Mathematical Methods for the Natural and Engineering Sciences 1st Edition by Ronald E Mickens ISBN 9789812387509 by Ronald E. Mickens 9789812387509, 9812387501 instant download after payment.

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ISBN 13: 9789812387509
Author: Ronald E Mickens

This book provides a variety of methods required for the analysis and solution of equations which arise in the modeling of phenomena from the natural and engineering sciences. It can be used productively by both undergraduate and graduate students, as well as others who need to learn and understand these techniques. A detailed discussion is also presented for several topics that are usually not included in standard textbooks at this level: qualitative methods for differential equations, dimensionalization and scaling, elements of asymptotics, difference equations, and various perturbation methods. Each chapter contains a large number of worked examples and provides references to the appropriate literature.

Mathematical Methods for the Natural and Engineering Sciences 1st Table of contents:

1. Introduction
1.1 Mathematical Modeling
1.2 Mathematical versus Physical Equations
1.3 Dimensionless Variables and Characteristic Scales
1.4 Construction of Mathematical Equations
1.4.1 Decay Equation
1.4.2 Logistic Equation
1.4.3 The Fisher Equation
1.4.4 Duffings’ Equation
1.4.5 Budworm Population Dynamics
1.5 Nonlinearity
Problems
Comments and References
Bibliography
Dimensionless Variables and Scaling
Mathematical Modeling
Nonlinearity
2. Trigonometric Relations and Fourier Analysis
2.1 Introduction
2.2 Euler’s Formula and DeMoivre’s Theorem
2.3 Derivation of Trigonometric Relations
2.4 Periodic, Even and Odd Functions
2.4.1 Periodic Functions
2.4.2 Even and Odd Functions
2.5 Fourier Series
2.6 Worked Examples for Fourier Series
2.6.1 Parseval's Identity
2.6.2 f(x) = |x|, - x
2.6.3 f(x) = x, - < x <
2.6.4 The Square Wave Function
2.6.5 Comparison of Problems 2.6.2 and 2.6.4
2.6.6 f(x) = sin x, 0 < x <
2.6.7 Fourier Series of x2, - < x < by Integration
2.6.8 Riemann-Lebesgue Theorem
2.7 Fourier Series for (cos ) and (sin )
2.8 Fourier Transforms
2.8.1 Definition of Fourier Transforms
2.8.2 Basic Properties of Fourier Transforms
2.9 Application of Fourier Transforms
2.9.1 Fourier Transform of the Square Pulse
2.9.2 Fourier Transform of the Gaussian Function
2.9.3 The Convolution Theorem
2.9.4 The Diffusion Equation
2.9.5 The Wave Equation
2.10 The Laplace Transform
Linearity Condition
Shift Theorems
Scaling Condition
Differentiation of a Transform
Transform of an Integral
Integration of the Transform
Transform of Derivatives
Convolution Theorem
Transform of a Periodic Function
2.11 Worked Problems Using the Laplace Transform
2.11.1 Laplace Transform of t-1/3
2.11.2 The Square Wave Function
2.11.3 The Dirac Delta Function
Problems
Comments and References
Bibliography
3. Gamma, Beta, Zeta, and Other Named Functions
3.1 Scope of Chapter
3.2 Gamma Function
3.3 The Beta Function
3.4 The Riemann Zeta Function
3.5 The Dirac Delta Function
3.6 Dirichlet Integrals
3.7 Applications
3.7.1 Additional Properties of (z)
3.7.2 A Definite Integral Containing Logarithms
3.7.3 A Class of Important Integrals
3.7.4 A Representation for x-p
3.7.5 Additional Properties of the Beta Function
3.7.6 Fermi-Dirac Integrals
3.7.7 An Integral Involving an Exponential
3.7.8 Fermi-Dirac Integrals Containing Logarithms
3.7.9 Magnetic Moment of the Electron
3.7.10 Relationship Between the Theta and Delta Functions
3.7.11 Evaluation of Integrals by Use of the Beta Function
3.8 Other Named Functions
3.9 Elliptic Integrals and Functions
3.9.1 Elliptic Integrals of the First and Second Kind
3.9.2 Jacobi Elliptic Functions
3.10 Evaluation of Integrals
Problems
Comments and References
Bibliography
4. Qualitative Methods for Ordinary Differential Equations
4.1 Introduction
4.2 One-Dimensional Systems
4.2.1 Definition
4.2.2 Fixed-Points
4.2.3 Sign of the Derivative
4.2.4 Linear Stability
4.3 Worked Examples
4.3.1 Examples A
4.3.2 Example B
4.3.3 Example C
4.3.4 Example D
4.3.5 Example E
4.4 Two-Dimensional Systems
4.4.1 Definition
4.4.2 Fixed- Points
4.4.3 Nullclines
4.4.4 First-Integrals and Symmetries
4.4.5 General Features of Two-Dimensional Phase Space
4.4.6 The Basic Procedure for Constructing Phase-Space Diagrams
4.4.7 Linear Stability
4.5 Worked Examples
4.5.1 Example A
4.5.2 Example B
4.5.3 Example C
4.5.4 Example D
4.5.5 Example E
4.5.6 Example F
4.5.7 Example G
4.5.8 Example H
Second Derivative Test
4.6 Bifurcations
4.6.1 Hopf- Bifurcations
Hopf-Bifurcation Theorem
Comments:
4.6.2 Two Examples
Problems
Comments and References
Bibliography
Bifurcations
Qualitative Methods
5. Difference Equations
5.1 Genesis of Difference Equations
5.1.1 Square-Root Iteration
5.1.2 Integral Depending on Two Parameters
5.2 Existence and Uniqueness Theorem
5.3 The Fundamental Operators
5.3.1 The Operator
5.3.2 The Shift Operator, E
5.3.3 Difference of Functions
Difference of a Product
Leibnitz's Theorem for Differences
Difference of a Quotient
Difference of a Finite Sum
5.3.4 The Operator -1
5.4 Worked Problems Based on Section 5.3
5.4.1 Examples A
5.4.2 Example B
5.4.3 Example C
5.4.4 Examples D
5.5 First-Order Linear Difference Equations
5.5.1 Example A
5.5.2 Example B
5.5.3 Examples C
5.6 General Linear Difference Equations
5.6.1 General Properties
5.6.2 Linearly Independent Functions
5.6.3 Fundamental Theorems for Homogeneous Equations
5.6.4 Inhomogeneous Equations
5.7 Worked Problems
5.7.1 Example A
5.7.2 Example B
5.7.3 Example C
5.7.4 Example D
5.8 Linear Difference Equations with Constant Coefficients
5.8.1 Homogeneous Equations
5.8.2 Inhomogeneous Equations
5.8.3 Worked Examples
Example A
Example B
Example C
Example D
Example E
Example F
5.9 Nonlinear Difference Equations
5.9.1 Homogeneous Equations
5.9.2 Riccati Equations
5.9.3 Clairaut’s Equation
5.9.4 Miscellaneous Forms
5.10 Worked Examples of Nonlinear Equations
5.10.1 Example A
5.10.2 Example B
5.10.3 Example C
5.10.4 Example D
5.10.5 Example E
5.11 Two Applications
5.11.1 Chebyshev Polynomials
5.11.2 A Discrete Logistic Equation
Problems
Comments and References
Bibliography
6. Sturm-Liouville Problems
6.1 Introduction
6.2 The Vibrating String
6.2.1 Fixed Ends
6.2.2 One Fixed and One Free Ends
6.2.3 Both Ends Free
6.2.4 Discussion
6.3 Sturm Separation and Comparison Theorem
6.3.1 Example A
6.3.2 Example B
6.3.3 Example C
6.4 Sturm-Liouville Problems
6.4.1 Fundamental Definition
6.4.2 Properties of Eigenvalues and Eigenfunctions
6.4.3 Orthogonality of Eigenfunctions
6.4.4 Expansion of Functions
6.4.5 The Completeness Relation
6.5 Applications
6.5.1 The Special Functions
6.5.2 Fourier Expansion of f(x) = x(1 - x)
6.5.3 Representation of (x) in Terms of Cosine Functions
6.5.4 Reality of the Eigenvalues
6.5.5 A Boundary Value Problem
6.6 Green’s Functions
6.7 Worked Examples for Green Functions
6.7.1 y”(x) = -f( ) : y(0) = y(L) = 0
6.7.2 y”(x) = -f(x) : y(0) = O,y’(L) = 0
6.7.3 y”(x) + k2y(x) = -f(x) : y(0) = y(L) = 0
6.7.4 y”(x) + y(x) = x : y(0) = y(1) = 0
6.8 Asymptotic Behavior of Solutions to Differential Equations
6.8.1 Elimination of First-Derivative Terms
6.8.2 The Liouville-Green Transformation
6.9 Worked Examples
6.9.1 The Airy Equation
6.9.2 The Bessel Equation
6.9.3 A General Expansion Procedure
Problems
Comments and References
Bibliography
Asymptotics for Differential Equations
Green’s Functions
Sturm-Liouville Problems
7. Special Functions and Their Properties
7.1 Introduction
7.2 Classical Orthogonal Polynomials
7.2.1 Differential Equation and Interval of Definition
7.2.2 Weight Functions and Rodrique's Formulas
7.2.3 Orthogonality Relations
7.2.4 Generating Function
7.2.5 Recurrence Relations
7.2.6 Differential Recurrences
7.2.7 Special Values
7.2.8 Zeros of COP
7.3 Legendre Polynomials: Pn(x)
Differential Equation
Interval and Weight Function
Generating Function
Rodrique 's Formula
Orthogonality Condition
Recurrence Relations
List of Po(x) to P5(x)
Special Properties and Values
7.4 Hermite Polynomials: Hn(x)
Differential Equation
Interval and Weight Function
Generating Function
Rodrique's Formula
Orthogonality Condition
Recurrence Relations
List of Ho(x) to H5(x)
Special Properties and Values
7.5 Chebyshev Polynomials: Tn(x) and Un(x)
Differential Equation
Interval and Weight Function
Generating Function
Rodrique 's Formula
Orthogonality Condition
Recurrence Relations
List of Tn(x) to Un(x) for n = 0 to 5
Special Properties and Values
7.6 Laguerre Polynomials: Ln(x)
Differential Equations
Interval and Weight Function
Generating Functions
Rodrique's Formula
Orthogonality Condition
Recurrence Relations
List of Lo(x) to L5(x)
7.7 Legendre Functions of the Second Kind and Associated Legendre Functions
7.8 Bessel Functions
7.9 Some Proofs and Worked Problems
7.9.1 Proof of Rodriques Formula for Pn(x)
7.9.2 Integrals of xm with Pn(x)
7.9.3 Expansion of (x) in Terms of Pn(x)
7.9.4 Laplace’s Equation in Spherical Coordinates
7.9.5 Sphere in a Uniform Flow
7.9.6 The Harmonic Oscillator
7.9.7 Matrix Elements of the Electric Dipole
7.9.8 General Solution Hermite's Differential Equation for n=O
7.9.9 Three Dimensional Harmonic Oscillator
7.9.10 Proof that J-n(x) = (-1)nJn(x)
7.9.11 Calculation of Jn(x) from J0(x)
7.9.12 An Integral Representation for Jn(x)
7.9.13 Expansion of Cosine in Terms of Jn(x)
7.9.14 Derivation of a Recurrence Relation from the Generating Function
7.9.15 An Alternative Representation for Yn(x)
7.9.16 Equations Reducible to Bessel’s Equation
Problems
Comments and References
Bibliography
Applications and General Topics
Bessel Functions
Generating Functions
Legendre Polynomials and Functions
8. Perturbation Methods for Oscillatory Systems
8.1 Introduction
8.2 The General Perturbation Procedure
8.3 Worked Examples Using the General Perturbation Procedure
8.3.1 Example A
8.3.2 Example B
8.4 First-Order Method of Averaging
8.4.1 The Method
8.4.2 Two Special Cases for f(x,dx/dt)
8.4.3 Stability of Limit-Cycles
8.5 Worked Examples for First-Order Averaging
8.5.1 Example A
8.5.2 Example B
8.5.3 Example C
8.6 The Lindstedt-Poincare Method
8.6.1 Secular Terms
8.6.2 The Formal Procedure
8.7 Worked Examples Using the Lindstedt-Poincare Method
8.7.1 Example A
8.7.2 Example B
8.8 Harmonic Balance
8.8.1 Direct Harmonic Balance
8.9 Worked Examples for Harmonic Balance
8.9.1 Example A
8.9.2 Example B
8.9.3 Example C
8.10 Averaging for Difference Equations
8.11 Worked Examples for Difference Equations
8.11.1 Example A
8.11.2 Example B
Problems
Comments and References
Bibliography
9. Approximations of Integrals and Sums
9.1 Resume of Asymptotics
9.2 Integration by Parts
9.3 Laplace Methods
9.3.1 Watson’s Lemma
9.3.2 Laplace's Method for Integrals
9.4 Worked Examples
9.4.1 Stirling's Formula
9.4.2 Integral Containing a Logarithmic Function
9.4.3 Integral Containing a Complex Exponential Structure
9.4.4 Cosine and Sine Integrals
9.5 Euler-Maclaurin Sum Formula
9.5.1 Bernoulli Functions and Numbers
9.5.2 Euler-Maclaurin Sum Formula
9.6 Worked Examples for the Euler-Maclaurin Sum Formula
9.6.1 Sums of Powers
9.6.2 Evaluation of ln(n!)
9.6.3 f(k) = x-1/2
Problems
Comments and References
Bibliography
10. Some Important Nonlinear Partial Differential Equations
10.1 Linear Wave Equations
10.2 Traveling Wave and Soliton Solutions
10.3 A Linear Advective, Nonlinear Reaction Equation
10.4 Burgers’ Equation
10.5 The Fisher Equation
10.6 The Korteweg-de Vries Equation
10.7 The Nonlinear Schrodinger Equation
10.8 Similarity Methods and Solutions
10.8.1 Similarity Methods
10.8.2 Examples
Example A
Example B
10.9 The Boltzmann Problem
10.10 The Nonlinear Diffusion Equation: uut = uxx
Problems

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