# Sturm Liouville System

Contents

## #2 Preliminaries Regarding the Differential Equations

### SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

The general form of an Ordinary Differential Equation (ODE) of order 2 may be taken as

Which may be linear or non-linear, homogeneous or non-homogeneous.

## #3 Introduction to Sturm-Liouville System

There are many second order, ordinary, Homogeneous, Linear Differential equations which appear in physical and Engineering problems: Some of these are

All these Differential Equations are special cases of more general type of second order, ordinary, homogeneous, Linear Differential Equation are known as Sturm-Liouville’s Equations and the Differential Equation together with appropriate (Suitable) boundary conditions is known as Sturm-Liouville system (Problem).

NoteJohn Sturm and Joseph Liouvi are French Mathematics who did work on such Differential Equations on 1830

## #4 Sturm-Liouville Equation (S.L Equation)

The second order, ordinary, homogeneous, linear differential equation of the form

Notation: In term of Linear Differential operator: If we define the operator L as

## #9 Regular & Periodic Sturm Liouville System

### Regular Sturm Liouville [S.L.] System

The Regular S.L. Equation together with linear, homogeneous, separated (.e., no mixing of) end point (boundary) conditions

Note: The system is regular when zero is not the point of the given interval.

### Periodic Sturm Liouville [S.L.] System

The periodic S.L. Equation together with linear, homogeneous mixed and periodic end point conditions