**MTH 509: Sturm-Liouville Theory (4)**

**Pre-requisites**: *MTH 306 Ordinary Differential Equations, MTH 404 Measure and Integration*

*Fourier Series:* Fourier series of a periodic function, question of point-wise convergence of such a series, behavior of the Fourier series under the operation of differentiation and integration , sufficient conditions for uniform and absolute convergence of a Fourier series, Fourier series on intervals, examples of boundary value problems for the one dimensional heat and wave equations illustrating the use of Fourier series in solving them by separating variables, a brief discussion on Cesaro summability and Gibbs phenomenon

*Orthogonal Expansions:* A quick review of L2 spaces on an interval, convergence, completeness, orthonormal systems, Bessel’s inequality, Parseval’s identity, dominated convergence theorem

*Sturm-Liouville Systems:* linear differential operators, formal adjoint of a linear operator, Lagrange’s identity, self-adjoint operators, regular and singular Sturm-Liouville systems, Sturm-Liouville series, Prufer substitution, Sturm comparison and oscillation theorems, eigenfunctions, Liouville normal form, distribution of eigenvalues, normalized eigenfunctions, Green’s functions, completeness of eigenfunctions

*Illustrative boundary value problems:* A technique to solve inhomogeneous equations using Sturm-Liouville expansions, one dimensional heat and wave equations with inhomogeneous boundary conditions, one dimensional inhomogeneous heat and wave equations, mixed boundary conditions, Dirichlet problem in a rectangle and a polar coordinate rectangle

*Maximum Principle and applications:* maximum principle for linear, second-order, ordinary differential equations, generalized maximum principle for such equations, applications to initial and boundary value problems, the eigenvalue problem, an extension of the principle to non-linear equations

*Orthogonal polynomials and their properties:* Legendre polynomials, Legendre equation, Legendre functions and spherical harmonics, Hermite polynomials, Hermite functions, Hermite equation, Laguerre polynomials, Laguerre equation, zeros of orthogonal polynomials on an interval, and a recurrence relation satisfied by them

*Bessel Functions:* Bessel’s equation, identities, asymptotics and zeros of Bessel functions

*Suggested Books*:

**Texts:**

- Birkhoff, G & Rota G.,
*Ordinary Differential Equations*, John Wiley & Sons - Folland, G.,
*Fourier Analysis & Its Applications,*AMS - Protter, M. & Weinberger, H.
*, Maximum Principles in Differential Equations*, Springer

**References:**

- Brown, J. & Churchill, R.,
*Fourier Series and Boundary Value Problems*, McGraw-Hill - Jackson, D.,
*Fourier Series and Orthogonal Polynomials*, Dover

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