Introduction to Partial Differential Equations

Abstract

The momentous revolution in science precipitated by Isaac Newton’s calculus soon revealed the central role of partial differential equations throughout mathematics and its manifold applications. Notable examples of fundamental physical phenomena modeled by partial differential equations, most of which are named after their discoverers or early proponents, include quantum mechanics (Schr¨odinger, Dirac), relativity (Einstein), electromagnetism (Maxwell), optics (eikonal, Maxwell–Bloch, nonlinear Schr¨odinger), fluid mechanics (Euler, Navier–Stokes, Korteweg–deVries, Kadomstev–Petviashvili), superconductivity (Ginzburg–Landau), plasmas (Vlasov), magneto-hydrodynamics (Navier–Stokes + Maxwell), elasticity (Lam´e, von Karman), thermodynamics (heat), chemical reactions (Kolmogorov–Petrovsky–Piskounov), finance (Black–Scholes), neuroscience (FitzHugh– Nagumo), and many, many more. The challenge is that, while their derivation as physical models — classical, quantum, and relativistic — is, for the most part, well established, [57, 69], most of the resulting partial differential equations are notoriously difficult to solve, and only a small handful can be deemed to be completely understood. In many cases, the only means of calculating and understanding their solutions is through the design of sophisticated numerical approximation schemes, an important and active subject in its own right. However, one cannot make serious progress on their numerical aspects without a deep understanding of the underlying analytical properties, and thus the analytical and numerical approaches to the subject are inextricably intertwined. This textbook is designed for a one-year course covering the fundamentals of partial differential equations, geared towards advanced undergraduates and beginning graduate students in mathematics, science, and engineering. No previous experience with the subject is assumed, while the mathematical prerequisites for embarking on this course of study will be listed below. For many years, I have been teaching such a course to students from mathematics, physics, engineering, statistics, chemistry, and, more recently, biology, finance, economics, and elsewhere. Over time, I realized that there is a genuine need for a well-written, systematic, modern introduction to the basic theory, solution techniques, qualitative properties, and numerical approximation schemes for the principal varieties of partial differential equations that one encounters in both mathematics and applications. It is my hope that this book will fill this need, and thus help to educate and inspire the next generation of students, researchers, and practitioners. While the classical topics of separation of variables, Fourier analysis, Green’s functions, and special functions continue to form the core of an introductory course, the inclusion of nonlinear equations, shock wave dynamics, dispersion, symmetry and similarity methods, the Maximum Principle, Huygens’ Principle, quantum mechanics and the Schr¨odinger equation, and mathematical finance makes this book more in tune with recent developments and trends. Numerical approximation schemes should also play an essential role in an introductory course, and this text covers the two most basic approaches: finite differences and finite elements.