Secondorder ordinary differential equations
Special functions, SturmLiouville theory and transforms
4.2
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ISBN: 9788776819729
1 edition
Pages : 181
 Price: 69.99 kr
 Price: €7.99
 Price: £7.99
 Price: ₹250
 Price: $9.99
 Price: 69.99 kr
 Price: 69.99 kr
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About the book
Description
Ordinary differential equations, and secondorder equations in particular, are at the heart of many mathematical descriptions of physical systems, as used by engineers, physicists and applied mathematicians. This text provides an introduction to all the relevant material normally encountered at university level: series solution, special functions (Bessel, etc.), SturmLiouville theory (involving the appearance of eigenvalues and eigenfunctions) and the definition, properties and use of various integral transforms (Fourier, Laplace, etc.). Numerous worked examples are provided throughout.
Content
 Powerseries solution of ODEs
 Series solution: essential ideas
 ODEs with regular singular points
 Exercises 1
 The method of Frobenius
 The basic method
 The two special cases
 Exercises
 The Bessel equation and Bessel functions
 First solution
 The second solution
 The modified Bessel equation
 The Legendre polynomials
 Exercises 4
 The Hermite polynomials
 Exercises 5
 Generating functions
 Legendre polynomials
 Hermite polynomials
 Bessel functions
 Exercises 6
 Answers
 Part II: An introduction to SturmLiouville theory
 Preface
 List of Equations
 Introduction and Background
 The secondorder equations
 The boundaryvalue problem
 Selfadjoint equations
 Exercises 1
 The SturmLiouville problem: the eigenvalues
 Real eigenvalues
 Simple eigenvalues
 Ordered eigenvalues
 Exercises 2
 The SturmLiouville problem: the eigenfunctions
 The fundamental oscillation theorem
 Using the fundamental oscillation theorem
 Orthogonality
 Eigenfunction expansions
 Exercises 3
 Inhomogeneous equations
 Exercise 4
 Answers
 Part III: Integral transforms
 Preface
 List of Problems
 Introduction
 The appearance of an integral transform from a PDE
 The appearance of an integral transform from an ODE
 Exercise 1
 The Laplace Transform
 LTs of some elementary functions
 Some properties of the LT
 Inversion of the Laplace Transform
 Applications to the solution of differential and integral equations
 Exercises 2
 The Fourier Transform
 FTs of some elementary functions
 Some properties of the FT
 Inversion of the Fourier Transform
 Applications to the solution of differential and integral equations
 Exercises 3
 The Hankel Transform
 HTs of some elementary functions
 Some properties of the HT
 Application to the solution of a PDE
 Exercises 4
 The Mellin Transform
 MTs of some elementary functions
 Some properties of the MT
 Applications to the solution of a PDE
 Exercises 5
 Tables of a few standard Integral Transforms
 Answers
 Index