Modern Mathematical Methods For Scientists And Engineers: A Street-smart Introduction

✍ Scribed by Athanassios S. Fokas and Efthimios Kaxiras

• Year: 2023
• Tongue: English
• Leaves: 568
• Edition: 1
• Category: Library

✦ Table of Contents

Contents
Preface
About the Authors
Acknowledgments
Part I Functions of Real Variables
Chapter 1 Functions of a Single Variable
1.1 Introduction: The various types of numbers
1.2 Elementary functions
1.2.1 Polynomials
1.2.2 The inverse function
1.2.3 Geometric shapes: Circle, ellipse, hyperbola
1.2.4 Trigonometric functions
1.2.5 Exponential, logarithm, hyperbolic functions
1.3 Continuity and derivatives
1.3.1 Continuity
1.3.2 Definition of derivatives
1.3.3 Geometric interpretation of derivatives
1.3.4 Finding roots by using derivatives
1.3.5 Chain rule and implicit differentiation
1.4 Integrals
1.4.1 The definite and indefinite integrals
1.4.2 Integration by change of variables
1.4.3 Integration by parts
1.4.4 The principal value integral
1.5 Norms and moments of a function
1.5.1 The norms of a function
1.5.2 The normalized Gaussian function
1.5.3 Moments of a function
1.6 Generalized functions: θ-function and δ-function
1.6.1 The θ-function or step-function
1.6.2 The δ-function
1.7 Application: Signal filtering, convolution, correlation
Chapter 2 Functions of Many Variables
2.1 General considerations
2.2 Coordinate systems and change of variables
2.2.1 Coordinate systems in two dimensions
2.2.2 Coordinate systems in three dimensions
2.2.3 Change of variables
2.3 Vector operations
2.3.1 The dot or scalar product
2.3.2 The cross or vector product
2.3.3 The wedge product
2.4 Differential operators
2.5 The vector calculus integrals
2.5.1 Simple curves and simply-connected domains
2.5.2 Green’s theorem
2.5.3 The divergence and curl theorems
2.5.4 Geometric interpretation of the divergence and curl theorems
2.6 Function optimization
2.6.1 Finding the extrema of a function
2.6.2 Constrained optimization: Lagrange multipliers
2.6.3 Calculus of variations
2.7 Application: Feedforward neural network
2.7.1 Definition of feedforward neural network
2.7.2 Training of the network
Chapter 3 Series Expansions
3.1 Infinite sequences and series
3.1.1 Series convergence tests
3.1.2 Some important number series
3.2 Series expansions of functions
3.2.1 Power series expansion: The Taylor series
3.2.2 Inversion of power series
3.2.3 Trigonometric series expansions
3.2.4 Polynomial series expansion
3.3 Convergence of series expansion of functions
3.3.1 Uniform convergence
3.3.2 Uniform convergence criteria
3.4 Truncation error in series expansions
3.5 Application: Wavelet analysis
3.5.1 Haar wavelet expansion
3.5.2 The wavelet transform
Part II Complex Analysis and the Fourier Transform
Chapter 4 Functions of Complex Variables
4.1 Complex numbers
4.2 Complex variables
4.2.1 The complex exponential
4.2.2 Multi-valuedness in polar representation
4.2.3 The complex logarithm
4.3 Continuity and derivatives
4.4 Analyticity
4.4.1 The d-bar derivative
4.4.2 Cauchy–Riemann relations and harmonic functions
4.4.3 Consequences of analyticity
4.4.4 Explicit formulas for computing an analytic function in terms of its real or imaginary parts
4.5 Complex integration: Cauchy’s theorem
4.5.1 Integration over a contour
4.5.2 Cauchy’s theorem
4.5.3 Contour deformation
4.5.4 Cauchy’s Integral Formula
4.6 Extensions of Cauchy’s theorem
4.6.1 General form of Cauchy’s theorem
4.6.2 Pompeiu’s formula
4.6.3 The δ-function of complex argument
4.7 Complex power series expansions
4.7.1 Taylor series
4.7.2 Laurent series
4.8 Application: The 2D Fourier transform
Chapter 5 Singularities, Residues, Contour Integration
5.1 Types of singularities
5.2 Residue theorem
5.3 Integration by residues
5.3.1 Integrands with simple poles
5.3.2 Integrands that are ratios of polynomials
5.3.3 Integrands with trigonometric or hyperbolic functions
5.3.4 Skipping a simple pole
5.4 Jordan’s lemma
5.5 Scalar Riemann–Hilbert problems
5.5.1 Derivation of the inverse Fourier transform
5.6 Branch points and branch cuts
5.7 The source–sink function and related problems
5.8 Application: Medical imaging
Chapter 6 Mappings Produced by Complex Functions
6.1 Mappings produced by positive powers
6.2 Mappings produced by 1/z
6.3 Mappings produced by the exponential function
6.4 Conformal mapping
6.5 Application: Fluid flow around an obstacle
Chapter 7 The Fourier Transform
7.1 Fourier series expansions
7.1.1 Real Fourier expansion
7.1.2 Odd and even function expansions
7.1.3 Handling of discontinuities in Fourier expansions
7.1.4 Fourier expansion of arbitrary range
7.1.5 Complex Fourier expansion
7.1.6 Fourier expansion of convolution and correlation
7.1.7 Conditions for the validity of a Fourier expansion
7.2 The Fourier transform
7.2.1 Definition of the Fourier transform
7.2.2 Properties of the Fourier transform
7.2.3 Symmetries of the Fourier transform
7.2.4 The sine- and cosine-transforms
7.3 Fourier transforms of special functions
7.3.1 The FT of the normalized Gaussian function
7.3.2 FT of the θ-function and of the δ-function
7.4 Application: Signal analysis
7.4.1 Aliasing: The Nyquist frequency
Part III Applications to Partial Differential Equations
Chapter 8 Partial Differential Equations: Introduction
8.1 General remarks
8.2 Overview of traditional approaches
8.2.1 Examples of ODEs
8.2.2 Eigenvalue equations
8.2.3 The Green’s function method
8.2.4 The Fourier series method
8.3 Evolution equations
8.3.1 Motivation of evolution equations
8.3.2 Separation of variables and initial value problems
8.3.3 Traditional integral transforms for problems on the half-line
8.3.4 Traditional infinite series for problems on a finite interval
8.4 The wave equation
8.4.1 Derivation of the wave equation
8.4.2 The solution of d’Alembert
8.4.3 Traditional transforms
8.5 The Laplace and Poisson equations
8.5.1 Motivation of the Laplace and Poisson equations
8.5.2 Integral representations through the fundamental solution
8.6 The Helmholtz and modified Helmholtz equations
8.7 Disadvantages of traditional integral transforms
8.8 Application: The Black–Scholes equation
Chapter 9 Unified Transform I: Evolution PDEs on the Half-lin
9.1 The unified transform is based on analyticity
9.2 The heat equation
9.3 The general methodology of the unified transform
9.3.1 Advantages of the unified transform
9.4 A PDE with a third-order spatial derivative
9.5 Inhomogeneous PDEs and other considerations
9.5.1 Robin boundary conditions
9.5.2 From PDEs to ODEs
9.5.3 Green’s functions
9.6 Application: Heat flow along a solid rod
9.6.1 The conventional solution
9.6.2 The solution through the unified transform
Chapter 10 Unified Transform II: Evolution PDEs on a Finite Interval
10.1 The heat equation
10.2 Advantages of the unified transform
10.3 A PDE with a third-order spatial derivative
10.4 Inhomogeneous PDEs and other considerations
10.4.1 Robin boundary conditions
10.4.2 From PDEs to ODEs
10.4.3 Green’s functions
10.5 Application: Detection of colloid concentration
Chapter 11 Unified Transform III: The Wave Equation
11.1 An alternative derivation of the global relation
11.2 The wave equation on the half-line
11.2.1 The Dirichlet problem
11.3 The wave equation on a finite interval
11.3.1 The Dirichlet–Dirichlet problem
11.4 The forced problem
Chapter 12 Unified Transform IV: Laplace, Poisson, and Helmholtz Equations
12.1 Introduction
12.1.1 Integral representations in the k-complex plane
12.2 The Laplace and Poisson equations
12.2.1 Global relations for a convex polygon
12.2.2 An integral representation in the k-complex plane
12.2.3 The approximate global relation for a convex polygon
12.2.4 The general case
12.3 The Helmholtz and modified Helmholtz equations
12.3.1 Global relations for a convex polygon
12.3.2 Novel integral representations for the modified Helmholtz equation
12.3.3 The general case
12.3.4 Computing the solution in the interior of a convex polygon
12.4 Generalizations
12.5 Application: Water waves
Part IV Probabilities, Numerical, and Stochastic Methods
Chapter 13 Probability Theory
13.1 Probability distributions
13.1.1 General concepts
13.1.2 Probability distributions and their features
13.2 Common probability distributions
13.2.1 Uniform distribution
13.2.2 Binomial distribution
13.2.3 Poisson distribution
13.2.4 Normal distribution
13.2.5 Arbitrary probability distributions
13.3 Probabilities, information and entropy
13.4 Multivariate probabilities
13.5 Composite random variables
13.5.1 Central Limit Theorem
13.6 Conditional probabilities
13.6.1 Bayes’ theorem
13.6.2 The Fokker–Planck equation
13.7 Application: Hypothesis testing
Chapter 14 Numerical Methods
14.1 Calculating with digital computers
14.2 Numerical representation of functions
14.2.1 Grid and spectral methods
14.2.2 Accuracy and stability
14.3 Numerical evaluation of derivatives
14.4 Numerical evaluation of integrals
14.4.1 The trapezoid formula
14.4.2 Simpson’s 1/3-rule formula
14.5 Numerical solution of ODEs
14.5.1 First-order ODEs
14.5.2 Second-order ODEs
14.6 Numerical solution of PDEs
14.7 Solving differential equations with neural networks
14.8 Computer-generated random numbers
14.9 Application: Monte Carlo integration
Chapter 15 Stochastic Methods
15.1 Stochastic simulation: Random walks and diffusion
15.2 Stochastic optimization
15.2.1 The method of “importance sampling”
15.2.2 The Metropolis algorithm
15.3 The “simulated annealing” method
15.4 Application: The traveling salesman problem
Appendix A Solution of the Black–Scholes Equation
Appendix B Gaussian Integral Table
Index