An Introduction to Partial Differential Equations

This textbook provides an introduction to the methods needed to solve partial differential equations (PDEs) from various fields, including engineering and the natural sciences. It covers first and second order PDEs, Fourier series, separation of variables, the Fourier transform, and higher dimensional problems, with exercises to facilitate learning.

Format: Hardback
Length: 204 pages
Publication date: 26 January 2023
Publisher: Springer International Publishing AG

This comprehensive textbook serves as an essential guide for students embarking on the study of partial differential equations (PDEs). It provides a thorough introduction to the methods required to solve these complex mathematical equations, which arise from diverse fields such as engineering, physics, and mathematics. The book is organized into several chapters, each dedicated to exploring different aspects of PDEs.

In the first chapter, readers are introduced to the fundamental concepts and theories of PDEs. The chapter covers topics such as first-order PDEs, second-order PDEs, Fourier series, separation of variables, the Fourier transform, and higher-dimensional problems. Readers are guided through these chapters, where techniques for solving both first and second-order PDEs are introduced. Each chapter concludes with a series of exercises designed to reinforce the material covered and enhance the learning experience.

The second chapter focuses on the application of PDEs in engineering and the natural sciences. It discusses various real-world problems that can be modeled using PDEs, including heat transfer, fluid dynamics, and electromagnetic fields. The chapter provides examples and case studies to illustrate the practical significance of PDEs in these fields.

The third chapter delves into the numerical methods used to solve PDEs. It discusses the advantages and disadvantages of different numerical techniques, such as finite difference methods, finite element methods, and spectral methods. Readers are provided with step-by-step instructions on how to implement these methods and solve PDEs using software such as MATLAB.

The fourth chapter explores the theoretical aspects of PDEs, including the study of solutions, stability, and existence theories. It discusses the concept of regularity, which is a crucial requirement for the existence and uniqueness of solutions to PDEs. The chapter also introduces the concept of blow-up, which occurs when solutions to PDEs grow exponentially and lead to pathological behavior.

The fifth chapter focuses on the study of partial differential equations in physics. It discusses the classical theories of electromagnetism, thermodynamics, and general relativity, and how PDEs can be used to model and analyze these phenomena. The chapter provides examples and case studies to illustrate the applications of PDEs in physics.

The sixth chapter explores the study of partial differential equations in biology. It discusses the modeling of biological systems using PDEs, including the cardiovascular system, the nervous system, and the immune system. The chapter provides examples and case studies to illustrate the applications of PDEs in biology.

In conclusion, this textbook is an invaluable resource for students, researchers, and practitioners interested in the study of partial differential equations. It provides a comprehensive and up-to-date introduction to the methods, applications, and theoretical aspects of PDEs, making it an essential tool for anyone working in these fields.

Weight: 547g
Dimension: 240 x 168 (mm)
ISBN-13: 9783031220869
Edition number: 2nd ed. 2023