Executive Summary : | In a series of articles dating from 1836-37, Sturm and Liouville independently worked on the problem of heat conduction through a metal bar. In the process of developing techniques to solve this problem, they created a whole new theory in mathematical analysis. Later, the whole theory was known as Sturm-Liouville (SL) theory. It is observed and realized that SL theory has many applications in real world. Probably, everyone has heard or played an instrument, for example a guitar. The sound of guitar comes from plucking a string. The vibration of the plucked string changes to initial position and initial velocity. The motion of the vibrating string can be described by a wave equation which is a typical problem that can be solved using SL theory. Exploring, some of its properties gives us insight in the harmonics of the instrument which clarifies the pleasing sound of a guitar. Not only sound of a guitar, but many other important physical processes and mechanical systems from classical physics and quantum physics can be described by means of SL theory. Considering these applications, the PI aims to study more generalized form of Sturm-Liouville problems defined using weight and another functions known as generalized Sturm-Liouville problems(GSLPs). Recently, Agrawal (2012) has presented a generalized derivative in term of weight $w(x)$ and another function $\psi(x)$. In special case, it reduces to the traditional integer order derivatives for choice of $\psi(x)=x$. We plan to investigate the properties of the eigenfunctions and eigenvalues for such GSLPs. Further, we wish to extend other properties such as Sturm comparison theorem and Sturm separation theorem for the GSLPs. |