A Numerical Investigation of Fractional-Order Differential Equations for Modeling Heat and Mass Transfer in Complex Engineering Systems

Abstract

Heat and mass transfer processes play a critical role in numerous engineering applications, including thermal energy systems, chemical processing, aerospace engineering, biomedical devices, and advanced manufacturing. Conventional integer-order differential equation models often fail to accurately describe the memory-dependent and non-local characteristics observed in complex transport phenomena, leading to reduced prediction accuracy under transient operating conditions. This study presents a comprehensive numerical investigation of fractional-order differential equations for modeling coupled heat and mass transfer in complex engineering systems. A fractional-order mathematical model is formulated using the Caputo fractional derivative to capture the hereditary behavior and long-term memory effects inherent in diffusion and thermal transport processes. The governing equations are discretized and solved using an efficient finite difference numerical scheme, and the stability and convergence of the proposed method are systematically analyzed. Numerical simulations are performed for different fractional-order parameters to examine their influence on temperature distribution, concentration profiles, heat transfer rate, and mass diffusion characteristics. The obtained results are compared with conventional integer-order models to evaluate improvements in prediction capability and computational performance. The findings demonstrate that the fractional-order model provides a more realistic representation of transport phenomena by effectively capturing anomalous diffusion and time-dependent thermal behavior while maintaining numerical stability and accuracy. Furthermore, sensitivity analysis reveals the significant influence of the fractional-order parameter on system dynamics, highlighting its importance in accurately characterizing complex engineering processes. The proposed numerical framework offers a reliable and computationally efficient approach for analyzing coupled heat and mass transfer problems and can be extended to various applications, including porous media, heat exchangers, thermal energy storage systems, microfluidic devices, and advanced manufacturing processes. The outcomes of this research contribute to the advancement of fractional calculus-based engineering models and provide valuable insights for the development of high-fidelity computational tools for next-generation thermal and transport system analysis.