Haar Wavelet Collocation Algorithm for the Numerical Solution of Volterra Integrodifferential Form of Emden-Fowler Type Equations

Abstract

This research investigates the application of non-dyadic Haar wavelets to analyze Emden-Fowler equations, which find diverse uses across scientific domains. The central challenge associated with these equations pertains to their singularity at the origin. Notably, no existing literature delves into the study of the Emden-Fowler equation's behavior utilizing scale-3 Haar wavelets. In this study, we convert the differential form of the Emden-Fowler equation into a Volterra integrodifferential form (VIDF). Through the implementation of Haar wavelets, this VIDF is further transformed into a system of algebraic equations. The resultant equations are amenable to iterative solution techniques. Notably, the Haar wavelet approach is found to effectively accommodate both initial and boundary constraints. Several illustrative examples show the algorithm's simplicity and practicality. To assess reliability, we compute the maximum absolute error, , and for four examples. These numerical results are compared against outcomes from established methodologies such as the Adomian decomposition method, Chebyshev wavelet, and the Variational iteration method.