An efficient Legendre-wavelet collocation technique for solving Emden-Fowler type equations

N. Saha^a, J. Shahni^a, R. Singh^a, V. Guleria^b, N. Sriwastav<sup>c

a</sup> Department of Mathematics, Birla Institute of Technology Mesra, Ranchi, 835215, India
^b Department of Applied Science and Humanities, National Institute of Advanced Manufacturing Technology Hatia, Ranchi, 834003, India
^c Department of Mathematics, Chandigarh University, Mohali, 140413, India

Abstract: Emden-Fowler-type equations are widely used in mathematical and physical modeling. They describe phenomena in various fields, including astrophysics, quantum mechanics, and nonlinear dynamics. Applications range from modeling stars' thermal behavior to species' distribution in a chemical reaction. Researchers continuously seek new methods to solve Emden-Fowler (EF)-type equations more efficiently and accurately due to their versatility and richness. This article presents a novel approach for solving the generalized EF equations subject to boundary conditions using the Legendre wavelet. First, we convert the problem into equivalent Fredholm integral equations. Next, we use a Legendre wavelet collocation approach and a Newton-Raphson iterative technique to solve the resulting integral equations. The formulation of the proposed algorithm is further supported by its convergence and error analysis. We examine the accuracy of the method by computing the numerical solution and errors for various examples. We compare our numerical outcomes to exact solutions and those achieved by techniques in the literature, such as the Haar wavelet and an optimal homotopy analysis method. The Legendre wavelet collocation method offers superior accuracy with fewer collocation points, making it advantageous.

Key words: Emden-Fowler equation, Legendre wavelet, Haar wavelet, uniqueness of solution, integral equation, collocation method.

MSC: 34B05, 34B15, 34B16, 34B18, 34B27

Received: 07.06.2024
Revised: 13.01.2025
Accepted: 04.03.2025

DOI: 10.15372/SJNM20250306