Analytical and Numerical Solutions of Nonlinear Partial Differential Equations Arising in Heat and Mass Transfer Problems

Abstract

Nonlinear partial differential equations (NPDEs) play a fundamental role in modeling complex heat and mass transfer phenomena encountered in engineering, industrial processing, environmental systems, and advanced manufacturing technologies. This study presents a comprehensive analysis of analytical, semi-analytical, numerical, and machine-learning-based techniques developed for solving NPDEs arising in nonlinear transport problems. The study discusses the mathematical foundations of nonlinear heat conduction, constitutive models for non-Newtonian and hybrid nanofluids, and dimensional reduction through similarity transformations and Lie symmetry analysis. Major semi-analytical methods, including the Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM), and Differential Transform Method (DTM), are critically examined alongside traditional numerical approaches such as Finite Difference, Finite Element, Finite Volume, and spectral methods. Recent developments in Physics-Informed Neural Networks (PINNs) and scientific machine learning are also highlighted for their computational efficiency and predictive capabilities. The review further explores non-Fourier heat conduction, nanofluid transport parameters, sustainability considerations, and future research directions in multiphysics and real-time thermal modeling.