Mathematical Methods for the Natural and Engineering Sciences

The following sections are included:

• Preface
• Contents

The need to study various mathematical techniques grows out of the requirement to understand, solve and/or evaluate the different types of equations that arise from the mathematical modeling of phenomena in the natural, social, and engineering sciences. The first equations written down for a particular system involve a number of parameters appearing in the equations relating variable quantities. What is generally required, at this point, is to convert these equations into structurally similar equations, but having the important property that all of the new parameters and variables have no physical dimensions. One of the purposes of this chapter is to provide some indication as to how to do this and illustrate the basic method by means of examples.

The following sections are included:

• Introduction
• Euler’s Formula and DeMoivre’s Theorem
• Derivation of Trigonometric Relations
• Periodic, Even and Odd Functions
  • Periodic Functions
  • Even and Odd Functions
• Fourier Series
• Worked Examples for Fourier Series
  • Parseval’s Identity
  • f(x) = |x|, −π ≤ x ≤ π
  • f(x) = x, −π < x < π
  • The Square Wave Function
  • Comparison of Problems 2.6.2 and 2.6.4
  • f(x) = sinx, 0 < x < π
  • Fourier Series of x², −π < x < π by Integration
  • Riemann-Lebesgue Theorem
• Fourier Series for (cos θ)^α and (sinθ)^α
• Fourier Transforms
  • Definition of Fourier Transforms
  • Basic Properties of Fourier Transforms
• Application of Fourier Transforms
  • Fourier Transform of the Square Pulse
  • Fourier Transform of the Gaussian Function
  • The Convolution Theorem
  • The Diffusion Equation
  • The Wave Equation
• The Laplace Transform
• Worked Problems Using the Laplace Transform
  • Laplace Transform of t^−1/3
  • The Square Wave Function
  • The Dirac Delta Function
• Comments and References
• Bibliography

The following sections are included:

• Scope of Chapter
• Gamma Function
• The Beta Function
• The Riemann Zeta Function
• The Dirac Delta Function
• Dirichlet Integrals
• Applications
  • Additional Properties of Γ(z)
  • A Definite Integral Containing Logarithms
  • A Class of Important Integrals
  • A Representation for x^−p
  • Additional Properties of the Beta Function
  • Fermi-Dirac Integrals
  • An Integral Involving an Exponential
  • Fermi-Dirac Integrals Containing Logarithms
  • Magnetic Moment of the Electron
  • Relationship Between the Theta and Delta Functions
  • Evaluation of Integrals by Use of the Beta Function
• Other Named Functions
  • Elliptic Integrals of the First and Second Kind
  • Jacobi Elliptic Functions
  • Evaluation of Integrals
  • Comments and References
  • Bibliography

The following sections are included:

• Introduction
• One-Dimensional Systems
  • Definition
  • Fixed-Points
  • Sign of the Derivative
  • Linear Stability
• Worked Examples
  • Examples A
  • Example B
  • Example C
  • Example D
  • Example E
• Two-Dimensional Systems
  • Definition
  • Fixed-Points
  • Nullclines
  • First-Integrals and Symmetries
  • General Features of Two-Dimensional Phase Space
  • The Basic Procedure for Constructing Phase-Space Diagrams
  • Linear Stability
• Worked Examples
  • Example A
  • Example B
  • Example C
  • Example D
  • Example E
  • Example F
  • Example G
  • Example H
• Bifurcations
  • Hopf-Bifurcations
  • Two Examples
• Comments and References
• Bibliography

The following sections are included:

• Genesis of Difference Equations
  • Square-Root Iteration
  • Integral Depending on Two Parameters
• Existence and Uniqueness Theorem
• The Fundamental Operators
  • The Δ Operator
  • The Shift Operator, E
  • Difference of Functions
  • The Operator Δ⁻¹
• Worked Problems Based on Section 5.3
  • Examples A
  • Example B
  • Example C
  • Examples D
• First-Order Linear Difference Equations
  • Example A
  • Example B
  • Example C
• General Linear Difference Equations
  • General Properties
  • Linearly Independent Functions
  • Fundamental Theorems for Homogeneous Equations
  • Inhomogeneous Equations
• Worked Examples
  • Example A
  • Example B
  • Example C
  • Example D
• Linear Difference Equations with Constant Coefficients
  • Homogeneous Equations
  • Inhomogeneous Equations
  • Worked Examples
• Nonlinear Difference Equations
  • Homogeneous Equations
  • Riccati Equations
  • Clairaut’s Equation
  • Miscellaneous Forms
• Worked Examples of Nonlinear Equations
  • Example A
  • Example B
  • Example C
  • Example D
  • Example E
• Two Applications
  • Chebyshev Polynomials
  • A Discrete Logistic Equation
• Comments and References
• Bibliography

The following sections are included:

• Introduction
• The Vibrating String
  • Fixed Ends
  • One Fixed and One Free Ends
  • Both Ends Free
  • Discussion
• Sturm Separation and Comparison Theorem
  • Examples A
  • Example B
  • Example C
• Sturm-Liouville Problems
  • Fundamental Definition
  • Properties of Eigenvalues and Eigenfunctions
  • Orthogonality of Eigenfunctions
  • Expansion of Functions
  • The Completeness Relation
• Applications
  • The Special Functions
  • Fourier Expansion of f(x) = x(1 − x)
  • Representation of δ(x) in Terms of Cosine Functions
  • Reality of the Eigenvalues
  • A Boundary Value Problem [3]
• Green’s Functions
• Worked Examples for Green Functions
  • y″(x) = –f(x); y(0) = y(L)=0
  • y″(x) = –f(x) : y(0) = 0, y′(L)=0
  • y″(x) + k²y(x) = −f(x) : y(0) = y(L)=0
  • y″(x) + y(x) = x : y(0) = y(1)=0
• Asymptotic Behavior of Solutions to Differential Equations
  • Elimination of First-Derivative Terms
  • The Liouville-Green Transformation
• Worked Examples
  • The Airy Equation
  • The Bessel Equation
  • A General Expansion Procedure
• Comments and References
• Bibliography

The following sections are included:

• Introduction
• Classical Orthogonal Polynomials
  • Differential Equation and Interval of Definition
  • Weight Functions and Rodrique’s Formulas
  • Orthogonality Relations
  • Generating Function
  • Recurrence Relations
  • Differential Recurrences
  • Special Values
  • Zeros of COP
• Legendre Polynomials: P_n(x)
• Hermite Polynomials: H_n(x)
• Chebyschev Polynomials: T_n(x) and U_n(x)
• Laguerre Polynomials: L_n(x)
• Legendre Functions of the Second Kind and Associated Legendre Functions
• Bessel Functions
• Some Proofs and Worked Problems
  • Proof of Rodriques Formula for P_n(x)
  • Integrals of x^m with P_n(x)
  • Expansion of δ(x) in Terms of P_n(x)
  • Laplace’s Equation in Spherical Coordinates
  • Sphere in a Uniform Flow
  • The Harmonic Oscillator
  • Matrix Elements of the Electric Dipole
  • General Solution Hermite’s Differential Equation for n=0
  • Three Dimensional Harmonic Oscillator
  • Proof that J_−n(x) = (−1)ⁿJ_n(x)
  • Calculation of J_n(x) from J₀(x)
  • An Integral Representation for J_n(x)
  • Expansion of Cosine in Terms of J_n(x)
  • Derivation of a Recurrence Relation from the Generating Function
  • An Alternative Representation for Y_n(x)
  • Equations Reducible to Bessel’s Equation
• Comments and References
• Bibliography

The following sections are included:

• Introduction
• The General Perturbation Procedure
• Worked Examples Using the General Perturbation Procedure
  • Example A
  • Example B
• First-Order Method of Averaging
  • The Method
  • Two Special Cases for f(x, dx/dt)
  • Stability of Limit-Cycles
• Worked Examples for First-Order Averaging
  • Example A
  • Example B
  • Example C
• The Lindstedt-PoincaréMethod
  • Secular Terms
  • The Formal Procedure
• Worked Examples Using the Lindstedt-Poincaré Method
  • Example A
  • Example B
• Harmonic Balance
  • Direct Harmonic Balance
• Worked Examples for Harmonic Balance
  • Example A
  • Example B
  • Example C
• Averaging for Difference Equations
• Worked Examples for Difference Equations
  • Example A
  • Example B
• Comments and References
• Bibliography

Functions defined by integrals often appear in the investigation of dynamical systems. For example, in chemistry, reaction rate coefficients take the form [1] where, β > 0, is inversely related to the temperature and the functions A(β) and σ(x) are specified for a given type of reaction. This particular expression has been used to study the behaviors of K(β) in the limits β→0 and β→∞[2].

Many of the equations modeling complex dynamical systems in the natural and engineering sciences are partial differential equations. For these equations the independent variables are usually the time and one or more space variables. It is important to note that a vast number of scientific and technologically interesting phenomena can be modeled in terms of a relatively small set of linear and nonlinear partial differential equations. Of further significance is the fact that several techniques exist to calculate special solutions to these equations and these particular solutions are often the ones needed to analyze the phenomena of interest.

The following sections are included:

• Scope of this Chapter
• Particle-in-a-Box
• Square Periodic Functions
  • Hamiltonian Formulation
  • SPF as Generalized Trigonometric Functions
• Analytical Periodic Solutions to f(t)² + g(t)² =1
  • Generalized Cosine and Sine Functions
  • Calculation of Phase Function – θ(t)
  • Differential Equations for f(t) and g(t)
  • An Explicit Example
  • Jacobi Elliptic Functions
  • Comments
• Leah-Cosine and -Sine Functions
  • Three Leah Periodic Functions
  • Determination of r(θ)
  • Comments
• Comments and References
• Bibliography

A functional equation is an equation for which the unknowns are one or more functions. In general, such an equation may relate the value of these functions at one point to values at other points.

This chapter presents a broad range of topics related to the natural and engineering sciences, especially as they relate to the modeling and analysis of the resulting mathematical equations. In each section, several references are given to the essential literature on the particular issue being studied. Also, it should be noted that the references are generally given in the introduction section to each topic rather than at the end of the chapter.

The following sections are included:

• Appendix A: Mathematical Relations
• Bibliography
• Appendix B: Asymptotics Expansions

The following sections are included:

• Bibliography
• Index