# Module 1

51

### Examples of Sturm-Liouville systems

#### Lectures

 Examples of Sturm-Liouville systems Bessel differential equation Classification of second order linear PDEs Introduction to Sturm-Liouville theory Regular Sturm-Liouville problem Examples of regular Sturm-Liouville systems Non-homogeneous wave equation Legendre Polynomials Properties of Legendre Polynomials Power series solutions around a regular singular point Periodic and singular Sturm-Liouville Problems Frobenius solutions for Bessel Equation Generalized Fourier series Second order linear PDEs Reduction to Normal form-More examples Frobenius method of solutions Temperature in a finite rod Frobenius method of solutions (continued) Reduction to canonical form for equations with variable coefficients Vibration of a finite string Laplace equation over a rectangle with flux boundary conditions Laplace equation over circular Sectors Laplace equation over circular domains Laplace equation over a rectangle Properties of Bessel functions Examples on Frobenius method Vibration of a semi-infinite string Properties of Bessel functions (continued) Sturm-Liouville Problems Uniqueness of the boundary value problems for Laplace equation Finite length string vibrations Solutions of heat equation-Properties Examples of Sturm-Liouville systems (continued) Temperature in a semi-infinite rod Reduction to canonical form for equations with constant coefficients Uniqueness of solutions for wave equation Finite length string vibrations (continued) Vibration of a circular drum Temperature in an infinite rod Non-homogeneous heat equation Temperature in a finite rod with insulated ends Conclusions Lecture 01 - Introduction to Ordinary Differential Equations (ODE) Lecture 07 - Non-Exact Equations - Finding Integrating Factors Lecture 14 - Euler - Cauchy equation Lecture 15 - Non homogeneous ODEs Variation of Parameters Lecture 16 - Method of undetermined coefficients Lecture 17 - Demonstration of Method of undetermined coefficients Lecture 18 - Power Series and its properties Lecture 20 - Power Series Solutions ( Continued ) Lecture 21 - Legendre Differential Equatio
##### OER.UoCh.Edu.Pk (University of Chitral Open Educational Resources).

In the wake of the global emergency on Novel Coronavirus (COVID-19), the universities of Pakistan are planning to go online under the aegis of the Higher Education Commission of Pakistan (HEC). The University of Chitral has launched a comprehensive open learning knowledge hub to assist the faculty and students both at home and abroad to learn through the web. This includes a wide range of learning resources including free audio-visual lectures from the top universities and institutes of the world.