After many years of teaching graduate courses in applied mathematics, Youssef N. Raffoul saw a need among his students for a book reviewing topics from undergraduate courses to help them recall what they had learned, while his students urged him to publish a brief and approachable book on the topic. Thus, the author used his lecture notes from his graduate course in applied mathematical methods, which comprises three chapters on linear algebra, calculus of variations, and integral equations, to serve as the foundation for this work. These notes have undergone continuous revision.
Applied Mathematics for Scientists and Engineers is designed to be used as a graduate textbook for one semester. The five chapters in the book can be used by the instructor to create a one-semester, three-chapter course. The only prerequisites for this self-contained book are a basic understanding of calculus and differential equations. In order to make the book accessible to a broad audience, the author endeavored to strike a balance between rigor and presentation of the most challenging content in a simple format by adopting friendlier, more approachable notations and using numerous examples to clarify complex themes. The hope is both instructors and students will find, in this single volume, a refresher on topics necessary to further their courses and study.
Author(s): Youssef Raffoul
Series: Textbooks in Mathematics
Edition: 1
Publisher: Chapman and Hall/CRC
Year: 2023
Language: English
Pages: 425
City: Boca Raton, FL
Tags: Engineering; Technology; Mathematics; Statistics
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Author
1. Ordinary Differential Equations
1.1. Preliminaries
1.2. Separable Equations
1.2.1. Exercises
1.3. Exact Differential Equations
1.3.1. Integrating factor
1.3.2. Exercises
1.4. Linear Differential Equations
1.4.1. Exercises
1.5. Homogeneous Differential Equations
1.5.1. Exercises
1.6. Bernoulli Equation
1.6.1. Exercises
1.7. Higher-Order Differential Equations
1.7.1. Exercises
1.8. Equations with Constant Coefficients
1.8.1. Exercises
1.9. Nonhomogeneous Equations
1.9.1. Exercises
1.10. Wronskian Method
1.10.1. Exercises
1.11. Cauchy-Euler Equation
1.11.1. Exercises
2. Partial Differential Equations
2.1. Introduction
2.1.1. Exercises
2.2. Linear Equations
2.2.1. Linear equations with constant coefficients
2.2.2. Exercises
2.2.3. Equations with variable coefficients
2.2.4. Exercises
2.3. Quasi-Linear Equations
2.3.1. Exercises
2.4. Burger’s Equation
2.4.1. Shock path
2.4.2. Exercises
2.5. Second-Order PDEs
2.5.1. Exercises
2.6. Wave Equation and D’Alembert’s Solution
2.6.1. Exercises
2.6.2. Vibrating string with fixed ends
2.6.3. Exercises
2.7. Heat Equation
2.7.1. Solution of the heat equation
2.7.2. Heat equation on semi-infinite domain: Dirichlet condition
2.7.3. Heat equation on semi-infinite domain: Neumann condition
2.7.4. Exercises
2.8. Wave Equation on Semi-Infinite Domain
2.8.1. Exercises
3. Matrices and Systems of Linear Equations
3.1. Systems of Equations and Gaussian Elimination
3.2. Homogeneous Systems
3.2.1. Exercises
3.3. Matrices
3.3.1. Exercises
3.4. Determinants and Inverse of Matrices
3.4.1. Application to least square fitting
3.4.2. Exercises
3.5. Vector Spaces
3.5.1. Exercises
3.6. Eigenvalues-Eigenvectors
3.6.1. Exercises
3.7. Inner Product Spaces
3.7.1. Exercises
3.8. Diagonalization
3.8.1. Exercises
3.9. Quadratic Forms
3.9.1. Exercises
3.10. Functions of Symmetric Matrices
3.10.1. Exercises
4. Calculus of Variations
4.1. Introduction
4.2. Euler-Lagrange Equation
4.2.1. Exercises
4.3. Impact of y′ on Euler-Lagrange Equation
4.3.1. Exercises
4.4. Necessary and Sufficient Conditions
4.4.1. Exercises
4.5. Applications
4.5.1. Exercises
4.6. Generalization of Euler-Lagrange Equation
4.6.1. Exercises
4.7. Natural Boundary Conditions
4.8. Impact of y′′ on Euler-Lagrange Equation
4.8.1. Exercises
4.9. Discontinuity in Euler-Lagrange Equation
4.9.1. Exercises
4.10. Transversality Condition
4.10.1. Problem of Bolza
4.10.2. Exercises
4.11. Corners and Broken Extremal
4.11.1. Exercises
4.12. Variational Problems with Constraints
4.12.1. Exercises
4.13. Isoperimetric Problems
4.13.1. Exercises
4.14. Sturm-Liouville Problem
4.14.1. The First Eigenvalue
4.14.2. Exercises
4.15. Rayleigh Ritz Method
4.15.1. Exercises
4.16. Multiple Integrals
4.16.1. Exercises
5. Integral Equations
5.1. Introduction and Classifications
5.1.1. Exercises
5.2. Connection between Ordinary Differential Equations and Integral Equations
5.2.1. Exercises
5.3. The Green’s Function
5.3.1. Exercises
5.4. Fredholm Integral Equations and Green’s Function
5.4.1. Exercises
5.4.2. Beam problem
5.4.3. Exercises
5.5. Fredholm Integral Equations with Separable Kernels
5.5.1. Exercises
5.6. Symmetric Kernel
5.6.1. Exercises
5.7. Iterative Methods and Neumann Series
5.7.1. Exercises
5.8. Approximating Non-Degenerate Kernels
5.8.1. Exercises
5.9. Laplace Transform and Integral Equations
5.9.1. Frequently used Laplace transforms
5.9.2. Exercises
5.10. Odd Behavior
5.10.1. Exercises
Appendices
A. Fourier Series
A.1. Preliminaries
A.2. Finding the Fourier Coefficients
A.3. Even and Odd Extensions
A.4. Applications of Fourier Series
A.5. Laplacian in Polar, Cylindrical and Spherical Coordinates
Bibliography
Index
