Integral Equations and Integral Transforms

This comprehensive textbook on linear integral equations and integral transforms is aimed at senior undergraduate and graduate students of mathematics and physics. The book covers a range of topics including Volterra and Fredholm integral equations, the second kind of integral equations with symmetric kernels, eigenvalues and eigen functions, the Hilbert–Schmidt theorem, and the solution of Abel integral equations by using an elementary method. In addition, the book covers various integral transforms including Fourier, Laplace, Mellin, Hankel, and Z-transforms. One of the unique features of the book is a general method for the construction of various integral transforms and their inverses, which is based on the properties of delta function representation in terms of Green's function of a Sturm–Liouville type ordinary differential equation and its applications to physical problems. The book is divided into two parts: integral equations and integral transforms.Each chapter is supplemented with numerous illustrative examples to aid in understanding. The clear and concise presentation of the topics covered makes this book an ideal resource for students, researchers, and professionals interested in the theory and application of linear integral equations and integral transforms. Preface Contents About the Authors Part I Integral Equations 1 Integral Equations: An Introduction 1.1 Introduction 1.1.1 What is an Integral Equation? 1.1.2 Classifications of Integral Equations 1.2 Occurrence of Integral Equations 1.2.1 Occurrence of Volterra Integral Equations 1.2.2 Occurrence of Fredholm Integral Equations References 2 Fredholm Integral Equation of the Second Kind with Degenerate Kernel 2.1 Integral Equation with Degenerate Kernel 2.2 Homogeneous Equations 2.3 Nonhomogeneous Equations References 3 Integral Equations of Second Kind with Continuous and Square Integrable Kernel 3.1 Fredholm Integral Equations of Second Kind with Continuous Kernel 3.2 Volterra Integral Equations of Second Kind with Continuous Kernel 3.3 Illustrative Examples 3.4 Iterated Kernels 3.5 Fredholm Theory for Integral Equation with Continuous Kernel 3.6 Fredholm Integral Equations of Second Kind with Square Integrable Kernel 3.6.1 Some Important Properties of Square Integrable Functions 3.6.2 Method of Solution of Integral Equation with Square Integrable Kernel 3.7 Fredholm Theory for Integral Equation with Square Integrable Kernel References 4 Integral Equations of the Second Kind with a Symmetric Kernel 4.1 Symmetric Kernel 4.2 Properties of Integral Equations with a Symmetric Kernel 4.3 Hilbert–Schmidt Theorem References 5 Abel Integral Equations 5.1 Solution Based on Elementary Integration 5.2 Solution Based on Laplace Transform References Part II Integral Transform 6 Fourier Transform 6.1 Integral Transform: An Introduction 6.2 Fourier Integral Theorem 6.3 Rigorous Justification of Fourier Integral Theorem 6.4 Fourier Cosine and Sine Transforms 6.5 Fourier Transforms of Some Simple Functions 6.6 Properties of Fourier Transform 6.7 Convolution Theorem and Parseval Relation 6.8 Fourier Transforms in Two or More Dimensions 6.9 Application of Fourier Transforms in Solving Linear Ordinary ... 6.10 Application of Fourier Sine and Cosine Transforms in Solving ... 6.11 Application to Partial Differential Equations 6.12 Application of Fourier Sine and Cosine Transform to the Solution ... References 7 Laplace Transform 7.1 Derivation of Laplace Transform from Fourier Integral Theorem 7.2 Laplace Inversion 7.3 Operational Properties of Laplace Transform 7.4 Laplace Convolution Integral 7.5 Tauberian Theorems 7.6 Method of Evaluation of Inverse Laplace Transform 7.7 Application of Laplace Transform in Solving Ordinary Differential Equations 7.8 Application Laplace Transform in Solving Partial Differential Equations References 8 Mellin Transform 8.1 Introduction 8.2 Formal Derivation of Mellin Transform 8.3 Theorem on Inversion of Mellin Transform 8.4 Properties of Mellin Transform 8.5 Mellin Transform of Some Simple Functions References 9 Hankel Transform 9.1 Formal Derivation of Hankel Transform 9.2 Properties of Hankel Transform 9.3 Hankel Transform of Some Known Functions References 10 mathcalZ Transform 10.1 Introduction 10.2 Properties of mathcalZ-Transform 10.3 Inverse mathcalZ-Transform References 11 Formal Construction of Integral Transforms and Their Inverses 11.1 The Method of Construction 11.2 Construction of Some Familiar Integral Transforms 11.2.1 Fourier Integral Transform 11.2.2 Fourier Cosine Transform 11.2.3 Fourier Sine Transform 11.2.4 Havelock Transform (Mixed Fourier Transform, Hybrid Fourier Transform) 11.2.5 Finite Fourier Sine Transform 11.2.6 Finite Fourier Cosine Transform 11.2.7 Finite Fourier Transform 11.2.8 Laplace Transform References