Exploring Solution Strategies for Volterra Integro-Differential Lane–Emden Equations in Astrophysics Using Haar Scale 3 Wavelets

The current research introduces a novel approach to address the computational challenges associated with solving the Lane–Emden-type equations by transforming them from their conventional differential form to the corresponding integro-differential form. These equations have wide-ranging applications in physical sciences, including modeling diffusion phenomena and thermal gradients. We utilize the Volterra integro-differential (VID) form to resolve computational challenges due to singularity issues. Through the Scale 3 Haar Wavelet (S3-HW) algorithm, we transform the VID equations into algebraic form and obtain solutions using the Gauss-elimination method. The quasilinearization technique is implemented whenever a nonlinearity is encountered. Comparative analysis against various techniques demonstrates the superior accuracy and efficiency of our method. Despite challenges such as the discontinuity of Scale 3 Haar Wavelets and singularity issues of Lane–Emden-type equations, our algorithm paves the way for extending its application to a wide range of physical problems.