This paper presents a novel and highly effective numerical method for solving the Lane-Emden equation, significantly expanding its applicability to a broad class of nonlinear differential equations. Our approach leverages the power of Generalized Hat functions and their operational matrix of integration to transform the target equation into a manageable, block-structured nonlinear system, which is then efficiently solved via forward substitution. Critically, a key innovation of our method is the use of quintic Hermite interpolation for constructing the final solution. This departure from the typical reliance on primary function bases results in a markedly more accurate approximation. A key strength of this method lies in its remarkable robustness. Unlike many existing techniques, its accuracy remains consistent regardless of the length of the solution interval. Furthermore, its adaptability is exceptional: with minimal modifications, it can be readily extended to tackle fractional-order Lane-Emden equations and a wide variety of other nonlinear ordinary differential equations. While several solution approximation methods are possible, we demonstrate the superior accuracy of Hermite interpolation. The paper provides a thorough analysis, including detailed error assessments, and showcases the method's accuracy, efficiency, and versatility through compelling numerical examples. We believe this innovative approach offers a significant advancement in the numerical solution of these important equations.
Samareh Hashemi, S. A. and Hatamian, R. (2025). A Hybrid Approach to Solving the Lane-Emden Differential Equation Using Generalized Hat Functions and Hermite Interpolation. Computational Methods for Differential Equations, (), -. doi: 10.22034/cmde.2025.67759.3240
MLA
Samareh Hashemi, S. A. , and Hatamian, R. . "A Hybrid Approach to Solving the Lane-Emden Differential Equation Using Generalized Hat Functions and Hermite Interpolation", Computational Methods for Differential Equations, , , 2025, -. doi: 10.22034/cmde.2025.67759.3240
HARVARD
Samareh Hashemi, S. A., Hatamian, R. (2025). 'A Hybrid Approach to Solving the Lane-Emden Differential Equation Using Generalized Hat Functions and Hermite Interpolation', Computational Methods for Differential Equations, (), pp. -. doi: 10.22034/cmde.2025.67759.3240
CHICAGO
S. A. Samareh Hashemi and R. Hatamian, "A Hybrid Approach to Solving the Lane-Emden Differential Equation Using Generalized Hat Functions and Hermite Interpolation," Computational Methods for Differential Equations, (2025): -, doi: 10.22034/cmde.2025.67759.3240
VANCOUVER
Samareh Hashemi, S. A., Hatamian, R. A Hybrid Approach to Solving the Lane-Emden Differential Equation Using Generalized Hat Functions and Hermite Interpolation. Computational Methods for Differential Equations, 2025; (): -. doi: 10.22034/cmde.2025.67759.3240
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