Advanced Engineering Mathematics and Analysis-Volume 1 offers a straightforward approach to understanding the theory of several engineering tools that are used to compute, evaluate, and analyze practical problems. It is a mathematics textbook that can be used by students, instructors, and technical carriers. Throughout the five chapters of the book, besides the pure mathematical examples, several practical issues from different fields are modeled and solved to illustrate the relation between the theory and its applications. The book elucidates the subjects in a self-contained style. This volume contains the basics and advanced topics of linear algebra and matrix theory, two-chapter ordinary differential equations to elaborate many classes, Laplace transforms with fundamental applications, and a complete engineering course of numerical methods. Each chapter ends with exercises that are arranged according to the chapter sections. The readers will find the answers at the end of the book.
Author(s): Rami A. Maher
Series: Mathematics Research Developments
Publisher: Nova Science Publishers
Year: 2021
Language: English
Pages: 394
City: New York
Preface
Acknowledgments
Chapter 1
Linear Algebra and Matrices
1.1. Introduction
1.2. Motivation
1.3. Basics and Definitions of Matrices
1.4. Computation by Elementary Operations
1.5. The Solution of Linear Algebraic Systems
1.6. Basics of Linear Algebra
1.7. Linear Transformations
1.8. Eigenvalue and Eigenvector Problems
1.9. Cayley-Hamilton Theorem and Minimum Polynomial of a Matrix
1.10. Symmetric and Orthogonal Matrices
1.11. Singular Values, Singular Vectors, and Norms of Matrices
1.12. Matrix Factorizations
Exercises
Sections 1.1-1.5
Sections 1.6-1.7
Sections 1.8-1.9
Sections 1.10-1.12
Chapter 2
Differential Equations – Part I
2.1. Introduction
2.2. Classification of Differential Equations
2.3. Ordinary Differential Equations in Engineering Problems
2.4. Solution Concepts
2.5. First-Order Ordinary Differential Equations
2.6. Second-Order Ordinary Differential Equations
2.7. High-Order Ordinary Differential Equations
2.8. The Solution of a First-Order System of Differential Equations
Exercises
Sections 2.1-2.4
Section 2.5
Section 2.6
Sections 2.7-2.8
Chapter 3
Differential Equations – Part II
3.1. Introduction
3.2. Second-Order Variable-Coefficients Ordinary Differential Equations
3.3. Special Second-Order Ordinary Differential Equations
3.4. Nonlinear Second-Order Differential Equations
3.5. Method of a Parameter
3.6. Solution Method Based on Jacobi Elliptic Functions
Exercises
Section 3.2
Section 3.3
Section 3.4-3.6
Chapter 4
Laplace Transforms
4.1. Introduction
4.2. Definitions and Theorems
4.3. Laplace Inverse
4.4. The Solution of Differential Equations Using Laplace Transformation
4.5. Laplace Transform of Especial Functions
4.6. Convolution Theorem
4.7. Advanced Applications
Exercises
Section 4.2
Section 4.3-4.4
Section 4.5
Section 4.6-4.7
Chapter 5
Numerical Methods
5.1. Introduction
5.2. Definitions and Concepts
5.3. The Solution of Nonlinear Algebraic Equations
5.3.1. The Solution of Transcendental Equations
5.3.2. The Solution of Polynomial Equations
5.3.3. The Solution of Nonlinear Systems
5.3.4. Solutions by MATLAB
5.4. Polynomial Approximation and Interpolation
5.4.1. Approximation Based on Least-Squares Error Method
5.4.2. Linear and Quadratic Regressions
5.4.3. Interpolation and Finite Differences
5.4.4. Lagrange Interpolation Formula
5.5. Numerical Differentiation
5.6. Numerical Integration
5.7. Numerical Methods for Solving Ordinary Differential Equations
5.7.1. Euler’s Method
5.7.2. Heun’s Method – Predictor Corrector Approach
5.7.3. Runge – Kutta Methods
5.7.4. Numerical Solution of Higher-Order Differential Equations
5.8. Numerical Solution of Boundary Value Problem
5.8.1. Shooting Method
5.8.2. Finite Difference Method
Exercises
Section 5.3
Section 5.4
Sections 5.5-5.6
Sections 5.7-5.8
Answers to Selected Exercises
Sections 1.1–1.5
Sections 1.6–1.7
Sections 1.8–1.9
Section 1.10–1.12
Section 2.1–2.4
Section 2.5
Section 2.6
Sections 2.7-2.8
Section 3.2
Section 3.3
Section 4.2
Section 4.3–4.4
Section 5.3
Section 5.4
Section 5.5–5.6
Section 5.7–5.8
References
About the Author
Index
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