Advanced Mathematical Methods in Science and Engineering, Second Edition

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ADVANCED MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING, Second Edition

© 2010 by Taylor & Francis Group, LLC
ISBN 978-1-4200-8197-8 (Hardback)

TABLE OF CONTENTS

PREFACE

ACKNOWLEDGMENTS

1 ORDINARY DIFFERENTIAL EQUATIONS
1.1 Definitions
1.2 Linear Differential Equations of First Order
1.3 Linear Independence and the Wronskian
1.4 Linear Homogeneous Differential Equation of Order n with Constant Coefficients
1.5 Euler's Equation
1.6 Particular Solutions by Method of Undetermined Coefficients
1.7 Particular Solutions by the Method of Variations of Parameters
1.8 Abel's Formula for the Wronskian
1.9 Initial Value Problems
PROBLEMS

2 SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
2.1 Introduction
2.2 Power Series Solutions
2.3 Classification of Singularities
2.4 Frobenius Solution
PROBLEMS

3 SPECIAL FUNCTIONS
3.1 Bessel Functions
3.2 Bessel Function of Order Zero
3.3 Bessel Function of an Integer Order n
3.4 Recurrence Relations for Bessel Functions
3.5 Bessel Functions of Half Orders
3.6 Spherical Bessel Functions
3. 7 Hankel Functions
3.8 Modified Bessel Functions
3.9 Generalized Equations Leading to Solutions in Terms of Bessel Functions
3.10 Bessel Coefficients
3.11 Integral Representation of Bessel Functions
3.12 Asymptotic Approximations of Bessel Functions for Small Arguments
3.13 Asymptotic Approximations of Bessel Functions for Large Arguments
3.14 Integrals of Bessel Functions
3.15 Zeroes of Bessel Functions
3.16 Legendre Functions
3.17 Legendre Coefficients
3.18 Recurrence Formulae for Legendre Polynomials
3.19 Integral Representation for Legendre Polynomials
3.20 Integrals of Legendre Polynomials
3.21 Expansions of Functions in Terms of Legendre Polynomials
3.22 Legendre Function of the Second Kind Q0 (x)
3.23 Associated Legendre Functions
3.24 Generating Function for Associated Legendre Functions
3.25 Recurrence Formulae for P:
3.26 Integrals of Associated Legendre Functions
3.27 Associated Legendre Function of the Second Kind Q:
PROBLEMS

4 BOUNDARY VALUE PROBLEMS AND EIGENVALUE PROBLEMS
4.1 Introduction
4.2 Vibration, Wave Propagation or Whirling of Stretched Strings
4.3 Longitudinal Vibration and Wave Propagation in Elastic Bars
4.4 Vibration, Wave Propagation, and Whirling of Beams
4.5 Waves in Acoustic Horns
4.6 Stability of Compressed Columns
4.7 Ideal Transmission Lines (Telegraph Equation)
4.8 Torsional Vibration of Circular Bars
4.9 Orthogonality and Orthogonal Sets of Functions
4.10 Generalized Fourier Series
4.11 Adjoint Systems
4.12 Boundary Value Problems
4.13 Eigenvalue Problems
4.14 Properties of Eigenfunctions of Self-Adjoint Systems
4.15 Sturm-Liouville System
4.16 Sturm-Liouville System for Fourth-Order Equations
4.17 Solution of Non-Homogeneous Eigenvalue Problems
4.18 Fourier Sine Series
4.19 Fourier Cosine Series
4.20 Complete Fourier Series
4.21 Fourier-Bessel Series
4.22 Fourier-Legendre Series
PROBLEMS

5 FUNCTIONS OF A COMPLEX VARIABLE
5.1 Complex Numbers
5.1.1 Complex Conjugate
5.1.2 Polar Representation
5.1.3 Absolute Value
5.1.4 Powers and Roots of a Complex Number
5.2 Analytic Functions
5.2.1 Neighborhood of a Point
5.2.2 Region
5.2.3 Functions of a Complex Variable
5.2.4 Limits
5.2.5 Continuity
5.2.6 Derivatives
5.2.7 Cauchy-Reimann Conditions
5.2.8 Analytic Functions
5.2.9 Multi-Valued Functions, Branch Cuts and Branch Points
5.3 Elementary Functions
5.3.1 Polynomials
5.3.2 Exponential Function
5.3.3 Circular Functions
5.3.4 Hyperbolic Functions
5.3.5 Logarithmic Function
5.3.6 Complex Exponents
5.3. 7 Inverse Circular and Hyperbolic Functions
5.4 Integration in the Complex Plane
5.4.1 Green's Theorem
5.5 Cauchy's Integral Theorem
5.6 Cauchy's Integral Formula
5.7 Infinite Series
5.8 Taylor's Expansion Theorem
5.9 Laurent's Series
5.10 Classification of Singularities
5.11 Residues and Residue Theorem
5.11.1 Residue Theorem
5.12 Integrals of Periodic Functions
5.13 Improper Real Integrals
5.14 Improper Real Integral Involving Circular Functions
5.15 Improper Real Integrals of Functions Having Singularities on the Real Axis
5.16 Theorems on Limiting Contours
5.16.1 Generalized Jordan's Lemma
5.16.2 Small Circle Theorem
5.16.3 Small Circle Integral
5.17 Evaluation of Real Improper Integrals by Non-Circular Contours
5.18 Integrals of Even Functions Involving log x
5.19 Integrals of Functions Involving x3
5.20 Integrals of Odd or Asymmetric Functions
5.21 Integrals of Odd or Asymmetric Functions Involving log x
5.22 Inverse Laplace Transforms
PROBLEMS

6 Partial Differential Equations of Mathematical Physics
6.1 Introduction
6.2 The Diffusion Equation
6.2.1 Heat Conduction in Solids
6.2.2 Diffusion of Gases
6.2.3 Diffusion and Absorption of Particles
6.3 The Vibration Equation
6.3.1 The Vibration of One-Dimensional Continua
6.3.2 The Vibration of Stretched Membranes
6.3.3 The Vibration of Plates
6.4 The Wave Equation
6.4.1 Wave Propagation in One-Dimensional Media
6.4.2 Wave Propagation in Two-Dimensional Media
6.4.3 Wave Propagation in Surface of Water Basin
6.4.4 Wave Propagation in an Acoustic Medium
6.5 Helmholtz Equation
6.5.1 Vibration in Bounded Media
6.5.2 Harmonic Waves
6.6 Poisson and Laplace Equations
6.6.1 Steady State Temperature Distribution
6.6.2 Flow of Ideal Incompressible Fluids
6.6.3 Gravitational (Newtonian) Potentials
6.6.4 Electrostatic Potential
6.7 Classification of Partial Differential Equations
6.8 Uniqueness of Solutions
6.8.1 Laplace and Poisson Equations
6.8.2 Helmholtz Equation
6.8.3 Diffusion Equation
6.8.4 Wave Equation
6.9 The Laplace Equation
6.10 The Poisson Equation
6.11 The Helmholtz Equation
6.12 The Diffusion Equation
6.13 The Vibration Equation
6.14 The Wave Equation
6.14.1 Wave Propagation in an Infinite, One-Dimensional Medium
6.14.2 Spherically Symmetric Wave Propagation in an Infinite Medium
6.14.3 Plane Harmonic Waves
6.14.4 Cylindrical Harmonic Waves
6.14.5 Spherical Harmonic Waves
PROBLEMS

7 INTEGRAL TRANSFORMS
7.1 Fourier Integral Theorem
7.2 Fourier Cosine Transform
7.3 Fourier Sine Transform
7.4 Complex Fourier Transform
7.5 Multiple Fourier Transform
7.6 Hankel Transform of Order Zero
7.7 Hankel Transform of Order v
7.8 General Remarks about Transforms Derived from the Fourier Integral Theorem
7.9 Generalized Fourier Transform
7.10 Two-Sided Laplace Transform
7.11 One-Sided Generalized Fourier Transform
7.12 Laplace Transform
7.13 Mellin Transform
7.14 Operational Calculus with Laplace Transforms
7.14.1 The Transform Function
7.14.2 Shift Theorem
7.14.3 Convolution (Faltung) Theorems
7.14.4 Laplace Transform of Derivatives
7 .14.5 Laplace Transform of Integrals
7.14.6 Laplace Transform of Elementary Functions
7.14.7 Laplace Transform of Periodic Functions
7.14.8 Heaviside Expansion Theorem
7.14.9 The Addition Theorem
7.15 Solution of Ordinary and Partial Differential Equations by Laplace Transforms
7.16 Operational Calculus with Fourier Cosine Transform
7.16.1 Fourier Cosine Transform of Derivatives
7.16.2 Convolution Theorem
7.16.3 Parseval Formula
7.17 Operational Calculus with Fourier Sine Transform
7.17.1 Fourier Sine Transform of Derivatives
7.17.2 Convolution Theorem
7.17.3 Parseval Formula
7.18 Operational Calculus with Complex Fourier Transform
7.18.1 Complex Fourier Transform of Derivatives
7.18.2 Convolution Theorem
7.18.3 Parseval Formula
7.19 Operational Calculus with Multiple Fourier Transform
7.19.1 Multiple Transform of Partial Derivatives
7.19.2 Convolution Theorem
7.20 Operational Calculus with Hankel Transform
7.20.1 Hankel Transform of Derivatives
7.20.2 Convolution Theorem
7.20.3 Parseval Formula
PROBLEMS

8 GREEN'S FUNCTIONS
8.1 Introduction
8.2 Green's Function for Ordinary Differential Boundary Value Problems
8.3 Green's Function for an Adjoint System
8.4 Symmetry of the Green's Functions and Reciprocity
8.5 Green's Function for Equations with Constant Coefficients
8.6 Green's Functions for Higher-Ordered Sources
8.7 Green's Function for Eigenvalue Problems
8.8 Green's Function for Semi-Infinite One-Dimensional Media
8.9 Green's Function for Infinite One-Dimensional Media
8.10 Green's Function for Partial Differential Equations
8.11 Green's Identities for the Laplacian Operator
8.12 Green's Identity for the Helmholtz Operator
8.13 Green's Identity for Hi-Laplacian Operator
8.14 Green's Identity for the Diffusion Operator
8.15 Green's Identity for the Wave Operator
8.16 Green's Function for Unbounded Media-Fundamental Solution
8.17 Fundamental Solution for the Laplacian
8.17.1 Three-Dimensional Space
8.17.2 Two-Dimensional Space
8.17.3 One-Dimensional Space
8.17.4 Development by Construction
8.17.5 Behavior for LargeR
8.18 Fundamental Solution for the Hi-Laplacian
8.19 Fundamental Solution for the Helmholtz Operator
8.19.1 Three-Dimensional Space
8.19.2 Two-Dimensional Space
8.19.3 One-Dimensional Space
8.19.4 Behavior for Large R
8.20 Fundamental Solution for the Operator,- V2 + J.L2
8.20.1 Three-Dimensional Space
8.20.2 Two-Dimensional Space
8.20.3 One-Dimensional Space
8.21 Causal Fundamental Solution for the Diffusion Operator
8.21.1 Three-Dimensional Space
8.21.2 Two-Dimensional Space
8.21.3 One-Dimensional Space
8.22 Causal Fundamental Solution for the Wave Operator
8.22.1 Three-Dimensional Space
8.22.2 Two-Dimensional Space
8.22.3 One-Dimensional Space
8.23 Fundamental Solutions for the Bi-Laplacian Helmholtz Operator
8.24 Green's Function for the Laplacian Operator for Bounded Media
8.24.1 Dirichlet Boundary Condition
8.24.2 Neumann Boundary Condition
8.24.3 Robin Boundary Condition
8.25 Construction of the Auxiliary Function-Method of Images
8.26 Green's Function for the Laplacian for Half-Space
8.26.1 Dirichlet Boundary Condition
8.26.2 Neumann Boundary Condition
8.27 Green's Function for the Laplacian by Eigenfunction Expansion for Bounded Media
8.28 Green's Function for a Circular Area for the Laplacian
8.28.1 Interior Problem
8.28.2 Exterior Problem
8.29 Green's Function for Spherical Geometry for the Laplacian
8.29.1 Interior Problem
8.29.2 Exterior Problem
8.30 Green's Function for the Helmholtz Operator for Bounded Media
8.31 Green's Function for the Helmholtz Operator for Half-Space
8.31.1 Three-Dimensional Half-Space
8.31.2 Two-Dimensional Half-Space
8.31.3 One-Dimensional Half-Space
8.32 Green's Function for a Helmholtz Operator in Quarter-Space
8.33 Causal Green's Function for the Wave Operator in Bounded Media
8.34 Causal Green's Function for the Diffusion Operator for Bounded Media
8.35 Method of Summation of Series Solutions in Two-Dimensional Media
8.35.1 Laplace's Equation in Cartesian Coordinates
8.35.2 Laplace's Equation in Polar Coordinates
PROBLEMS

9 ASYMPTOTIC METHODS
9.1 Introduction
9.2 Method of Integration by Parts
9.3 Laplace's Integral
9.4 Steepest Descent Method
9.5 Debye's First-Order Approximation
9.6 Asymptotic Series Approximation
9.7 Method of Stationary Phase
9.8 Steepest Descent Method in Two Dimensions
9.9 Modified Saddle Point Method: Subtraction of a Simple Pole
9.10 Modified Saddle Point Method: Subtraction of Pole of Order N
9.11 Solution of Ordinary Differential Equations for Large Arguments
9.12 Classification of Points at Infinity
9.13 Solutions of Ordinary Differential Equations with Regular Singular Points
9.14 Asymptotic Solutions of Ordinary Differential Equations with Irregular Singular Points of Rank One
9.14.1 Normal Solutions
9.14.2 Subnormal Solutions
9.15 The Phase Integral and WKBJ Method for an Irregular Singular Point of Rank One
9.16 Asymptotic Solutions of Ordinary Differential Equations with Irregular Singular Points of Rank Higher th_an One
9.17 Asymptotic Solutions of Ordinary Differential Equations with Large Parameters
9.17.1 Formal Solution in Terms of Series in x and A.
9.17.2 Formal Solutions in Exponential Form
9.17.3 Asymptotic Solutions of Ordinary Differential Equations with Large Parameters by the WKBJ Method
PROBLEMS

10 NUMERICAL METHODS
10.1 Introduction
10.2 Roots of Non-Linear Equations
10.2.1 Bisection Method
10.2.2 Newton-Raphson Method
10.2.3 Secant Method
10.2.4 Iterative Method
10.3 Roots of a System of Nonlinear Equations
10.3.1 Iterative Method
10.3.2 Newton's Method
10.4 Finite Differences
10.4.1 Forward Difference
10.4.2 Backward Difference
10.4.3 Central Difference
10.5 Numerical Differentiation
10.5.1 Forward Differentiation
10.5.2 Backward Differentiation
10.5.3 Central Differentiation
10.6 Numerical Integration
10.6.1 Trapezoidal Rule
10.6.2 Simpson's Rule
10.6.3 Romberg Integration
10.6.4 Gaussian Quadrature
10.7 Ordinary Differential Equations-Initial Value Problems
10.7.1 Euler's Method for First-Order ODE
10.7.2 Euler Prediction-Corrector Method
10.7.3 Runge-Kutta Methods
10.7.4 Adams Method
10.7.5 System of First-Order Simultaneous ODE
10.7.6 High-Ordered ODE
10.7.7 Correction-Extrapolation ofResults
10.8 ODE-Boundary Value Problems (BVP)
10.8.1 One-Dimensional BVP
10.8.2 Shooting Method
10.8.3 Equilibrium Method
10.9 ODE-Eigenvalue Problems
10.10 Partial Differential Equations
10.10.1 Laplace Equation
10.10.2 Poison's Equation
10.10.3 The Laplacian in Cylindrical Coordinates
10.10.4 Helmholtz Equation
10.10.5 Diffusion Equation
10.10.6 Wave Equation
Problems


APPENDIX A INFINITE SERIES
A.1 Introduction
A.2 Convergence Tests
A.2.1 Comparison Test
A.2.2 Ratio Test: (d'Aiembert's)
A.2.3 Root Test: (Cauchy's)
A.2.4 Raabe's Test
A.2.5 Integral Test
A.3 Infinite Series of Functions of One Variable
A.3.1 Uniform Convergence
A.3.2 Weierstrass's Test for Uniform Convergence
A.3.3 Consequences of Uniform Convergence
A.4 Power Series
A.4.1 Radius of Convergence
A.4.2 Properties of Power Series
PROBLEMS

APPENDIX B SPECIAL FUNCTIONS
B.1 The Gamma Function r(x)
B.2 PSI Function w(x)
B.3 Incomplete Gamma Function y (x,y)
B.4 Beta Function B(x,y)
B.5 Error Function erf(x)
B.6 Fresnel Functions C(x), S(x), and F(x)
B.7 Exponential Integrals Ei(x) and En(x)
B.8 Sine and Cosine Integrals Si(x) and Ci(x)
B.9 Tchebyshev Polynomials T n(x) and Un(x)
B.l0 Laguerre Polynomials L 0 (x)
B.11 Associated Laguerre Polynomials L':: (x)
B.12 Hermite Polynomials H0 (x)
B.13 Hypergeometric Functions F(a, b; c; x)
B.14 Confluent Hypergeometric Functions M(a,c,x) and U(a,c,x)
B.15 Kelvin Functions (berv (x), beiv (x), kerv (x), kei(x))

APPENDIX C ORTHOGONAL COORDINATE SYSTEMS
C.1 Introduction
C.2 Generalized Orthogonal Coordinate Systems
C.3 Cartesian Coordinates
C.4 Circular Cylindrical Coordinates
C.5 Elliptic-Cylindrical Coordinates
C.6 Spherical Coordinates
C.7 Prolate Spheroidal Coordinates
C.7.1 Prolate Spheroidal Coordinates I
C.7.2 Prolate Spheroidal Coordinates II
C.8 Oblate Spheroidal Coordinates
C.8.1 Oblate Spherical Coordinates I
C.8.2 Oblate Spheroidal Coordinates II

APPENDIX D DIRAC DELTA FUNCTIONS
D.1 Dirac Delta Function
D.1.1 Definitions and Integrals
D.1.2 Integral Representations
D.1.3 Transformation Property
D.1.4 Concentrated Field Representations
D.2 Dirac Delta Function of Order One
D.3 Dirac Delta Function of Order N
D.4 Equivalent Representations of Distributed Functions
D.5 Dirac Delta Functions inn-Dimensional Space
D.5.l Definitions and Integrals
D.5.2 Representation by Products of Dirac Delta Functions
D.5.3 Dirac Delta Function in Linear Transformation
D.6 Spherically Symmetric Dirac Delta Function Representation
D.7 Dirac Delta Function of Order N inn-Dimensional Space
PROBLEMS

APPENDIX E PLOTS OF SPECIAL FUNCTIONS
E.1 Bessel Functions of the First and Second Kind of Order 0, 1, 2
E.2 Spherical Bessel Functions of the First and Second Kind of Order 0, 1, 2
E.3 Modified Bessel Function of the First and Second Kind of Order 0, 1, 2
E.4 Bessel Function of the First and Second Kind of Order 112
E.5 Modified Bessel Function of the First and Second Kind of Order 1/2

APPENDIX F VECTOR ANALYSIS
F.1 Definitions and Index Notation
F.2 Vector Algebra
F.3 Scalar and Vector Products
F.4 Vector Fields
F.5 Gradient of a Scalar
F.6 Divergence of a Vector
F.7 Curl of a Vector
F.8 Divergence (Green's) Theorem
F.9 Stoke's Theorem
F.10 Representation of Vector Fields
PROBLEMS

APPENDIX G MATRIX ALGEBRA
G.1 Definitions
G.2 Properties of Matrices
G.3 Determinants of Square Matrices
G.4 Properties of Determinants of Square Matricies
G.5 Solution of Linear Algebraic Equations
G.6 Eigenvalues of Hermetian Matrices
G.7 Properties of Eigenvalues and Eigenvectors
PROBLEMS

REFERENCES

ANSWERS

INDEX