# Advanced Topics in Applied Mathematics by Sudhakar Nair

Download Advanced Topics in Applied Mathematics – This book(Advanced Topics in Applied Mathematics) is ideal for engineering, physical science, and applied mathematics students and professionals who want to enhance their mathematical knowledge. Advanced Topics in Applied Mathematics covers four essential applied mathematics topics: Green’s functions, Integral equations, Fourier transforms, and Laplace transforms.

Table of Contents

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Also included is a useful discussion of topics such as the Wiener-Hopf method, Finite Hilbert transforms, Cagniard-De Hoop method, and the proper orthogonal decomposition. This book reflects Sudhakar Nair’s long classroom experience and includes numerous examples of differential and integral equations from engineering and physics to illustrate the solution procedures. The text includes exercise sets at the end of each chapter and a solutions manual, which is available for instructors.

## Book Description – Advanced Topics in Applied Mathematics

This book is ideal for engineering, physical science, and applied mathematics students to enhance their mathematical knowledge. This book reflects Sudhakar Nair’s long classroom experience and includes numerous examples of differential and integral equations from engineering and physics to illustrate solution procedures. The text includes end-of-chapter exercises and a solutions manual, available to instructors.

## Table of Contents

Contents

Preface page ix

1 Green’s Functions …………………………… 1

1.1 Heaviside Step Function 1

1.2 Dirac Delta Function 3

1.2.1 Macaulay Brackets 6

1.2.2 Higher Dimensions 7

1.2.3 Test Functions, Linear Functionals, and

Distributions 7

1.2.4 Examples: Delta Function 8

1.3 Linear Differential Operators 10

1.3.1 Example: Boundary Conditions 10

1.4 Inner Product and Norm 11

1.5 Green’s Operator and Green’s Function 12

1.5.1 Examples: Direct Integrations 13

1.6 Adjoint Operators 16

1.6.1 Example: Adjoint Operator 17

1.7 Green’s Function and Adjoint Green’s Function 18

1.8 Green’s Function for L 19

1.9 Sturm-Liouville Operator 20

1.9.1 Method of Variable Constants 22

1.9.2 Example: Self-Adjoint Problem 23

1.9.3 Example: Non-Self-Adjoint Problem 24

1.10 Eigenfunctions and Green’s Function 26

1.10.1 Example: Eigenfunctions 28

1.11 Higher-Dimensional Operators 28

1.11.1 Example: Steady-State Heat Conduction

in a Plate 32

1.11.2 Example: Poisson’s Equation in a Rectangle 32

1.11.3 Steady-State Waves and the Helmholtz Equation 33

v

vi Contents

1.12 Method of Images 34

1.13 Complex Variables and the Laplace Equation 36

1.13.1 Nonhomogeneous Boundary Conditions 38

1.13.2 Example: Laplace Equation in a Semi-infinite

Region 38

1.13.3 Example: Laplace Equation in a Unit Circle 39

1.14 Generalized Green’s Function 39

1.14.1 Examples: Generalized Green’s Functions 42

1.14.2 A Récipé for Generalized Green’s Function 43

1.15 Non-Self-Adjoint Operator 44

1.16 More on Green’s Functions 47

2 Integral Equations …………………………… 56

2.1 Classification 56

2.2 Integral Equation from Differential Equations 58

2.3 Example: Converting Differential Equation 59

2.4 Separable Kernel 60

2.5 Eigenvalue Problem 62

2.5.1 Example: Eigenvalues 63

2.5.2 Nonhomogeneous Equation with a Parameter 64

2.6 Hilbert-Schmidt Theory 65

2.7 Iterations, Neumann Series, and Resolvent Kernel 67

2.7.1 Example: Neumann Series 68

2.7.2 Example: Direct Calculation of the Resolvent

Kernel 69

2.8 Quadratic Forms 70

2.9 Expansion Theorems for Symmetric Kernels 71

2.10 Eigenfunctions by Iteration 72

2.11 Bound Relations 73

2.12 Approximate Solution 74

2.12.1 Approximate Kernel 74

2.12.2 Approximate Solution 74

2.12.3 Numerical Solution 75

2.13 VolterraEquation 76

2.13.1 Example: Volterra Equation 77

2.14 Equations of the First Kind 78

2.15 Dual Integral Equations 80

2.16 Singular Integral Equations 81

2.16.1 Examples: Singular Equations 82

2.17 Abel Integral Equation 82

Contents vii

2.18 Boundary Element Method 84

2.18.1 Example: Laplace Operator 86

2.19 Proper Orthogonal Decomposition 88

3 Fourier Transforms ………………………….. 98

3.1 Fourier Series 98

3.2 Fourier Transform 99

3.2.1 Riemann-Lebesgue Lemma 102

3.2.2 Localization Lemma 103

3.3 Fourier Integral Theorem 104

3.4 Fourier Cosine and Sine Transforms 105

3.5 Properties of Fourier Transforms 108

3.5.1 Derivatives of F 108

3.5.2 Scaling 109

3.5.3 Phase Change 109

3.5.4 Shift 109

3.5.5 Derivatives of f 109

3.6 Properties of Trigonometric Transforms 110

3.6.1 Derivatives of Fc and Fs 110

3.6.2 Scaling 110

3.6.3 Derivatives of f 110

3.7 Examples: Transforms of Elementary Functions 111

3.7.1 Exponential Functions 111

3.7.2 Gaussian Function 113

3.7.3 Powers 117

3.8 Convolution Integral 119

3.8.1 Inner Products and Norms 120

3.8.2 Convolution for Trigonometric Transforms 121

3.9 Mixed Trigonometric Transform 122

3.9.1 Example: Mixed Transform 123

3.10 Multiple Fourier Transforms 124

3.11 Applications of Fourier Transform 124

3.11.1 Examples: Partial Differential Equations 124

3.11.2 Examples: Integral Equations 137

3.12 Hilbert Transform 142

3.13 Cauchy Principal Value 143

3.14 Hilbert Transform on a Unit Circle 145

3.15 Finite Hilbert Transform 146

3.15.1 Cauchy Integral 146

3.15.2 Plemelj Formulas 149

viii Contents

3.16 Complex Fourier Transform 151

3.16.1 Example: Complex Fourier Transform of x2 154

3.16.2 Example: Complex Fourier Transform of e|x| 154

3.17 Wiener-Hopf Method 155

3.17.1 Example: Integral Equation 155

3.17.2 Example: Factoring the Kernel 159

3.18 Discrete Fourier Transforms 162

3.18.1 Fast Fourier Transform 165

4 Laplace Transforms ………………………….. 174

4.1 Inversion Formula 175

4.2 Properties of the Laplace Transform 176

4.2.1 Linearity 176

4.2.2 Scaling 177

4.2.3 Shifting 177

4.2.4 Phase Factor 177

4.2.5 Derivative 178

4.2.6 Integral 178

4.2.7 Power Factors 179

4.3 Transforms of Elementary Functions 179

4.4 Convolution Integral 180

4.5 Inversion Using Elementary Properties 181

4.6 Inversion Using the Residue Theorem 182

4.7 Inversion Requiring Branch Cuts 183

4.8 Theorems of Tauber 186

4.8.1 Behavior of f(t) as t → 0 186

4.8.2 Behavior of f(t) as t → ∞ 187

4.9 Applications of Laplace Transform 187

4.9.1 Ordinary Differential Equations 187

4.9.2 Boundary Value Problems 191

4.9.3 Partial Differential Equations 191

4.9.4 Integral Equations 196

4.9.5 Cagniard–De Hoop Method 198

4.10 Sequences and the Z-Transform 203

4.10.1 Difference Equations 205

4.10.2 First-Order Difference Equation 206

4.10.3 Second-Order Difference Equation 207

4.10.4 Brilluoin Approximation for Crystal Acoustics 210

Author Index 219

Subject Index

## Preface – Advanced Topics in Applied Mathematics

This text is aimed at graduate students in engineering, physics, and applied mathematics. I have included four essential topics: Green’s functions, integral equations, Fourier transforms, and Laplace transforms. As background material for understanding these topics, a course in complex variables with contour integration and analytic continuation and a second course in differential equations are assumed. One may point out that these topics are not all that advanced – the expected advanced-level knowledge of complex variables and a familiarity with the classical partial differential equations of physics may be used as a justification for the term “advanced.” Most graduate students in engineering satisfy these prerequisites. Another aspect of this book that makes it “advanced” is the expected maturity of the students to handle the fast pace of the course. The fours topics covered in this book can be used for a one-semester course, as is done at the Illinois Institute of Technology (IIT). As an application-oriented course, I have included techniques with a number of examples at the expense of rigor. Materials for further reading are included to help students further their understanding in special areas of individual interest. With the advent of multiphysics computational software, the study of classical methods is in general on a decline, and this book is an attempt to optimize the time allotted in the curricula for applied mathematics. I have included a selection of exercises at the end of each chapter for instructors to choose as weekly assignments. A solutions manual for ix x Preface these exercises is available on request. The problems are numbered in such a way as to simplify the assignment process, instead of clustering anumber of similar problems under one number. Classical books on integral transforms by Sneddon and on mathematical methods by Morse and Feshbach and by Courant and Hilbert form the foundation for this book. I have included sections on the Boundary Element Method and Proper Orthogonal Decomposition under integral equations – topics of interest to the current research community. Advanced Topics in Applied Mathematics

The Cagniard–De Hoop method for inverting combined Fourier-Laplace transforms is well known to researchers in the area of elastic waves, and I feel it deserves exposure to applied mathematicians in general. Discrete Fourier transform leading to the fast Fourier algorithm and the Z-transform are included. I am grateful to my numerous students who have read my notes and corrected me over the years. My thanks also go to my colleagues, who helped to proofread the manuscript, Kevin Cassel, Dietmar Rempfer, Warren Edelstein, Fred Hickernell, Jeff Duan, and Greg Fasshauer, who have been persistent in instilling applied mathematics to believers and nonbelievers at IIT, and, especially, for training the students who take my course.Advanced Topics in Applied Mathematics

I am also indebted to my late colleague, Professor L. N. Tao, who shared the applied mathematics teaching with me for more than twenty-five years. The editorial assistance provided by Peter Gordon and Sara Black is appreciated. The MathematicaTM package from Wolfram Research was used to generate the number function plots. My wife, Celeste, has provided constant encouragement throughout the preparation of the manuscript, and I am always thankful to her.

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## About the Author

Sudhakar Nair is the Associate Dean for Academic Affairs of the Graduate College, Professor of Mechanical Engineering and Aerospace Engineering and Professor of Applied Mathematics at the Illinois Institute of Technology in Chicago. He is a Fellow of the ASME, an Associate Fellow of the AIAA and a member of the American Academy of Mechanics as well as Tau Beta Pi and Sigma Xi. Professor Nair is the author of numerous research articles and Introduction to Continuum Mechanics (2009).