Fourier Series and Boundary Value Problems with Engineering Applications

1. Fourier Series              

1.1 Fourier Series

1.2 Sine and Cosine Series

1.3 Fourier Series on Arbitrary Periods

1.4 Complex Form of Fourier Series

2. Theory of Fourier Series         

2.1 Bessel’s Inequality and its Consequences

2.2 Pointwise Convergence of Fourier Series

2.3 Gibb’s Phenomenon

3. Uniform Convergence and Operations on Fourier Series         

3.1 Preliminaries

3.2 Uniform and Absolute Convergence of Fourier Series

3.3 More on the Dirichlet Kernel

3.4 Integrations and Differentiations of Fourier Series

3.5 Generalized Fourier Series and Orthonormal Functions

3.6 Best Approximation in the Mean

4. Heat Equation and Boundary Value Problems

4.1 Introduction to PDEs               

4.2 Derivation of the Heat Equation          

4.3 Separable Equations

4.4 BVP-Heat Equation

4.5 Insulated Homogeneous Heat Problem with Homogeneous Boundaries

4.6 Insulated Homogeneous Heat Problem with Non-homogeneous Boundaries

4.7 Non-insulated Homogeneous Heat Problem with Homogeneous Boundaries 

4.8 Non-insulated Homogeneous Heat Problem with Non-homogeneous Boundaries

4.9 Heat Source

4.10 Heat Flow with Time and Space Dependent Heat Source and Time Dependent Boundary Conditions 

4.11 Method of Eigenfunction Expansion Using Green’s Formula

4.12 Laplace Equation on Rectangles and Unbounded Domains

4.13 Laplacian in Polar, Cylindrical and Spherical Coordinates

4.14 The Laplacian on an Annulus and a Disk: Analysis and Applications

4.15 Poisson’s Equation with a Source

5. The Wave Equation and Boundary Value Problems   

5.1 Derivation of the Wave Equation

5.2 Vibrating String with Fixed Ends

5.3 Forced Vibrating String at Resonance 

5.4 Vibrating String and Double Fourier Series    

5.5 Vibration of a Beam

6.Sturm-Liouville Systems and Applications      

6.1 Sturm-Liouville Problem

6.2 Expansion in Series of Eigenfunctions

6.3 Rayleigh Quotient

6.4 Reformulation of the Zero Eigenvalue and Minimum Eigenvalue Concepts

6.5 Applications to Nonuniform String

6.6 Robin’s Boundary Conditions in Heat Flow

6.7 Robin’s Boundary Conditions in Laplacian

6.8 Bessel’s Function

6.9 Orthogonality of Bessel’s Functions

6.10 Vibrations of Circular Membrane; Bessel’s Equation

6.11 Legendre Polynomials

6.12 The Generating Function and Eigenfunctions Expansion

6.13 Applications of Legendre’s Equation

6.14 Non-homogeneous Sturm-Liouville Problem

7. Fourier Integrals and Applications     

7.1 Transition from Fourier Series to Fourier Integral

7.2 Fourier Integral Theorem

7.3 Complex Form of Fourier Integral Representation

7.4 Heat Flow in an Infinite Rod

7.5 Wave Motion for Infinite String

7.6 Fourier Transforms

7.7 Properties of Fourier Transform

7.8 Parseval’s Formula

7.9 Normalized Auto-Correlation Function

7.10 Application of Fourier Transform

7.10.5 Exercises

8. Classic PDEs

8.1 Linear Equations 

8.2 Equations with Variable Coefficients

8.3 Quasi-linear Equations

8.4 Burger’s Equation

8.5 Second Order PDEs

8.6 Inequalities and Energy Estimates

9. Heat Equation on Unbounded Domains          

9.1 Uniqueness of Solution

9.2 Solution of the Heat Equation

9.3 Heat Equation on Semi-infinite Domain; Dirichlet Condition

9.4 Heat Equation on Semi-infinite Domain; Neumann Condition

10. Wave Equation on Unbounded Domains      

10.1 Wave equation and D’Alembert’s Solution

10.2 Non-homogeneous Wave Equation

10.3 Vibrating String with Fixed Ends

10.4 Wave Equation on Semi-infinite Domain

10.5 Duhamel’s Principle for the Wave Equation

Biography

Youssef N. Raffoul is Professor and Graduate Program Director at the University of Dayton. He holds a Ph.D. from Southern Illinois University. He is the recipient of the University of Dayton College of Arts and Sciences’ Award for Outstanding Scholarship (twice), and the University of Dayton Alumni Award in Scholarship. Professor Raffoul has published over one hundred and sixty articles in prestigious journals in the area of functional differential equations, difference equations, and dynamical systems on time scales.  He was honored by the Lebanese government with the Career in Science Award. The Archbishop of Lebanon awarded him the Lifetime Achievement Award. Most notably, and he is the recipient of the Order of Merit, Silver Medal with Distinction and awarded by the president of Lebanon, General Aoun.