Equivalence of Ratio and Residual Approaches in the Homotopy Analysis Method and Some Applications in Nonlinear Science and Engineering

Abstract

A ratio approach based on the simple ratio test associated with the terms of homotopy series was proposed by the author in the previous publications. It was shown in the latter through various comparative physical models that the ratio approach of identifying the range of the convergence control parameter and also an optimal value for it in the homotopy analysis method is a promising alternative to the classically used h-level curves or to the minimizing the residual (squared) error. A mathematical analysis is targeted here to prove the equivalence of both the ratio approach and the traditional residual approach, especially regarding the root-finding problems via the homotopy analysis method. Examples are provided to further justify this. Moreover, it is conjectured that every nonlinear differential equation can be considered as a root-finding problem by plugging a parameter in it from a physical viewpoint. Two examples from the boundary and initial and value problems are provided to verify this assertion. Hence, besides the advantages as deciphered in the previous publications, the feasibility of the ratio approach over the traditional residual approach is made clearer in this paper.

Theory and Semi-Analytical Study of Micropolar Fluid Dynamics through a Porous Channel

“…We consider the following additional boundary conditions which lead the solution of the Eqs. ( 22)- (24) as…”

“…However, there are some nonlinear situations that do not have a perturbation quantity like HAM [17,18], HPM [19], OHAM [20] and ADM [21]. Similarly some further updated versions of the aforementioned methods have been applied to study various problems in fluid mechanics, we refer few as [22][23][24][25].…”

Heat transmission in Darcy-Forchheimer flow of Sutterby nanofluid containing gyrotactic microorganisms

“…Platt (1961) gave the concept of bioconvection. The most relevant studies have been reported by Bhatti et al (2021), Khan et al (2017), Turkyilmazoglu (2019, 2022), Balla et al (2020), Khan and Makinde (2014) and Khan et al (2014).…”

Transient heat transfer and electro-osmotic flow of Carreau–Yasuda non-Newtonian fluid through a rectangular microchannel

“…EOF is simulated by Chang and Tsao (2007). Few staple investigations on different non-Newtonian fluid models can be seen in Das and Chakraborty (2006), Zhao et al (2008), Zhao and Yang (2011), Zhao and Yang (2012), Zhao and Yang (2013b), Berli and Olivares (2008), Berli (2010), Turkyilmazoglu (2022), Othman et al (2019), Riaz et al (2021a), Zhang et al (2021), Goodarzi et al (2020), Turkyilmazoglu (2019), Saffarian et al (2020), Ellahi et al (2019), Awan et al (2022), Ellahi (2013) and Elelamy et al (2020); however, a hybrid model for non-Newtonian fluids in the said studies is more general form of the Carreau non-Newtonian model. At low and intermediate shear rate, a Carreau fluid behaves as a Newtonian and power law fluid, respectively (Alqarni et al , 2022; Riaz and Sadiq, 2020; Nabwey et al , 2022a; Jalali et al , 2020).…”