An improved variational iteration method with convergence analysis for solving higher-order nonlinear Emden–Fowler equations

Sattam Alharbi; Taruni Alekhya Sarvasuddi; Prashanth Maroju; Ramandeep Behl; Sattam Alharbi; Taruni Alekhya Sarvasuddi; Prashanth Maroju; Ramandeep Behl

doi:10.3934/math.20261381

AIMS Mathematics

2026, Volume 11, Issue 10: 35631-35654. doi: 10.3934/math.20261381

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Research article

An improved variational iteration method with convergence analysis for solving higher-order nonlinear Emden–Fowler equations

1.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
2.
Department of Mathematics, School of Advanced Sciences, VIT-AP University, Amaravati-522237, Andhra Pradesh, India
3.
Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received: 26 August 2026 Revised: 06 October 2026 Accepted: 20 October 2026 Published: 30 October 2026
MSC : 65L05, 34B16, 65L20, 34A34, 65L10, 65L80

In this study, we present new results on the solution of higher-order, nonlinear Emden–Fowler equations (EFEs) of arbitrary order $ N $, which frequently arise across various physical domains, including astrophysics, fluid dynamics, and thermal explosions. These equations are characterized by strong local singularities and complex composite nonlinearities, making accurate analytical or stable numerical treatments highly challenging. We develop an improved variational iteration method (IVIM) which enhances the classical variational iteration framework by introducing a dynamic, residual-minimizing parameter-selection rule. This optimal selection rule establishes a rigorous error-dampening feedback layer directly inside the integration kernel, systematically accelerating computational efficiency while ensuring stable error contractive properties for higher-order continuous and multisingular systems. To establish the validity, accuracy, and reproducibility of the proposed approach, we provide a thorough quantitative evaluation across second-order, third-order, fourth-order, and sixth-order frameworks evaluated over a uniform discrete spatial grid. A comprehensive comparative analysis conducted against both the classical, unrelaxed variational iteration method (VIM) and the homotopy perturbation method (HPM) demonstrates that IVIM systematically bounds consecutive grid-based Cauchy iterate errors down to an exceptionally accurate baseline (reaching as low as $ 0.0056 $ in volatile higher-order configurations) while reducing CPU runtime execution costs by up to an order of magnitude. These results verify that IVIM operates as a robust, mathematically sound, and computationally dominant tool for tackling complex higher-order nonlinear singular physical systems.
- improved variational iteration method (IVIM),
- Emden–Fowler equations,
- convergence analysis,
- nonlinear differential equations,
- numerical methods
Citation: Sattam Alharbi, Taruni Alekhya Sarvasuddi, Prashanth Maroju, Ramandeep Behl. An improved variational iteration method with convergence analysis for solving higher-order nonlinear Emden–Fowler equations[J]. AIMS Mathematics, 2026, 11(10): 35631-35654. doi: 10.3934/math.20261381

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Abstract

In this study, we present new results on the solution of higher-order, nonlinear Emden–Fowler equations (EFEs) of arbitrary order $ N $, which frequently arise across various physical domains, including astrophysics, fluid dynamics, and thermal explosions. These equations are characterized by strong local singularities and complex composite nonlinearities, making accurate analytical or stable numerical treatments highly challenging. We develop an improved variational iteration method (IVIM) which enhances the classical variational iteration framework by introducing a dynamic, residual-minimizing parameter-selection rule. This optimal selection rule establishes a rigorous error-dampening feedback layer directly inside the integration kernel, systematically accelerating computational efficiency while ensuring stable error contractive properties for higher-order continuous and multisingular systems. To establish the validity, accuracy, and reproducibility of the proposed approach, we provide a thorough quantitative evaluation across second-order, third-order, fourth-order, and sixth-order frameworks evaluated over a uniform discrete spatial grid. A comprehensive comparative analysis conducted against both the classical, unrelaxed variational iteration method (VIM) and the homotopy perturbation method (HPM) demonstrates that IVIM systematically bounds consecutive grid-based Cauchy iterate errors down to an exceptionally accurate baseline (reaching as low as $ 0.0056 $ in volatile higher-order configurations) while reducing CPU runtime execution costs by up to an order of magnitude. These results verify that IVIM operates as a robust, mathematically sound, and computationally dominant tool for tackling complex higher-order nonlinear singular physical systems.

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