Interdisciplinary Applications of Applied Mathematics and Partial Differential Equations in Modeling Complex Multi-Scale Phenomena

Abstract

Background: In disciplines involving the modeling of complex phenomena of multiple scales, there is a need to have robust models that have the potential of capturing the interaction at multiple spatial and temporal domain. PDEs provide an effective solution to model a dynamic system, and the combination with applied mathematics yielded remarkable contributions across sectors like biomedical engineering and environmental science to computational modeling. Objective: This paper set out to understand how applied mathematics and PDEs are used interdisciplinarity in modelling of complex systems with many scales of lower focus on existing trends, modelling strategies, challenges, and computational methods within a wide range of sciences. Method: To reach the conclusion, a qualitative systematic review of peer-reviewed articles has been carried out in database Scopus, Web of Science, and ScienceDirect published between 2021 and 2025. Thirty studies of high caliber were chosen depending on relevance and methodological superiority. Thematic analysis was done to stratify the PDEs according to types, their uses in domains, intended modeling, representation in scale and what computing strategies. Results: The PDE app area with the largest percentage of applications (30 percent) was in biomedical sciences followed by environmental modelling and material sciences. Algebraic Navier-Stokes and reaction-diffusion equations were used in the greatest frequency. The most important modeling can be done to have simulation, optimization and pattern formation. It was noted that such obstacles as computer cost, scale bridging and data sparsity can be major. There was the appearance of the hybrid approaches that combine PDEs with machine learning (e.g., PINNs) as a good solution. Conclusion: PDE-based modeling paradigm is an essential interdisciplinary paradigm that is used in modeling complex systems on different scales. Its growing synchronization with data-driven techniques heralds a major transition to more adaptive and predictive elucidation frameworks of science.

Keywords: Partial Differential Equations, Multi-Scale Modeling, Applied Mathematics, Interdisciplinary Modeling, Simulation, Hybrid Computation, Reaction-Diffusion Systems, Navier-Stokes Equations.