Applied Mathematics for Scientists and Engineers

✍ Scribed by Youssef N. Raffoul

• Publisher: CRC Press
• Year: 2024
• Tongue: English
• Leaves: 442
• Series: Textbooks in Mathematics
• Category: Library

✦ Table of Contents

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