This textbook is an introduction to the methods needed to solve partial differential equations (PDEs). Readers are introduced to PDEs that come from a variety of fields in engineering and the natural sciences. The chapters include the following topics: 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 first and second order PDEs are introduced. Each chapter ends with series of exercises to facilitate learning as well as illustrate the material presented in each chapter.
Author(s): Daniel Arrigo
Series: Synthesis Lectures on Mathematics & Statistics
Publisher: Springer
Year: 2023
Language: English
Pages: 207
City: Cham
Preface
Preface to the Second Edition
Acknowledgements
Contents
1 Introduction
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1.1 Model Equations
1.1.1 Advection Equation
1.1.2 Diffusion Equation
1.1.3 Laplace's Equation
1.1.4 Wave Equation
1.2 PDEs Are Everywhere
2 First Order PDEs
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2.1 Constant Coefficient Equations
2.2 Linear Equations
2.3 Method of Characteristics
2.4 Quasilinear Equations
2.5 Higher Dimensional Equations
2.6 Fully Nonlinear First Order Equations
2.6.1 Method of Characteristics
2.6.2 Charpit's Method
3 Second Order Linear PDEs
3.1 Introduction
3.2 Standard Forms
3.2.1 Parabolic Standard Form
3.2.2 Hyperbolic Standard Form
3.2.3 Modified Hyperbolic Form
3.2.4 Regular Hyperbolic Form
3.2.5 Elliptic Standard Form
3.3 The Wave Equation
4 Fourier Series
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4.1 Fourier Series
4.2 Fourier Series on [- π, π]
4.3 Fourier Series on [- L, L]
4.4 Odd and Even Extensions
4.4.1 Sine Series
4.4.2 Cosine Series
5 Separation of Variables
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5.1 The Heat Equation
5.1.1 Nonhomogeneous Boundary Conditions
5.1.2 Nonhomogeneous Equations
5.1.3 Equations with a Solution Dependent Source Term
5.1.4 Equations with a Solution Dependent Convective Term
5.2 Laplace's Equation
5.2.1 Laplace's Equation on an Arbitrary Rectangular Domain
5.2.2 Laplace's Equation on a Disc
5.3 The Wave Equation
6 Fourier Transform
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6.1 Fourier Transform
6.2 Fourier Sine and Cosine Transforms
7 Higher Dimensional Problems
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7.1 2+1 Dimensional Heat Equation
7.2 2+1 Dimensional Diffusion Equation in a Disc
7.2.1 With Angular Symmetry
7.2.2 Without Angular Symmetry
7.3 Laplaces Equation
7.3.1 Cartesian Coordinates
7.3.2 Cylindrical Polar Coordinates
7.3.3 Spherical Polar Coordinates
7.4 Special Functions
7.4.1 Bessel Functions
7.4.2 Legendre Functions
Solutions
Index
