Partial Differential Equations: An Introduction to Analytical and Numerical Methods

This textbook provides an integrated approach to the study of partial differential equations, combining analytical and numerical methods. It begins with motivating examples from the physical sciences, engineering, and economics and then introduces elementary methods to solve hyperbolic, elliptic, and parabolic equations. Hilbert spaces and the spectral theorem are also covered, and numerical methods are discussed using finite difference and finite element methods. Three fundamental examples are studied with different tools, and the Black-Scholes equation from mathematical finance is one of several opportunities for extension. The book is suitable for courses in real analysis and linear algebra at the upper-undergraduate level.

Format: Hardback
Length: 452 pages
Publication date: 04 January 2023
Publisher: Springer International Publishing AG

This comprehensive textbook delves into the study of partial differential equations (PDEs) through a blend of analytical and numerical methods. By seamlessly integrating these two complementary approaches, the authors provide an ideal foundation for advanced exploration. Motivating examples from the realms of physical sciences, engineering, and economics enrich the text, showcasing the practical applications of PDEs.

The book begins with a series of models, illustrating how PDEs emerge in real-world problems such as heat transfer, vibration analysis, fluid dynamics, and financial market dynamics. Key characterizing properties are employed to classify mathematical similarities, paving the way for elementary methods to solve a range of hyperbolic, elliptic, and parabolic equations.

From this foundational stage, the text introduces Hilbert spaces and the spectral theorem, laying the groundwork for advanced methods. Sobolev spaces, initially explored in one dimension, are extended to arbitrary dimensions to study elliptic equations. An extensive chapter dedicated to numerical methods covers finite difference and finite element methods, with computer-aided calculation using Maple™ providing a practical application.

Throughout the book, three fundamental examples are studied using diverse tools: the Poissons equation, the heat equation, and the wave equation on Euclidean domains. The Black–Scholes equation from mathematical finance offers a valuable opportunity for extension.

Partial Differential Equations is designed to cater to students embarking on their journey into the realm of PDEs. It combines analytical and numerical tools with modeling, forming a versatile toolbox suitable for both pure and applied mathematics. The text is accompanied by illuminating illustrations and engaging exercises, enhancing the learning experience. Courses in real analysis and linear algebra at the upper-undergraduate level are assumed as prerequisites.

Weight: 875g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031133787
Edition number: 1st ed. 2023