Partial Differential Equations: Concepts, Applications, and Future Directions

Liuyang Cheng

doi:10.54254/2755-2721/2025.LD29470

Applied and Computational EngineeringOpen access

Partial Differential Equations: Concepts, Applications, and Future Directions

Research Article

Open Access

Partial Differential Equations: Concepts, Applications, and Future Directions

Liuyang Cheng ^1*

¹ Department of Mathematics, University of California, Los Angeles, Los Angeles, California, United States

^*Corresponding author: lygchengx@g.ucla.edu

Published on 11 November 2025

ACE Vol.204

ISSN (Print): 2755-273X

ISSN (Online): 2755-2721

ISBN (Print): 978-1-80590-517-2

ISBN (Online): 978-1-80590-518-9

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Abstract

Partial Differential Equations (PDEs) comprise one of the most basic mathematical frameworks for describing phenomena occurring in both space and time. From classical equations for heat and waves to recent applications in physics, engineering, and computer science, PDEs provide the framework for describing dynamics often modeled as heat conduction, fluid flow, electromagnetic fields, and image analysis. Although they date back a number of centuries, they remain entirely relevant today, especially as numerical methods and computational tools have enabled the study of complex, real-world systems. This paper will review PDEs in a number of ways. First, the paper reviews the fundamentals of theory and classification in PDEs, specifically by distinguishing PDEs into elliptic, parabolic and hyperbolic types. Second, applications, particularly privileging the use of PDEs in image processing, particularly spatial denoising, edge detection and reconstruction, as well as the physical sciences, like quantum mechanics and fluid dynamics. Finally, this paper will highlight limitations and future directions of PDEs, highlighting how PDE-based models may be improved through machine learning, or more generally, data-driven approaches. This paper seeks to combine the theoretical aspects of PDEs with practical application to demonstrate both the mathematical richness of the field as well as the interdisciplinary purpose in terms of furthering both scientific understanding and technological innovations.

Keywords:

Partial Differential Equations, Image Processing, Interdisciplinary

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References

[1]. Evans, L. C. (2022). Partial differential equations (Vol. 19). American mathematical society.

[2]. Strauss, W. A. (2007). Partial differential equations: An introduction. John Wiley & Sons.

[3]. LeVeque, R. J. (2007). Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. Society for Industrial and Applied Mathematics.

[4]. Trefethen, L. N. (2000). Spectral methods in MATLAB. Society for industrial and applied mathematics.

[5]. Perona, P., & Malik, J. (1990). Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7), 629–639.

[6]. Weickert, J. (1998). Anisotropic diffusion in image processing (Vol. 1, pp. 59-60). Stuttgart: Teubner.

[7]. Chan, T. F., & Shen, J. (2005). Image processing and analysis: variational, PDE, wavelet, and stochastic methods. Society for Industrial and Applied Mathematics.

[8]. Batchelor, G. K. (2000). An introduction to fluid dynamics. Cambridge university press.

[9]. Griffiths, D. J., & Schroeter, D. F. (2018). Introduction to Quantum Mechanics (3rd ed.). Cambridge University Press.

[10]. Courant, R., & Hilbert, D. (2008). Methods of Mathematical Physics, Vol. II: Partial Differential Equations. Wiley-VCH.

[11]. E, W., & Yu, B. (2018). The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics, 6(1), 1–12.

[12]. Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707.