Partial Differential Equations: Overview, Applications, and Numerical Solutions
- 1Department of Computer Science and Mathematics, Southwest Minnesota State University, Marshall, MN 56258, USA
- 2Department of Mathematical Sciences, University of South Africa, Pretoria 0003, South Africa
Abstract
Partial differential equations (PDEs) provide a fundamental mathematical framework for describing phenomena that vary across space and time. However, understanding their formulation, analysis, and numerical solution requires familiarity with concepts that are often presented separately across mathematical and computational disciplines. This review offers an integrated and accessible introduction to PDEs, connecting foundational theory with mathematical modeling and computational practice. Beginning with partial derivatives and a historical overview, the article examines differentiability, smoothness, test functions, and function spaces, including Hilbert and Sobolev spaces, to establish the analytical foundations of PDE theory. It then discusses the classification of PDEs and the roles of initial and boundary conditions in defining mathematical problems and interpreting their solutions. A structured account of mathematical modeling explains how ordinary differential equations (ODEs) and PDEs can be formulated from assumptions, governing principles, and relationships among model variables. Applications illustrate the relevance of these equations to problems in science and engineering. The review also surveys established numerical approaches, including the finite difference, finite volume, and finite element methods, alongside deep learning approaches for approximating PDE solutions. Relevant software tools are introduced to support computational implementation and further exploration. By bringing these topics together within a coherent framework, the article provides a foundation for understanding the connections among model formulation, analytical concepts, and numerical approximation. It serves as an introductory resource for students and early-career researchers and as a consolidated reference for practitioners pursuing further work in mathematical modeling, numerical analysis, and applied mathematics.
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BibTeX
@article{babatunde2026partial,
title = {{Partial Differential Equations: Overview, Applications, and Numerical Solutions}},
author = {Oluleye H. Babatunde and Justin M. W. Munganga and Emile Franc Doungmo Goufo},
journal = {Mathematics},
year = {2026},
month = nov,
doi = {10.3390/math14214077},
url = {https://www.mdpi.com/2227-7390/14/21/4077},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Partial Differential Equations: Overview, Applications, and Numerical Solutions
AU - Oluleye H. Babatunde
AU - Justin M. W. Munganga
AU - Emile Franc Doungmo Goufo
PY - 2026
DA - 2026/11/06
JO - Mathematics
DO - 10.3390/math14214077
UR - https://www.mdpi.com/2227-7390/14/21/4077
AB - Partial differential equations (PDEs) provide a fundamental mathematical framework for describing phenomena that vary across space and time. However, understanding their formulation, analysis, and numerical solution requires familiarity with concepts that are often presented separately across mathematical and computational disciplines. This review offers an integrated and accessible introduction to PDEs, connecting foundational theory with mathematical modeling and computational practice. Beginning with partial derivatives and a historical overview, the article examines differentiability, smoothness, test functions, and function spaces, including Hilbert and Sobolev spaces, to establish the analytical foundations of PDE theory. It then discusses the classification of PDEs and the roles of initial and boundary conditions in defining mathematical problems and interpreting their solutions. A structured account of mathematical modeling explains how ordinary differential equations (ODEs) and PDEs can be formulated from assumptions, governing principles, and relationships among model variables. Applications illustrate the relevance of these equations to problems in science and engineering. The review also surveys established numerical approaches, including the finite difference, finite volume, and finite element methods, alongside deep learning approaches for approximating PDE solutions. Relevant software tools are introduced to support computational implementation and further exploration. By bringing these topics together within a coherent framework, the article provides a foundation for understanding the connections among model formulation, analytical concepts, and numerical approximation. It serves as an introductory resource for students and early-career researchers and as a consolidated reference for practitioners pursuing further work in mathematical modeling, numerical analysis, and applied mathematics.
SN - 2227-7390
ER -