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.
Author(s): Atul Kumar Razdan, V. Ravichandran
Publisher: Springer
Year: 2022
Language: English
Pages: 558
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
