Fundamentals of Partial Differential Equations

Description

The book serves as a primary textbook of partial differential equations (PDEs), with due attention to their importance to various physical and engineering phenomena. The book focuses on maintaining a balance between the mathematical expressions used and the significance they hold in the context of some physical problem. The book has wider outreach as it covers topics relevant to many different applications of ordinary differential equations (ODEs), PDEs, Fourier series, integral transforms, and applications. It also discusses applications of analytical and geometric methods to solve some fundamental PDE models of physical phenomena such as transport of mass, momentum, and energy. As far as possible, historical notes are added for most important developments in science and engineering. Both the presentation and treatment of topics are fashioned to meet the expectations of interested readers working in any branch of science and technology. Senior undergraduates in mathematics and engineering are the targeted student readership, and the topical focus with applications to real-world examples will promote higher-level mathematical understanding for undergraduates in sciences and engineering. Preface References Contents List of Figures 1 Introduction References 2 Classical Vector Analysis 2.1 Multivariable Calculus 2.2 Classical Theory of Surfaces and Curves 2.3 Vector Calculus References 3 Ordinary Differential Equations 3.1 Introduction 3.2 First Order Differential Equations 3.2.1 Integrable Forms 3.2.2 Picard–Lindelöf Theorem 3.3 Higher Order Linear Differential Equations 3.3.1 The Case of Constant Coefficients 3.3.2 Power Series Solution 3.4 Boundary Value Problems 3.4.1 Green's Functions and Nonhomogeneous Problems 3.4.2 Sturm–Liouville Theory 3.4.3 Eigenfunctions Expansions 3.5 First Order System of Differential Equations 3.5.1 Existence and Uniqueness Theorem 3.5.2 Linear Systems References 4 Partial Differential Equation Models 4.1 Mathematical Modelling 4.2 Three Prototypical Equations 4.3 Models for Transport Phenomena References 5 Partial Differential Equations 5.1 Preliminaries 5.2 Classification and Canonical Forms 5.3 Classical Solution 5.4 Initial-Boundary Value Problems 5.5 Uniqueness Theorems and Stability Issues References 6 General Solution and Complete Integral 6.1 Characteristics Coordinates 6.2 Lagrange's Method 6.3 Linear Equations with Constant Coefficients 6.4 Lagrange–Charpit Method 7 Method of Characteristics 7.1 Linear and Semilinear Equations 7.2 Quasilinear Equations 7.3 Fully Nonlinear Equation References 8 Separation of Variables 8.1 Vibrating String Controversy 8.2 Fourier Series 8.3 Separation of Variables References 9 Method of Eigenfunctions Expansion 9.1 Generalised Fourier Series 9.2 Nonhomogeneous Boundary Value Problems 9.3 Poisson Equations References 10 Fourier Transforms 10.1 Introduction 10.2 Fourier's Transform Pair 10.3 Transforms of Generalised Functions 10.4 Fundamental Properties 10.5 Applications to Partial Differential Equations References 11 Laplace Transform 11.1 Basic Theorems and Examples 11.2 Properties of Laplace Transform 11.3 Inverse Laplace Transform 11.4 Applications to Differential Equations Reference Appendix A Supplements A.1 Banach Fixed Point Theorem A.2 Maxwell and Helmholtz Equations A.3 Generalised Functions A.4 Signals and (LTI) Systems Index