Haar wavelets collocation on a class of Emden-Fowler equation via Newton's quasilinearization and Newton-Raphson techniques

Abstract

In this paper we have considered generalized Emden-Fowler equation,subject to the following boundary conditionswhere γ, β and σ are real numbers, γ < −2, β > 1. We propsoed to solve the above singular nonlinear BVPs with the aid of Haar wavelet coupled with quasilinearization approach as well as Newton-Raphson approach. We have also considered the special case of Emden-Fowler equation (σ = −1,γ = −1 2 and β = 3 2 ) which is popularly, known as Thomas-Fermi equation. We have analysed different cases of generalised Emden-Fowler equation and compared our results with existing results in literature. We observe that small perturbations in initial guesses does not affect the the final solution significantly.

A Comparative Study using Scale-2 and Scale-3 Haar Wavelet for the Solution of Higher Order Differential Equation

“…For these kinds of problem convergence is directly affected by the small parameters. So, Different numerical solutions have been investigated by the researchers for the solution of this particular kind of problems, homotopy analysis method (Singh, 2018), Genocchi operational matrix for solving Emden-fowler equations (Isah and Phang, 2020) it includes quartic polynomial spline method, fourth order B-spline method (Caglar et al, 1999;Akram, 2011), modifies Adomian decomposition method, differential transformation method, variational iteration method (Khuri, 2001;Hasan and Zhu, 2009;Aruna and Kanth, 2013;Wazwaz, 2015a;Wazwaz, 2015b), In the research article Haar scale wavelet method is discussed for obtaining the approximate solution of linear Emdenfowler equations (Alkan, 2017), Emden fowler equation is solved by using HSWM combined with Newton Raphson method and for solving nonlinearity quasi-linearisation technique is used and discussed some special cases of Emden-fowler equation (Verma and Kumar, 2019), Fourth order Emden-fowler equation is discussed using Haar scale collocation method and by converting the differential equation in to set of algebraic equations and through various examples discuss the applicability of the proposed technique (Khan et al, 2017),Third order differential equations are solved using Haar scale wavelet method, through different examples discussed the effectiveness of the method for solving higher order differential equations (Singh and Kaur, 2021), in the study the author developed the solution of higher order boundary value problem using Haar scale wavelet method (Heydari et al, 2022). Fourth order Lane-Emden fowler equation also described by two different methods adomain decomposition and quintic B-spline method (Ali et al, 2022).…”

Neuro-swarm computational heuristic for solving a nonlinear second-order coupled Emden–Fowler model

“…These models are not easy to solve due of this singular model, nonlinearity and stiff nature, and only a few techniques are available in the literature to solve these models. Few of them are Legendre spectral wavelets scheme (Dizicheh et al 2020), Adomian decomposition scheme (Abdullah Alderremy et al 2019), Haar quasilinearization wavelet scheme (Singh et al 2020;Verma and Kumar 2019), an analytic algorithm approach (Arqub et al 2020), rational Legendre approximation scheme (Dizicheh et al 2020), modified variational iteration scheme (Verma et al 2021), differential transformation scheme (Xie et al 2019), fourth-order B-spline collocation scheme (Roul and Thula 2019), Chebyshev operational matrix scheme (Sharma et al 2019) and variation of parameters scheme with an auxiliary parameter (Khalifa and Hassan 2019). Beside these, the numerical methodologies introduced in (Abdelrahman and Alharbi 2021; Alharbi et al 2020;Almatrafi et al 2021;Lotfy 2019;Sabir 2022a, Sabir et al 2022c) can be exploited for EF equations-based systems.…”