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Advanced Engineering Mathematics With MATLAB by Duffy

Contents of Advanced Engineering Mathematics With MATLAB

• Chapter 1: Complex Variables
• Complex Numbers
• Finding Roots
• The Derivative in the Complex Plane:
• The Cauchy-Riemann Equations
• Line Integrals
• The Cauchy-Goursat Theorem
• Cauchy’s Integral Formula
• Taylor and Laurent Expansions and Singularities
• Theory of Residues
• Evaluation of Real Definite Integrals
• Cauchy’s Principal Value Integral
• Chapter 2: First-Order Ordinary
• Differential Equations
• Classification of Differential Equations
• Separation of Variables
• Homogeneous Equations
• Exact Equations
• Linear Equations
• Graphical Solutions
• Numerical Methods
• Chapter 3: Higher-Order Ordinary
• Differential Equations
• Homogeneous Linear Equations
• with Constant Coefficients
• Simple Harmonic Motion
• Damped Harmonic Motion
• Method of Undetermined Coefficients
• Forced Harmonic Motion
• Variation of Parameters
• Euler-Cauchy Equation
• Phase Diagrams
• Numerical Methods amplitude spectrum (ft) times
• Bay bridge and tunnel
• Chapter 4: Fourier Series
• Fourier Series
• Properties of Fourier Series
• Half-Range Expansions
• Fourier Series with Phase Angles
• Complex Fourier Series
• The Use of Fourier Series in the Solution of Ordinary
• Differential Equations
• Finite Fourier Series
• Chapter 5: The Fourier Transform
• Fourier Transforms
• Fourier Transforms Containing the Delta Function
• Properties of Fourier Transforms
• Inversion of Fourier Transforms
• Convolution
• Solution of Ordinary Differential Equations by Fourier Transforms
• Chapter 6: The Laplace Transform
• Definition and Elementary Properties
• The Heaviside Step and Dirac Delta Functions
• Some Useful Theorems
• The Laplace Transform of a Periodic Function
• Inversion by Partial Fractions:
• Heaviside’s Expansion Theorem
• Convolution
• Integral Equations
• Solution of Linear Differential Equations
• with Constant Coefficients
• Inversion by Contour Integration
• Ratio of quadrature amplitudes to ideal integration
• Trapezoidal
• Simpson’s
• Chapter 7: The Z-Transform
• The Relationship of the Z-Transform to the Laplace
• Transform
• Some Useful Properties
• Inverse Z-Transforms
• Solution of Difference Equations
• Stability of Discrete-Time Systems
• Chapter 8: The Hilbert Transform
• Definition
• Some Useful Properties
• Analytic Signals
• Causality: The Kramers-Kronig Relationship
• Chapter 9: The Sturm-Liouville
• Problem
• Eigenvalues and Eigenfunctions
• Orthogonality of Eigenfunctions
• Expansion in Series of Eigenfunctions
• A Singular Sturm-Liouville Problem:
• Legendre’s Equation
• Another Singular Sturm-Liouville Problem:
• Bessel’s Equation
• Finite Element Method
• Chapter 10: The Wave Equation
• The Vibrating String
• Initial Conditions: Cauchy Problem
• Separation of Variables
• D’Alembert’s Formula
• The Laplace Transform Method
• Numerical Solution of the Wave Equation
• Chapter 11: The Heat Equation
• Derivation of the Heat Equation
• Initial and Boundary Conditions
• Separation of Variables
• The Laplace Transform Method
• The Fourier Transform Method
• The Superposition Integral
• Numerical Solution of the Heat Equation
• Chapter 12: Laplace’s Equation
• Derivation of Laplace’s Equation
• Boundary Conditions
• Separation of Variables
• The Solution of Laplace’s Equation
• on the Upper Half-Plane
• Poisson’s Equation on a Rectangle
• The Laplace Transform Method
• Numerical Solution of Laplace’s Equation
• Finite Element Solution of Laplace’s Equation
• Chapter 13: Green’s Functions
• What Is a Green’s Function?
• Ordinary Differential Equations
• Joint Transform Method
• Wave Equation
• Heat Equation
• Helmholtz’s Equation
• Chapter 14: Vector Calculus
• Review
• Divergence and Curl
• Line Integrals
• The Potential Function
• Surface Integrals
• Green’s Lemma
• Stokes’ Theorem
• Divergence Theorem
• Chapter 15: Linear Algebra
• Fundamentals of Linear Algebra
• Determinants
• Cramer’s Rule
• Row Echelon Form and Gaussian Elimination
• Eigenvalues and Eigenvectors
• Systems of Linear Differential Equations
• Matrix Exponential
• Chapter 16: Probability
• Review of Set Theory
• Classic Probability
• Discrete Random Variables
• Continuous Random Variables
• Mean and Variance
• Some Commonly Used Distributions
• Joint Distributions
• Chapter 17: Random Processes
• Fundamental Concepts
• Power Spectrum
• Differential Equations Forced by Random Forcing
• Two-State Markov Chains
• Birth and Death Processes
• Poisson Processes
• Random Walk

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