Second-order ordinary differential equations Special functions, Sturm-Liouville theory and transforms
Publisher: Bookboon.com
Published: 2013
ISBN-13: 9788776819729
Table of contents: title: Content; title: 1. Power-series solution of ODEs; title: 1.1. Series solution: essential ideas; title: 1.2. ODEs with regular singular points; title: 1.3. Exercises 1; title: 2. The method of Frobenius; title: 2.1. The basic method; title: 2.2. The two special cases; title: 2.3. Exercises; title: 3. The Bessel equation and Bessel functions; title: 3.1. First solution; title: 3.2. The second solution; title: 3.3. The modified Bessel equation; title: 4. The Legendre polynomials; title: 4.1. Exercises 4; title: 5. The Hermite polynomials; title: 5.1. Exercises 5; title: 6. Generating functions; title: 6.1. Legendre polynomials; title: 6.2. Hermite polynomials; title: 6.3. Bessel functions; title: 6.4. Exercises 6; title: 7. Answers; title: 8. Part II: An introduction to Sturm-Liouville theory; title: 9. Preface; title: 10. List of Equations; title: 11. Introduction and Background; title: 11.1. The second-order equations; title: 11.2. The boundary-value problem; title: 11.3. Self-adjoint equations; title: 11.4. Exercises 1; title: 12. The Sturm-Liouville problem: the eigenvalues; title: 12.1. Real eigenvalues; title: 12.2. Simple eigenvalues; title: 12.3. Ordered eigenvalues; title: 12.4. Exercises 2; title: 13. The Sturm-Liouville problem: the eigenfunctions; title: 13.1. The fundamental oscillation theorem; title: 13.2. Using the fundamental oscillation theorem; title: 13.3. Orthogonality; title: 13.4. Eigenfunction expansions; title: 13.5. Exercises 3; title: 14. Inhomogeneous equations; title: 14.1. Exercise 4; title: 15. Answers; title: 16. Part III: Integral transforms; title: 17. Preface; title: 18. List of Problems; title: 19. Introduction; title: 19.1. The appearance of an integral transform from a PDE; title: 19.2. The appearance of an integral transform from an ODE; title: 19.3. Exercise 1; title: 20. The Laplace Transform; title: 20.1. LTs of some elementary functions; title: 20.2. Some properties of the LT; title: 20.3. Inversion of the Laplace Transform; title: 20.4. Applications to the solution of differential and integral equations; title: 20.5. Exercises 2; title: 21. The Fourier Transform; title: 21.1. FTs of some elementary functions; title: 21.2. Some properties of the FT; title: 21.3. Inversion of the Fourier Transform; title: 21.4. Applications to the solution of differential and integral equations; title: 21.5. Exercises 3; title: 22. The Hankel Transform; title: 22.1. HTs of some elementary functions; title: 22.2. Some properties of the HT; title: 22.3. Application to the solution of a PDE; title: 22.4. Exercises 4; title: 23. The Mellin Transform; title: 23.1. MTs of some elementary functions; title: 23.2. Some properties of the MT; title: 23.3. Applications to the solution of a PDE; title: 23.4. Exercises 5; title: 24. Tables of a few standard Integral Transforms; title: 25. Answers; title: 26. Index