1st Edition

# Integral and Discrete Transforms with Applications and Error Analysis

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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 *n*th-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