Mathematical Methods in Physics Partial Differential Equations, Fourier Series, and Special Functions

This book is a text on partial differential equations (PDEs) of mathematical physics and boundary value problems, trigonometric Fourier series, and special functions. This is the core content of many courses in the fields of engineering, physics, mathematics, and applied mathematics. The accompanying software provides a laboratory environment that allows the user to generate and model different physical situations and learn by experimentation. From this standpoint, the book along with the software can also be used as a reference book on PDEs, Fourier series and special functions for students and professionals alike.

Fourier Series Periodic Processes and Periodic Functions Fourier Formulas Orthogonal Systems of Functions Convergence of Fourier Series Fourier Series for Non-periodic Functions Fourier Expansions on Intervals of Arbitrary Length Fourier Series in Cosine or Sine Functions The Complex Form of the Fourier Series Complex Generalized Fourier Series Fourier Series for Functions of Several Variables Uniform Convergence of Fourier Series The Gibbs Phenomenon Completeness of a System of Trigonometric Functions General Systems of Functions: Parseval’s Equality and Completeness Approximation of Functions in the Mean Fourier Series of Functions Given at Discrete Points Solution of Differential Equations by Using Fourier Series Fourier Transforms The Fourier Integral Problems Sturm-Liouville Theory The Sturm-Liouville Problem Mixed Boundary Conditions Examples of Sturm-Liouville Problems Problems One-Dimensional Hyperbolic Equations Derivation of the Basic Equations Boundary and Initial Conditions Other Boundary Value Problems: Longitudinal Vibrations of a Thin Rod Torsional Oscillations of an Elastic Cylinder Acoustic Waves Waves in a Shallow Channel Electrical Oscillations in a Circuit Traveling Waves: D’Alembert Method Semi-infinite String Oscillations and the Use of Symmetry Properties Finite Intervals: The Fourier Method for One-Dimensional Wave Equations Generalized Fourier Solutions Energy of the String Problems Two-Dimensional Hyperbolic Equations Derivation of the Equations of Motion Oscillations of a Rectangular Membrane The FourierMethod Applied to Small Transverse Oscillations of a Circular Membrane Problems One-Dimensional Parabolic Equations Physical Problems Described by Parabolic Equations: Boundary Value Problems The Principle of the Maximum, Correctness, and the Generalized Solution The Fourier Method of Separation of Variables for the Heat Conduction Equation Heat Conduction in an Infinite Bar Heat Equation for a Semi-infinite Bar Problems Parabolic Equations for Higher-Dimensional Problems Heat Conduction in More than One Dimension Heat Conduction within a Finite Rectangular Domain Heat Conduction within a Circular Domain Problems Elliptic Equations Elliptic Partial Differential Equations and Related Physical Problems The Dirichlet Boundary Value Problem for Laplace’s Equation in a Rectangular Domain Laplace’s and Poisson’s Equations for Two-Dimensional Domains with Circular Symmetry Laplace’s Equation in Cylindrical Coordinates Problems Bessel Functions Boundary Value Problems Leading to Bessel Functions Bessel Functions of the First Kind Properties of Bessel Functions of the First Kind: Jn (x) Bessel Functions of the Second Kind Bessel Functions of the Third Kind Modified Bessel Functions The Effect of Boundaries on Bessel Functions Orthogonality and Normalization of Bessel Functions The Fourier-Bessel Series Further Examples of Fourier-Bessel Series Expansions Spherical Bessel Functions The Gamma Function Problems Legendre Functions Boundary Value Problems Leading to Legendre Polynomials Generating Function for Legendre Polynomials Recurrence Relations Orthogonality of Legendre Polynomials The Multipole Expansion in Electrostatics Associated Legendre Functions P m (x)nOrthogonality and the Normof Associated Legendre Functions Fourier-Legendre Series in Legendre Polynomials Fourier-Legendre Series in Associated Legendre Functions Laplace’s Equation in Spherical Coordinates and Spherical Functions Problems Eigenvalues and Eigenfunctions of the Sturm-Liouville ProblemAuxiliary Functions for Different Types of Boundary ConditionsThe Sturm-Liouville Problem and the Laplace Equation Vector Calculus How to Use the Software Associated with this Book Program Overview Examples Using the Program TrigSeries Examples Using the Program Waves Examples Using the Program Heat Examples Using the Program Laplace Examples Using the Program FourierSeries

Victor Henner, Department of Physics and Astronomy, University of Louisville, Kentucky, USA Tatyana Belozerova, Perm State University, Russia Kyle Forinash, Professor of Physics and Program Coordinator, Indiana University Southeast, New Albany, USA