Ordinary differential equations, and second-order 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.), Sturm-Liouville 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.
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Second-order ordinary differential equations
Special functions, Sturm-Liouville theory and transforms
Description
Content
- Power-series 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 Sturm-Liouville theory
- Preface
- List of Equations
- Introduction and Background
- The second-order equations
- The boundary-value problem
- Self-adjoint equations
- Exercises 1
- The Sturm-Liouville problem: the eigenvalues
- Real eigenvalues
- Simple eigenvalues
- Ordered eigenvalues
- Exercises 2
- The Sturm-Liouville 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