# An Introduction to Partial Differential Equations

## TL;DR

This paper introduces partial differential equations (PDEs) from an applied mathematics perspective, focusing on physical context, methods, and applications, suitable for advanced students with basic programming and differential equations background.

## Contribution

It provides an accessible overview of PDEs emphasizing methods and applications, bridging physical intuition with analytical and numerical techniques.

## Key findings

- Introduces basic PDEs like heat, wave, and Laplace equations.
- Discusses analytical and numerical solution methods.
- Provides educational material for advanced students.

## Abstract

The field of partial differential equations (PDEs) is vast in size and diversity. The basic reason for this is that essentially all fundamental laws of physics are formulated in terms of PDEs. In addition, approximations to these fundamental laws, that form a patchwork of mathematical models covering the range from the smallest to the largest observable space-time scales, are also formulated in terms of PDEs. The diverse applications of PDEs in science and technology testify to the flexibility and expressiveness of the language of PDEs, but it also makes it a hard topic to teach right. Exactly because of the diversity of applications, there are just so many different points of view when it comes to PDEs. These lecture notes view the subject through the lens of applied mathematics. From this point of view, the physical context for basic equations like the heat equation, the wave equation and the Laplace equation are introduced early on, and the focus of the lecture notes are on methods, rather than precise mathematical definitions and proofs. With respect to methods, both analytical and numerical approaches are discussed. These lecture notes has been succesfully used as the text for a master class in partial differential equations for several years. The students attending this class are assumed to have previously attended a standard beginners class in ordinary differential equations and a standard beginners class in numerical methods. It is also assumed that they are familiar with programming at the level of a beginners class in informatics at the university level.