PDE in Modern Science and Engineering: Cross-Cutting Practices and Challenges

Jincan Li

2025

Abstract

Partial differential equations (PDEs), as a core tool for mathematical modelling, have demonstrated remarkable universality in science and engineering by describing the spatio-temporal evolution laws of multivariate dynamical systems. From fluid motion (Navier-Stokes equations) and heat conduction (Fourier equations) in classical mechanics, to pricing of financial derivatives (Black-Scholes model), and prediction of tumor growth (reaction-diffusion equations) in biomedicine, PDEs provide a unifying theoretical framework for interdisciplinary complex problems. However, their applications face two core challenges: first, high-dimensional PDEs (e.g., the quantum many-body problem) lead to ‘dimensional catastrophe’, where the computational complexity of traditional numerical methods grows exponentially with the dimensionality; second, the deviation of the idealised physical assumptions (e.g., homogeneous medium, linear eigenstructure relationship) from the actual scenarios leads to the limitation of the accuracy of the model predictions. The study shows that interdisciplinary collaboration and algorithmic innovation are the keys to breaking through the existing limitations, and future research needs to find a balance between theoretical rigor, computational efficiency and engineering applicability, in order to promote the paradigm change of PDEs in the era of artificial intelligence and quantum.

Paper Citation

in Harvard Style

Li J. (2025). PDE in Modern Science and Engineering: Cross-Cutting Practices and Challenges. In Proceedings of the 2nd International Conference on Data Science and Engineering - Volume 1: ICDSE; ISBN 978-989-758-765-8, SciTePress, pages 317-323. DOI: 10.5220/0013688800004670

in Bibtex Style

@conference{icdse25,
author={Jincan Li},
title={PDE in Modern Science and Engineering: Cross-Cutting Practices and Challenges},
booktitle={Proceedings of the 2nd International Conference on Data Science and Engineering - Volume 1: ICDSE},
year={2025},
pages={317-323},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0013688800004670},
isbn={978-989-758-765-8},
}

in EndNote Style

TY - CONF

JO - Proceedings of the 2nd International Conference on Data Science and Engineering - Volume 1: ICDSE
TI - PDE in Modern Science and Engineering: Cross-Cutting Practices and Challenges
SN - 978-989-758-765-8
AU - Li J.
PY - 2025
SP - 317
EP - 323
DO - 10.5220/0013688800004670
PB - SciTePress