Advancing numerical solutions to analytical form through the hybrid analytical and numerical method Ali Ahmadi Azar 1, a a Department of Mechanical Engineering, NT.C., Islamic Azad University, Tehran, Iran Abstract Analytical solutions of differential equations pro- vide exact representations of physical phenomena, enhance computational efficiency, and offer deeper theoretical in- sights than purely numerical approaches. In mathematical physics, such solutions are essential for uncovering funda- mental laws, accurately predicting system behavior, and de- veloping closed-form expressions that drive further theoret- ical advancements and physical interpretations. However, the complexity of real-world differential equations—with their inherent nonlinearities, variable coefficients, and intri- cate boundary conditions—often precludes the attainment of exact analytical solutions, resulting in the predominance of numerical methods that offer computational feasibility al- beit at the cost of precision. This study aims to overcome these limitations by employing the hybrid analytical and nu- merical method (HAN method) to derive an analytical solu- tion for the nonlinear differential equation (NDE) governing Jeffery–Hamel flow. Initially, numerical solutions of the governing equations are obtained to construct a comprehen- sive dataset, which, together with the boundary conditions, facilitates the extraction and formulation of an exact analyt- ical solution. This HAN method effectively upgrades nu- merical approximations into analytical forms, thereby brid- ging the gap between computational practicality and theo- retical rigor in the analysis of complex NDEs. 1 Introduction 1.1 Significance of the HAN method Realistic modelling of natural phenomena often leads to highly nonlinear differential equations characterized by co- plex nonlinearities, variable coefficients, and challenging boundary conditions. Although numerical methods are prevalently used to approximate solutions for these equa- tions, their outputs are typically approximate and may lack the precise theoretical insights offered by analytical solu- tions. The HAN method addresses this gap by initially 1 E-mail: aliahmadiazar.mech@gmail.com, a.ahmadi.azar@iau-tnb.ac.ir employing numerical techniques to obtain approximate so- lutions and subsequently converting these results into closed-form analytical expressions. This approach leverages the computational efficiency of numerical methods while ul- timately providing the exactness and interpretability of ana- lytical solutions, thereby offering a unique and effective framework for solving complex NDEs. 1.2 Literature review A growing body of literature has focused on the HAN method as a novel approach to overcoming the challenges posed by NDEs. Several recent studies have demonstrated that the HAN method not only enhances computational ef- ficiency by leveraging numerical approximations but also facilitates the extraction of precise analytical solutions that offer deeper theoretical insights. These investigations col- lectively underscore the potential of the HAN method to bridge the gap between conventional numerical techniques and exact analytical formulations in mathematical physics. A pioneering study [1] introduces the HAN method that co- nverts numerical approximations of the NDEs into closed - form analytical solutions. The study investigates complex physical phenomena such as heat and mass transfer, mo- mentum transport, and electromagnetic effects in fluid sys- tems, where intricate nonlinearities and challenging bound- ary conditions often hinder direct analytical treatment. By leveraging numerical data to systematically extract analyti- cal expressions, the HAN method overcomes these difficul- ties while delivering enhanced computational efficiency and precision. This approach bridges the gap between the prac- tical benefits of numerical techniques and the deep theoret- ical insights offered by analytical solutions, thereby advanc- ing the understanding of the underlying physics in mathe- matical modeling. In another study [2], the combined effects of thermo-diffusion, electric fields, and nonlinear thermal radiation on the steady flow of an incompressible non-Darcy Casson fluid over a vertical permeable stretchable plate were examined. The researchers employed the HAN method Received 26 May 2025 / Accepted: 25 August 2025 / Published: 26 August 2025