Journal of Applied & Computational Mathematics

Multiphase flow simulation in porous media is a critical area of research with significant implications for various fields, including petroleum engineering, hydrology, and environmental science. The behavior of fluids in porous materials is complex, particularly when multiple fluid phases, such as oil, water, and gas, coexist and interact within the porous medium. Understanding and predicting the dynamics of these flows is essential for optimizing extraction processes, managing reservoirs, and mitigating environmental impacts. Computational methods play a crucial role in modeling and simulating multiphase flow in porous media, providing insights into the intricate physical processes that govern these systems. The challenge of simulating multiphase flow in porous media arises from the complex interplay of fluid dynamics, capillarity, and the heterogeneous nature of the porous medium. Porous media are typically composed of solid matrices with interconnected pores through which fluids move. The flow of fluids within these pores is influenced by various factors, including pore size distribution, fluid viscosity, surface tension, and wetting properties.

Partial differential equations are fundamental mathematical tools used to describe a wide range of physical phenomena, from fluid dynamics and heat conduction to quantum mechanics and financial modeling. Solving PDEs is crucial for understanding and predicting the behavior of these systems, but traditional numerical methods, such as finite difference, finite element, and spectral methods, often encounter significant challenges when dealing with complex, high-dimensional problems. In recent years, machine learning has emerged as a powerful alternative or complement to classical numerical methods, offering new approaches for efficiently solving PDEs. Machine learning-driven numerical solutions to PDEs have the potential to revolutionize computational science by providing more accurate, faster, and scalable solutions. One of the key motivations for integrating machine learning with numerical PDE solvers is the ability of ML models to approximate complex functions and their derivatives with high accuracy. Neural networks, particularly deep learning models, have demonstrated remarkable success in learning intricate patterns and relationships within large datasets.

Mathematical models for epidemic spread play a crucial role in understanding and predicting the behavior of infectious diseases. These models provide valuable insights into how diseases propagate through populations, helping public health officials and policymakers make informed decisions to control outbreaks. With advancements in computational techniques and predictive analytics, researchers can now simulate and analyze epidemic dynamics with greater accuracy and detail. This article explores the various mathematical models used to study epidemic spread, their computational insights, and the role of predictive analytics in managing public health crises.