Applied Engineering Mathematics

Cover
Half Title	#2,0,-32767Title Page	#4,0,-32767Copyright Page	#5,0,-32767Table of Contents	#6,0,-32767Preface	#12,0,-32767About the Author
Chapter 1 Overview
	1.1 Objectives
	1.2 Educational Philosophy
	1.3 Physical Processes
	1.4 Mathematical Models
		1.4.1 Algebraic Equations
		1.4.2 Ordinary Differential Equations
		1.4.3 Partial Differential Equations
	1.5 Solution Methods
	1.6 Software
Chapter 2 Physical Processes
	2.1 Physical Phenomena
	2.2 Fundamental Principles
	2.3 Conservation Laws
		2.3.1 Conservation of Mass: Continuity
		2.3.2 Conservation of Momentum: Newton’s Second Law
		2.3.3 Conservation of Energy: First Law of Thermodynamics
	2.4 Rate Equations
		2.4.1 Heat Conduction: Fourier’s Law
		2.4.2 Heat Convection: Newton’s Law of Cooling
		2.4.3 Thermal Radiation
		2.4.4 Viscous Fluid Shear: Newton’s Viscosity Law
		2.4.5 Binary Mass Diffusion: Fick’s Law
		2.4.6 Electrical Conduction: Ohm’s Law
		2.4.7 Stress-Strain: Hooke’s Law
	2.5 Diffusion Analogies
Chapter 3 Modeling of Physical Processes
	3.1 Cause and Effect
		3.1.1 General Physical Process
		3.1.2 Thermal Processes
		3.1.3 Mechanical Processes
	3.2 Mathematical Modeling
	3.3 Complete Mathematical Model
		3.3.1 Mechanical Vibrations
		3.3.2 Heat Conduction
	3.4 Dimensionless Formulation
		3.4.1 General Procedure
		3.4.2 Mechanical Vibrations
		3.4.3 Steady Heat Conduction
	3.5 Inverse and Parameter Estimation Problems
		3.5.1 Direct Problem
		3.5.2 Inverse Problem
		3.5.3 Parameter Estimation Problem
	3.6 Mathematical Classification of Physical Problems
	Problems
Chapter 4 Calculus
	4.1 Derivatives
		4.1.1 Basic Concept of a Derivative
		4.1.2 Velocity from Displacement
		4.1.3 Derivative of tn
		4.1.4 Chain Rule
		4.1.5 Product Rule
		4.1.6 Partial Derivatives
	4.2 Numerical Differentiation: Taylor Series
		4.2.1 Taylor Series Expansion
		4.2.2 First Derivatives Using Taylor Series
		4.2.3 Second Derivatives Using Taylor Series
	4.3 Integrals
		4.3.1 Basic Concept of an Integral
		4.3.2 Geometric Interpretation of an Integral: Area Under a Curve
		4.3.3 Mean Value Theorem
		4.3.4 Integration by Parts
		4.3.5 Leibniz Rule: Derivatives of Integrals
	4.4 Summary of Derivatives and Integrals
	4.5 The Step, PULSE, and Delta Functions
		4.5.1 The Step Function
		4.5.2 The Unit Pulse Function
		4.5.3 The Delta Function
	4.6 Numerical Integration
		4.6.1 Trapezoid Rule
		4.6.2 Trapezoid Rule for Unequal Segments
		4.6.3 Simpson’s Rule
		4.6.4 Simpson’s 3/8 Rule
		4.6.5 Gauss Quadrature
	4.7 Multiple Integrals
	Problems
Chapter 5 Linear Algebra
	5.1 Introduction
	5.2 Cause and Effect
	5.3 Applications
		5.3.1 Networks
		5.3.2 Finite Difference Equations
	5.4 Geometric Interpretations
		5.4.1 Row Interpretation
		5.4.2 Column Interpretation
	5.5 Possibility of Solutions
	5.6 Characteristics of Square Matrices
	5.7 Square, Overdetermined, and Underdetermined Systems
		5.7.1 Overdetermined Systems
		5.7.2 Underdetermined Systems
		5.7.3 Square Systems
	5.8 Row Operations
	5.9 The Determinant and Cramer’s Rule
	5.10 Gaussian Elimination
		5.10.1 Naïve Gaussian Elimination
		5.10.2 Pivoting
		5.10.3 Tridiagonal Systems
	5.11 LU Factorization
	5.12 Gauss–Seidel Iteration
	5.13 Matrix Inversion
	5.14 Least Squares Regression
	Problems
Chapter 6 Nonlinear Algebra: Root Finding
	6.1 Introduction
	6.2 Applications
		6.2.1 Simple Interest
		6.2.2 Thermodynamic Equations of State
		6.2.3 Heat Transfer: Thermal Radiation
		6.2.4 Design of an Electric Circuit
	6.3 Root Finding Methods
	6.4 Graphical Method
	6.5 Bisection Method
	6.6 False Position Method
	6.7 Newton–Raphson Method
	6.8 Secant Method
	6.9 Roots of Simultaneous Nonlinear Equations
	Problems
Chapter 7 Introduction to Ordinary Differential Equations
	7.1 Classification of Ordinary Differential Equations
		7.1.1 Autonomous versus Nonautonomous Systems
		7.1.2 Initial Value and Boundary Value Problems
	7.2 First-Order Ordinary Differential Equations
		7.2.1 First-Order Phase Portraits
		7.2.2 Nonautonomous Systems
		7.2.3 First-Order Linear Equations
		7.2.4 Lumped Thermal Models
		7.2.5 RC Electrical Circuit
		7.2.6 First-Order Nonlinear Equations
		7.2.7 Population Dynamics
	7.3 Second-Order Initial Value Problems
		7.3.1 Second-Order Phase Portraits
		7.3.2 Second-Order Linear Equations
		7.3.3 Mechanical Vibrations
		7.3.4 Mechanical and Electrical Circuits
		7.3.5 Second-Order Nonlinear Equations
		7.3.6 The Pendulum
		7.3.7 Predator–Prey Models
	7.4 Second-Order Boundary Value Problems
	7.5 Higher-Order Systems
	Problems
Chapter 8 Laplace Transforms
	8.1 Definition of the Laplace Transform
	8.2 Laplace Transform Pairs
	8.3 Properties of the Laplace Transform
	8.4 The Inverse Laplace Transformation
		8.4.1 Partial-Fraction Expansion Method
		8.4.2 Partial-Fraction Expansion for Distinct Poles
		8.4.3 Partial-Fraction Expansion for Multiple Poles
	8.5 Solutions of Linear Ordinary Differential Equation
		8.5.1 General Strategy
		8.5.2 First-Order Ordinary Differential Equations
		8.5.3 Second-Order Ordinary Differential Equations
	8.6 The Transfer Function
		8.6.1 The Impulse Response
		8.6.2 First-Order Ordinary Differential Equations
	Problems
Chapter 9 Numerical Solutions of Ordinary Differential Equations
	9.1 Introduction to Numerical Solutions
	9.2 Runge–Kutta Methods
		9.2.1 Euler’s Method
		9.2.2 Heun’s Method
		9.2.3 Higher-Order Runge–Kutta Methods
		9.2.4 Numerical Comparison of Runge–Kutta Schemes
	9.3 Coupled Systems of First-Order Differential Equations
	9.4 Second-Order Initial Value Problems
	9.5 Implicit Schemes
	9.6 Second-Order Boundary Value Problems: The Shooting Method
	Problems
Chapter 10 First-Order Ordinary Differential Equations
	10.1 Stability of the Fixed Points
		10.1.1 RC Electrical Circuit
		10.1.2 Population Model
	10.2 Characteristics of Linear Systems
	10.3 Solution Using Integrating Factors
	10.4 First-Order Nonlinear Systems and Bifurcations
		10.4.1 Saddle-Node Bifurcation
		10.4.2 Transcritical Bifurcation
		10.4.3 Example of a Transcritical Bifurcation: Laser Threshold
		10.4.4 Supercritical Pitchfork Bifurcation
		10.4.5 Subcritical Pitchfork Bifurcation
	Problems
Chapter 11 Second-Order Ordinary Differential Equations
	11.1 Linear Systems
	11.2 Classification of Linear Systems
	11.3 Classical Spring-Mass-Damper
	11.4 Stability Analysis of the Fixed Points
	11.5 Pendulum
		11.5.1 Fixed Points: No Forcing, No Damping
		11.5.2 Fixed Points: General Case
	11.6 Competition Models
		11.6.1 Coexistence
		11.6.2 Extinction
	11.7 Limit Cycles
		11.7.1 van der Pol Oscillator
		11.7.2 Poincare–Bendixson Theorem
	11.8 Bifurcations
		11.8.1 Saddle-Node Bifurcation
		11.8.2 Transcritical Bifurcation
		11.8.3 Supercritical Pitchfork Bifurcation
		11.8.4 Subcritical Pitchfork Bifurcation
		11.8.5 Hopf Bifurcations
		11.8.6 Supercritical Hopf Bifurcation
		11.8.7 Subcritical Hopf Bifurcation
	11.9 Coupled Oscillators
	Problems: LINEAR SYSTEMS
	Problems: NONLINEAR SYSTEMS
Index