An accurate method for solving a singular second-order fractional Emden-Fowler problem

Abstract

In this paper, we study a singular second-order fractional Emden-Fowler problem. The reproducing kernel Hilbert space method (RKHSM) is employed to compute an approximation to the proposed problem. The construction of the reproducing kernel based on orthonormal shifted Legendre polynomials is presented. The validity of the RKHSM is ascertained by presenting several examples. We prove the existence of solution of the singular second-order fractional Emden-Fowler problem. The convergence of the approximate solution using the proposed method is investigated. The uniform convergence of the approximate solution to the exact solution is presented. Error estimation to the proposed method is proven. The results reveal that the proposed analytical method can achieve excellent results in predicting the solutions of such problems.

A new operational matrix based on Boubaker polynomials for solving fractional Emden-Fowler problem

“…(2002); Das S. (2011); Diethelm K. (2010) ). One of these equations describing many phenomena in mathematical physics and astrophysics such as, the thermal behaviour of a spherical, cloud of gas, isothermal gas sphere and theory of stellar structure, theory of thermionic currents among many others, is called the singular Emden-Fowler equations of fractional order formulated as: Syam, M. (2018); Syam et al (2018); Huan et al (2017); Rebenda and Smarda . (1978) D 2α u(x) + λ x α D α u(x) + s(x)g(u(x)) = h(x), x ∈ (0, 1), λ > 0,…”

“…The problem (1) was studied by using the Residual Power Series Method by Syam, M. (2018), Homotopy analysis method (HAM) by Huan et al (2017), Reproducing kernel Hilbert space method by Syam et al (2018), The fractional differential transformation (FDT) Rebenda and Smarda . (1978), Polynomial Least Squares Method by Caruntu et al (2019), Shifted Legendre Operational Matrix by Tripathi N. (2019), Chebyshev wavelets by Kazemi Nasab et al (2018), Orthonormal Bernoulli's polynomials by Sahu and Mallick.…”