Advanced Mathematics for Engineering Students: The Essential Toolbox

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