An Efficient Legendre-Wavelet Collocation Technique for Solving Emden–Fowler Type Equations

• Published in: Numerical analysis and applications Vol. 18; no. 3; pp. 249 - 267
• Main Authors: Saha, N., Shahni, J., Singh, R., Guleria, V., Sriwastav, N.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.09.2025; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Approximation; Astrophysics; Banach spaces; Boundary conditions; Boundary value problems; Chemical reactions; Collocation methods; Error analysis; Exact solutions; Fredholm equations; Homotopy theory; Integral equations; Mathematics; Mathematics and Statistics; Methods; Modelling; Nonlinear dynamics; Numerical Analysis; Quantum mechanics; Quantum physics; Thermodynamic properties; Uniqueness of Solution; Legendre wavelet; Integral Equation; Collocation method; Emden‑Fowler equation; Haar wavelet
• ISSN: 1995-4239; 1995-4247
• DOI: 10.1134/S1995423925030061

Emden–Fowler 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) 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 the Legendre wavelet collocation approach and the 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 the exact solution and those achieved by techniques in the literature, such as the Haar wavelet and the optimal homotopy analysis method. The Legendre wavelet collocation method offers superior accuracy with fewer collocation points, making it advantageous.