Partial Differential Equations: Analyitical and Numerical Methods

This book contains the material for the course "Partial Differential Equations: Analytical and Numerical Methods" taught at ICTP–SAIFR in May 2025.

If you come across any errors or have suggestions for improving the book's structure, please don't hesitate to contact the authors. Your feedback is greatly appreciated.

Introduction and motivation

In this book, we aim to introduce the reader to the expansive field of Partial Differential Equations (PDEs). These equations serve as powerful tools for modeling a remarkable spectrum of natural phenomena—ranging from quantum effects at nanometer spatial scales and femtosecond timescales, to fluid dynamics at scales familiar from everyday life, and even up to the evolution of galaxies and cosmic structures across billions of years.

There are many excellent books on PDEs. Some focus on their analytical aspects, often specializing in certain subclasses or specific equations. Others address numerical methods for obtaining sufficiently accurate approximations to solutions of particular problems. Additionally, there are books dedicated to the necessary numerical analysis tools that ensure these approximations are robust and credible.

Here, perhaps at the expense of some depth, we propose a comprehensive survey of the different types of PDEs. We will not only study their analytical properties, but also explore their numerical approximations, which will, in turn, help us visualize these properties and behaviors more clearly. To this end, we will delve into fundamental analytical tools that are essential not only for demonstrating the existence, uniqueness, and stability of solutions for a given type of problem, but also for establishing the convergence of the numerical approximations we will be producing.

Ultimately, the aim of this book is to provide a balanced understanding of both the theory and practical techniques for solving PDEs. By combining analytical and numerical approaches, we seek to equip the reader with the tools necessary for addressing real-world problems and advancing their study of these powerful equations.