Engineering Analysis

Advanced Mathematical Methods for Engineers

1st Edition - May 20, 2024
Author: Zhihe Jin
Language: English
Paperback ISBN:
9 7 8 - 0 - 3 2 3 - 9 5 3 9 7 - 9
eBook ISBN:
9 7 8 - 0 - 3 2 3 - 9 5 3 9 6 - 2

Engineering Analysis: Advanced Mathematical Methods for Engineers introduces graduate engineering students to the fundamental but advanced mathematics tools used in engineeri… Read more

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Engineering Analysis: Advanced Mathematical Methods for Engineers introduces graduate engineering students to the fundamental but advanced mathematics tools used in engineering application, especially in mechanical, aerospace, and civil engineering. Most engineering problems are described by differential equations, particularly partial differential equations (PDEs). Deformation and failure in solid structures, fluid flow, heat transfer, and mass diffusion are all governed by PDEs in general. Many physical quantities in engineering are tensors, including deformation gradient, strain rates, stresses, elastic stiffness, and thermal conductivity of composite materials. This book helps engineering graduate students develop the skills to establish the mathematical models of engineering problems and to solve the problems described by the mathematical models.

Cover image
Title page
Table of Contents
Copyright
Dedication
About the Author
Acknowledgments
Preface
Part I: Ordinary Differential Equations – Advanced Topics
Chapter 1. Ordinary Differential Equations and Power Series Solutions
Abstract
1.1 Review of First-Order Linear Ordinary Differential Equations
1.2 Review of Second-Order Linear Ordinary Differential Equations
1.3 Power Series Solutions
1.4 Legendre’s Equation and Legendre Polynomials
1.5 Summary
Problems
Chapter 2. The Frobenius Method
Abstract
2.1 Ordinary Differential Equations with a Regular Singular Point
2.2 Frobenius Series Solutions
2.3 Bessel’s Equation and Bessel Functions
2.4 Summary
Problems
Chapter 3. The Laplace Transform Method
Abstract
3.1 The Laplace Transform
3.2 Solutions of Initial Value Problems for Ordinary Differential Equations
3.3 The Shifting Theorems
3.4 Convolution and Dirac Delta Function
3.5 Forced Vibration of a Mass–Spring–Damper System
3.6 Forced Vibration of a Two-Degree-of-Freedom Mass–Spring System
3.7 Laplace Transform by MATLAB®
3.8 Summary
Problems
Chapter 4. Numerical Solutions of Ordinary Differential Equations
Abstract
4.1 Introduction
4.2 Euler’s Method
4.3 The Runge–Kutta Method
4.4 Convergence and Stability of Numerical Methods
4.5 System of First-Order ODEs and Higher-Order ODEs
4.6 Boundary Value Problems: The Finite Difference Method
4.7 Boundary Value Problems: The Shooting Method
4.8 Summary
Problems
Part II: Fourier Analysis
Chapter 5. Fourier Series
Abstract
5.1 Fourier Series of a Function
5.2 Fourier Series of a Periodic Function
5.3 The Gibbs Phenomenon
5.4 Fourier Cosine and Sine Series
5.5 Integration and Differentiation of Fourier Series
5.6 Vibration of a Mass–Spring–Damper System Excited by a Periodic Force
5.7 Summary
Problems
Chapter 6. Fourier Transforms
Abstract
6.1 The Fourier Integral
6.2 The Fourier Transform
6.3 Fourier Sine and Cosine Transforms
6.4 Fourier Transform by MATLAB®
6.5 Summary
Problems
Part III: Partial Differential Equations
Chapter 7. Partial Differential Equations in Engineering
Abstract
7.1 Partial Differential Equations
7.2 The Wave Equation
7.3 The Heat Equation
7.4 Laplace’s Equation
7.5 Partial Differential Equations in Cylindrical and Spherical Coordinates
7.6 Classification of Second-Order Partial Differential Equations
7.7 The Superposition Principle for Linear Problems
7.8 Wave Propagation in an Infinite String—d’Alembert’s Solution
7.9 Summary
Problems
Chapter 8. Separation of Variables Method
Abstract
8.1 Free Vibration of a Finite String with Fixed Ends
8.2 Forced Vibration of a Finite String with Fixed Ends
8.3 Heat Conduction in a Finite Bar with a Convection End
8.4 Nonhomogeneous Heat Conduction Problems
8.5 Free Vibration of a Rectangular Membrane
8.6 Heat Conduction in a Rectangular Plate
8.7 Laplace’s Equation—Dirichlet Problem for a Rectangular Plate
8.8 Laplace’s Equation—The Neumann Problem for a Rectangular Plate
8.9 Sturm–Liouville Theory and Eigenfunction Expansions
8.10 Summary
Problems
Chapter 9. Separation of Variables Method—Circular and Spherical Regions
Abstract
9.1 Laplace’s Equation—Dirichlet Problem for a Circular Plate
9.2 Laplace’s Equation—Neumann Problem for an Infinite Plate with a Circular Hole
9.3 Steady State Heat Conduction in a Sphere
9.4 Heat Conduction in an Infinite Circular Cylinder
9.5 Vibration of a Circular Membrane
9.6 Fourier–Legendre and Fourier–Bessel Series Expansions
9.7 Summary
Problems
Chapter 10. Integral Transform Methods
Abstract
10.1 Heat Conduction in a Semi-infinite Bar—Laplace Transform Solution
10.2 Heat Conduction in an Infinite Bar—Fourier Transform Solution
10.3 Wave Propagation in a Semi-infinite String
10.4 Steady State Heat Conduction in a Semi-infinite Plate
10.5 Steady State Heat Conduction in a Quarter-Plane
10.6 Summary
Problems
Chapter 11. The Finite Difference Method
Abstract
11.1 Laplace’s Equation
11.2 The Heat Equation
11.3 The Wave Equation
11.4 Summary
Problems
Part IV: Tensor Analysis
Chapter 12. Cartesian Tensors
Abstract
12.1 The Summation Convention
12.2 The Kronecker Delta and the Permutation Symbol
12.3 Index Notations for Vectors and Their Products
12.4 Transformation of Coordinates
12.5 Cartesian Tensors
12.6 Tensor Algebra
12.7 The Quotient Rule
12.8 Second-Order Tensors
12.9 Isotropic Tensors
12.10 Summary
Problems
Chapter 13. Tensor Analysis
Abstract
13.1 Tensor Functions
13.2 Derivatives of Tensor Functions
13.3 Tensor Fields
13.4 The Integral Theorems
13.5 The Heat and Wave Equations, Revisited
13.6 Summary
Problems
Bibliography
Index

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