1st Edition
Integral and Discrete Transforms with Applications and Error Analysis
This reference/text desribes the basic elements of the integral, finite, and discrete transforms - emphasizing their use for solving boundary and initial value problems as well as facilitating the representations of signals and systems.;Proceeding to the final solution in the same setting of Fourier analysis without interruption, Integral and Discrete Transforms with Applications and Error Analysis: presents the background of the FFT and explains how to choose the appropriate transform for solving a boundary value problem; discusses modelling of the basic partial differential equations, as well as the solutions in terms of the main special functions; considers the Laplace, Fourier, and Hankel transforms and their variations, offering a more logical continuation of the operational method; covers integral, discrete, and finite transforms and trigonometric Fourier and general orthogonal series expansion, providing an application to signal analysis and boundary-value problems; and examines the practical approximation of computing the resulting Fourier series or integral representation of the final solution and treats the errors incurred.;Containing many detailed examples and numerous end-of-chapter exercises of varying difficulty for each section with answers, Integral and Discrete Transforms with Applications and Error Analysis is a thorough reference for analysts; industrial and applied mathematicians; electrical, electronics, and other engineers; and physicists and an informative text for upper-level undergraduate and graduate students in these disciplines.
Preface
Guide to Course Adoption
1 Compatible Transforms
The Method of Separation of Variables and the Integral Transforms
Integral Transforms
Compatible Transforms
Examples of Compatible Transforms
Nonlinear Terms
Classification of the Transforms
Integral Transforms
Band-Limited Functions (or Transforms)
Finite Transforms—The Fourier Coefficients
The Truncation and Discretization (Sampling) Errors
The Discrete Transforms
Comments on the Inverse Transforms—Tables of the Transforms
Integral Equations—Basic Definitions
The Compatible Transform and the Adjoint Problem
The Adjoint Differential Operator
The Two Eigenvalue Problems
Constructing the Compatible Transforms for Self-Adjoint Problems—Second-Order Differential Equations
Examples of the Strum-Liouville and Other Transforms—Boundary Value Problems
The nth-Order Differential Operator
Relevant References to Chapter 1
Exercises
2 Integral Transforms
Laplace Transforms
Transform Pairs and Operations
The Convolution Theorem for Laplace Transforms
Solution of Initial Value Problems Associated with Ordinary and Partial Differential Equations
Applications to Volterra Integral Equations with Difference Kernels
The z-Transform
Fourier Exponential Transforms
Existence of the Fourier Transform and Its Inverse—the Fourier Integral Formula
Basic Properties and the Convolution Theorem
Boundary and Initial Value Problems—Solutions by Fourier Transforms
The Heat Equation on an Infinite Domain
The Wave Equation
The Schodinger Equation
The Laplace Equation
Signals and Linear Systems—Representation in the Fourier (Spectrum) Space
Linear Systems
Bandlimited Functions—the Sampling Expansion
Bandlimited Functions and B-Splines (Hill Functions)
Fourier Sine and Cosine Transforms
Compatibility of the Fourier Sine and Cosine Transforms with Even-Order Derivatives
Applications to Boundary Value Problems on Semi-Infinite Domain
Higher-Dimensional Fourier Transforms
Relation Between the Hankel Transform and the Multiple Fourier Transform—Circular Symmetry
The Double Fourier Transform of Functions with Circular Symmetry—The Jo-Hanckel Transform
A Double Fourier Transform Convolution Theorem for the Jo-Hankel Transform
The Hankel (Bessel) Transforms
Applications of the Hankel Transforms
Laplace Transform Inversion
Fourier Transform in the Complex Plane
The Laplace Transform in the Inversion Formula
The Numerical Inversion of the Laplace Transform
Applications
Other Important Integral Transforms
Hilbert Transform
Mellin Transform
The z-Transform and the Laplace Transform Relevant for Chapter 2
Exercises
3 Finite Transforms—Fourier Series and Coefficients
Fourier (Trigonometric) Series and General Orthogonal Expansion
Convergence of the Fourier Series
Elements of Infinite Series—Convergence Theorems
The Orthogonal Expansions—Bessel’s Inequality and Fourier Series
Fourier Sine and Cosine Transforms
Fourier (Exponential) Transforms
The Finite Fourier Exponential Transform and the Sampling Expansion
Hankel (Bessel) Transforms
Another Finite Hankel Transform
Classical Orthogonal Polynomial Transforms
Legendre Transforms
Laguerre Transform
Hermite Transforms
Tchebychev Transforms
The Generalized Sampling Expansion
Generalized Translation and Convolution Products
Impulse Train for Bessel Orthogonal Series Expansion for a (New) Bessel-Type Possion Summation Formula
A Remark on the Transform Methods and Nonlinear Problems
Relevant References to Chapter 3
Exercises
4 Discrete Transforms
Discrete Fourier Transforms
Fourier Integrals, Series, and the Discrete Transforms
Computing for Complex- Valued Functions
The Fast Fourier Transform
Construction and Basic Properties of the Discrete Transforms
Opertational Difference Calculus for the DFT and the z-Transform
Approximating Fourier Integrals and Series by Discrete Fourier Transforms
Examples of Computing Fourier Integrals and Series
Discrete Orthogonal Polynomial Transforms
Basic Properties and Illustrations
Properties of the Discrete Legendre Transforms
The Use of the Orthogonal Polynomial Transforms
Bessel-Type Possion Summation Formula (for the Bessel-Fourier Series and the Hankel Transforms)
Relevant References for Chapter 4
Exercises
Appendix A Basic Second-Order Differential Equations and Their (Series) Solutions—Special Functions
Introduction
Method of Variation of Parameters
Power Series Method of Solution
Frobenius Method of Solution- Power Series Expansion About a Regular Singular Point
Special Differential Equations and Their Soultions
Bessel’s Equation
Legendre’s Equation
Other Special Equations
Exercises
Appendix B Mathematical Modeling of Partial Differential Equations—Boundary and Initial Value Problems
Partial Differential Equations for Vibrating Systems
Diffusion (or Heat Conduction) Equation
Exercises
Appendix C Tables of Transforms
Laplace Transforms
Fourier Exponential Transforms
Fourier Sine Transforms
Fourier Cosine Transforms
Hankel Transforms
Mellin Transforms
Hilbert Transforms
Finite Exponential Transforms
Finite Since Transforms
Finite Cosine Transforms
Finite (First) Hankel Transforms, Jn(Ka)=0
Finite (Second) Hankel Transforms, kjn(ka)+hJn(ka)=0
Finite Legendre Transforms
Finite Tchebychev Transforms
Finite Hermite Transforms
z-Transforms
Bibliography
Index of Notations
Subject Index
Biography
Abdul Jerri