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<p>This review examines recent advances in semi-analytical and hybrid solution methodologies for nonlinear ordinary and partial differential equation (ODE/PDE) systems. Its primary focus is on the Adomian Decomposition Method (ADM), the Homotopy Perturbation Method (HPM), and the Akbari–Ganji Method (AGM). These approaches are categorized as semi-analytical because they combine analytical series expansions with iterative or numerical steps to obtain approximate solutions. Thereby occupying an intermediate position between fully analytical methods (i.e. exact closed-form solutions) and fully numerical techniques. The discussion systematically addresses their theoretical properties, including convergence behavior, computational requirements, and the role of hybrid extensions. Selected engineering applications are highlighted, with emphasis on nonlinear oscillators, heat transfer systems, biosensor modeling, and reaction–diffusion problems. A comparative assessment of strengths, limitations, and failure scenarios (e.g. stiffness, slow convergence) is provided to guide method selection. This study presents a structured synthesis that identifies problem classes, recommends appropriate solution methods, and highlights scenarios in which hybrid strategies can enhance stability and computational efficiency. Overall, this review provides researchers with a practical framework for selecting and applying semi-analytical methods to solve nonlinear problems efficiently.</p>

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Semi-Analytical and Hybrid Solution Methods for Nonlinear Ordinary and Partial Differential Equation Systems: Recent Advances and Applications

  • 期刊论文

  • 发表日期：

    2026年5月17日

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  • V. Sreelatha. Devi,
  • K. Saranya,
  • Raman Kumar,
  • Rahul Makwana,
  • Manjunatha R,
  • Ajay Sharma,
  • Abhishek Bhattacherjee,
  • Abhijit Bhowmik

摘要

This review examines recent advances in semi-analytical and hybrid solution methodologies for nonlinear ordinary and partial differential equation (ODE/PDE) systems. Its primary focus is on the Adomian Decomposition Method (ADM), the Homotopy Perturbation Method (HPM), and the Akbari–Ganji Method (AGM). These approaches are categorized as semi-analytical because they combine analytical series expansions with iterative or numerical steps to obtain approximate solutions. Thereby occupying an intermediate position between fully analytical methods (i.e. exact closed-form solutions) and fully numerical techniques. The discussion systematically addresses their theoretical properties, including convergence behavior, computational requirements, and the role of hybrid extensions. Selected engineering applications are highlighted, with emphasis on nonlinear oscillators, heat transfer systems, biosensor modeling, and reaction–diffusion problems. A comparative assessment of strengths, limitations, and failure scenarios (e.g. stiffness, slow convergence) is provided to guide method selection. This study presents a structured synthesis that identifies problem classes, recommends appropriate solution methods, and highlights scenarios in which hybrid strategies can enhance stability and computational efficiency. Overall, this review provides researchers with a practical framework for selecting and applying semi-analytical methods to solve nonlinear problems efficiently.