Transactions in Theoretical and Mathematical Physics

Transactions in Theoretical and Mathematical Physics

Advancing Numerical Solutions to Analytical Form Through the Hybrid Analytical and Numerical Method

Document Type : Original Article

Author
Ali Ahmadi Azar
Department of Mechanical Engineering, NT.C., Islamic Azad University, Tehran, Iran
10.30511/ttmp.2025.2061851.1055
Abstract
Analytical solutions of differential equations provide exact representations of physical phenomena, enhance computational efficiency, and offer deeper theoretical insights than purely numerical approaches. In mathematical physics, such solutions are essential for uncovering fundamental laws, accurately predicting system behavior, and developing closed-form expressions that drive further theoretical advancements and physical interpretations. However, the complexity of real-world differential equations—with their inherent nonlinearities, variable coefficients, and intricate boundary conditions—often precludes the attainment of exact analytical solutions, resulting in the predominance of numerical methods that offer computational feasibility albeit at the cost of precision. This study aims to overcome these limitations by employing the Hybrid Analytical and Numerical Method (HAN method) to derive an analytical solution for the nonlinear differential equation (NDE) governing Jeffery–Hamel flow. Initially, numerical solutions of the governing equations are obtained to construct a comprehensive dataset, which, together with the boundary conditions, facilitates the extraction and formulation of an exact analytical solution. This HAN method effectively upgrades numerical approximations into analytical forms, thereby bridging the gap between computational practicality and theoretical rigor in the analysis of complex NDEs.
Keywords
The Hybrid Analytical and Numerical Method
HAN Method
Jeffery&ndash
Hamel flow
Mathematical Physics
Nonlinear Dynamics
Subjects
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Transactions in Theoretical and Mathematical Physics
Volume 2, Issue 3
Summer 2025
Pages 132-147

  • Receive Date 26 May 2025
  • Accept Date 25 August 2025
  • First Publish Date 25 August 2025
  • Publish Date 01 August 2025