Advanced Mathematics for Engineering Students: The Essential Toolbox

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Advanced Mathematics for Engineering Students: The Essential Toolbox provides a concise treatment for applied mathematics. Derived from two semester advanced mathematics courses at the author’s university, the book delivers the mathematical foundation needed in an engineering program of study. Other treatments typically provide a thorough but somewhat complicated presentation where students do not appreciate the application. This book focuses on the development of tools to solve most types of mathematical problems that arise in engineering – a “toolbox” for the engineer. It provides an important foundation but goes one step further and demonstrates the practical use of new technology for applied analysis with commercial software packages (e.g., algebraic, numerical and statistical).

Author(s): Brent J. Lewis, E. Nihan Onder, Andrew Prudil
Edition: 1
Publisher: Butterworth-Heinemann
Year: 2021

Language: English
Pages: 432
Tags: applied, mathematics, engineering

Preface
Acknowledgments
Contents
List of tables
List of figures
1 Prologue
1.1 Introduction
1.1.1 History of differential equations
2 Ordinary differential equations
2.1 First-order equations
2.1.1 Summary of solution methods
2.2 Second-order linear differential equations
2.2.1 Homogeneous linear equations
Linear independence
2.2.2 Homogeneous equations with constant coefficients
2.3 Higher-order linear differential equations
2.4 Systems of differential equations
2.4.1 Basic concepts and theory
2.4.2 Homogeneous linear systems with constant coefficients
2.5 Series solutions and special functions
2.5.1 Legendre equation
Legendre polynomials
2.5.2 Frobenius method
2.5.3 Bessel's equation
Bessel function of the first kind
Bessel function of the second kind (or Neumann's function)
Modified Bessel equation
Generalized form of Bessel's equation
Problems
3 Laplace and Fourier transforms
3.1 Laplace transform methods
3.1.1 Definition of a Laplace transform
3.2 Fourier transform methods
3.2.1 Fourier integrals
Equivalent forms of Fourier integrals
3.2.2 Fourier transforms
3.3 Discrete Fourier transforms and fast Fourier transforms
3.3.1 Discrete Fourier transforms
Problems
4 Matrices, linear systems, and vector analysis
4.1 Determinants
4.2 Inverse of a matrix
4.3 Linear systems of equations
4.4 Vector analysis
4.4.1 Vectors and fields
Vectors
Fields
Problems
5 Partial differential equations
5.1 Derivation of important partial differential equations
5.2 Analytical methods of solution
5.2.1 General solutions
Superposition principle
General solution of linear nonhomogeneous partial differential equations
5.2.2 Separation of variables
Problems
6 Difference numerical methods
6.1 Ordinary differential equations
6.1.1 First-order equations
One-step methods
6.2 Partial differential equations
6.2.1 Poisson and Laplace difference equations
Problems
7 Finite element methods
7.1 COMSOL application methodology
7.1.1 Solved problems with COMSOL
Partial differential equations
7.2 General theory of finite elements
7.2.1 Finite elements and shape functions
One dimension
Linear elements
Problems
8 Treatment of experimental results
8.1 Definitions
8.2 Uses of statistics
8.2.1 Confidence interval
8.3 Regression analysis and software applications
8.4 Propagation of errors
8.4.1 Error propagation
Accumulation of determinate errors
Accumulation of indeterminate errors
Total uncertainty
Problems
9 Numerical analysis
9.1 Finding zeros of functions
9.1.1 Fixed point iteration
9.1.2 Newton's method
9.2 Interpolation
9.2.1 Lagrange interpolation
9.2.2 Newton's divided difference
9.3 Splines
9.4 Data smoothing
9.5 Numerical integration and differentiation
9.5.1 Trapezoidal rule
Problems
10 Introduction to complex analysis
10.1 Complex functions
10.2 Complex integration
10.3 Taylor and Laurent series
10.4 Residue integration method
10.5 Residue theorem
10.6 Applications of conformal mapping
Problems
11 Nondimensionalization
11.1 Dimensional analysis
11.2 Buckingham's π theorem – dimensional analysis
11.3 Similarity laws
11.4 Nondimensionalization technique
Problems
12 Nonlinear differential equations
12.1 Analytical solution
12.1.1 Inverse scattering transform
12.2 Numerical solution
12.2.1 Roots of a nonlinear function
12.2.2 Nonlinear initial and boundary value problems
Problems
13 Integral equations
13.1 Integral equations
13.1.1 Solution methods of integral equations
Successive approximation (Fredholm equation)
Successive substitution – resolvent method (Fredholm equation)
Successive approximation (Volterra equation of the second kind)
Laplace transform method
13.1.2 Integral/differential equation transformations
Reduction of an initial value problem to a Volterra equation
Reduction of the Volterra equation to an initial value problem
Reduction of a boundary value problem to a Fredholm equation
Picard's iteration method
13.2 Green's function
Problems
14 Calculus of variations
14.1 Euler–Lagrange equation
14.2 Lagrange multipliers
Problems
A Maple software package
A.1 Maple commands
B Geodesic formulation
B.1 Tensors
B.2 Lagrangian and action
Bibliography
Index