General approach on the best fitted linear operator and basis function for homotopy methods and application to strongly nonlinear oscillators

Tapas Roy and Dilip K. Maiti

Mathematics and Computers in Simulation (MATCOM), 2024, vol. 220, issue C, 44-64

Abstract: In this study, we propose a universal mathematical guideline for the selection of auxiliary linear operator L(u) and the basis function u0(x), which are the principal parts of Homotopy methods: Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM). We assume the coefficients of derivatives involved in L(u) as a function of auxiliary roots of L(u)=0. Based on the residual error minimization we compute unknown roots and thereby obtain the optimal linear operator at an order of approximation. Also, an effort is made to explore the role of the number of unknowns embedded into L(u) (for different choices of real roots of L(u)=0) on the rate of converges of the solution and CPU uses. Hence the best fitted linear-operator and corresponding basis function are proposed for a particular problem. Also, we propose to discretize the exact square residual using Simpson’s 1/3 algorithms, and compare its accessibility in comparison to other existing residual calculation methods. Initially, we applied our algorithms to three IVPs (Logistic, Stiff system and Riccati system) and three singular BVPs, and then finally to a strongly nonlinear oscillator (cubic-quintic Duffing equation). We compare our technique’s accuracy, efficiency and simplicity with other contemporary analytical and numerical methods. It is demonstrated that our optimal linear operator, which is updated at every order of approximation, is much more efficient, important (than the artificial parameters and functions of developed HAM), and self-sufficient for the convergence of series solutions. Our approach is more effective, straightforward and easy to use when applied to many nonlinear problems arising in mechanics and engineering. It is seen for the initial value problems that our technique provides the globally convergent series solution, while the solutions by HPM and optimal HAM are locally convergent. This novel development will make homotopy methods more powerful.

Keywords: Homotopy analysis method; Optimal auxiliary linear operator; Basis function; Series solution; Analytical methods (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0378475424000041
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:220:y:2024:i:c:p:44-64

DOI: 10.1016/j.matcom.2024.01.004

Access Statistics for this article

Mathematics and Computers in Simulation (MATCOM) is currently edited by Robert Beauwens

More articles in Mathematics and Computers in Simulation (MATCOM) from Elsevier
Bibliographic data for series maintained by Catherine Liu ().