions. The proposed HPLT solutions show that the present technique provides more accurate, efficient, fast convergence and comparatively small absolute errors for extensive finite range. The authors found no assumption for the constriction of this approach. The computer software Maple has been used to compute numerical results of BVPs. The results obtained from HPLT method are found in excellent agreement with the exact solutions. Research limitations/implications: This paper invokes these two main inspirations: firstly, Laplace transform is associated with homotopy perturbation method in a new manner, secondly, handling of boundary value problems with higher order. Practical implications: In this paper, the values of the approximate solution have excellent Promise with those of exact solutions. Social implications: This paper presents a valuable technique for handling the nonlinear higher order differential equations (ODEs) without involving any restrictions or hypothesis. Originality: The work in present article is original and advanced. Significantly, no such work has yet been published in the literature. © 2021, The Author(s), under exclusive licence to Springer Nature India Private Limited.