Fourier Series, Transforms, and Boundary Value Problems: Second Edition (Dover Books on Mathematics)
The chief focus of this text is the application of the Fourier method to physical problems, which are described mathematically in terms of boundary value problems. Problems involving separation of variables, Sturm-Liouville theory, superposition, and boundary complaints are addressed in a logical sequence. Multidimensional Fourier series solutions and Fourier integral solutions on unbounded domains are followed by the special functions of Bessel and Legendre, which are introduced to deal with the cylindrical and spherical geometry of boundary value problems. Students and professionals in mathematics, the physical sciences, and engineering will find this volume an excellent study guide and resource.
Why Read This Book
You will learn practical Fourier techniques that directly support spectral analysis, filter intuition, and PDE-based modeling used across DSP, audio, radar, and communications. The book emphasizes worked procedures and modeling steps so you can follow solution outlines and apply transform methods to real engineering boundary-value problems.
Who Will Benefit
Graduate students and practicing engineers in DSP, communications, acoustics, or radar who need a concise, application-focused grounding in Fourier series, transforms, and boundary-value methods for modeling and spectral analysis.
Level: Intermediate — Prerequisites: Single- and multivariable calculus, ordinary differential equations, basic complex numbers/analysis and linear algebra; familiarity with basic PDE ideas (helpful but not strictly required).
Key Takeaways
- Apply Fourier series and Fourier integrals to solve classic boundary value PDEs (heat, wave, Laplace) encountered in engineering models.
- Use transform theorems (convolution, Parseval/Plancherel, modulation) to perform spectral analysis and reason about energy and filtering.
- Derive and use eigenfunction expansions and Sturm–Liouville results to construct orthogonal bases for signal and field representations.
- Extend Fourier methods to multidimensional problems to analyze wave propagation and spatial spectra relevant to acoustics, radar, and antenna theory.
- Translate continuous-transform intuition into practical approaches for filter design, response analysis, and understanding aliasing/leakage phenomena.
Topics Covered
- 1. Introduction and Motivation: Fourier Methods in Physical Problems
- 2. Trigonometric Fourier Series — Formulation and Examples
- 3. Convergence, Gibbs Phenomenon, and Summation Methods
- 4. Orthogonality, Inner Products, and Sturm–Liouville Theory
- 5. Separation of Variables and Boundary Value Problems in One Dimension
- 6. Fourier Integrals and the Continuous Fourier Transform
- 7. Applications to the Heat, Wave, and Laplace Equations
- 8. Multidimensional Fourier Series and Fourier Integrals
- 9. Eigenfunction Expansions and Green's Functions
- 10. Transform Methods for Unbounded Domains and Radiation Problems
- 11. Selected Applications to Physical Modeling (acoustics, vibrations, electromagnetics)
- Appendices: Tables, Worked Solutions, and Selected Answers to Exercises
How It Compares
More applied and example-driven than Titchmarsh's theoretical treatment of Fourier integrals and more focused on classical PDE techniques than Haberman's modern pedagogical PDE texts; complements DSP texts like Oppenheim & Willsky by providing deeper continuous-transform and PDE foundations.
