Sturm Liouville System

#2 Preliminaries Regarding the Differential Equations

Differential Equations;

  • w.r.t. number of independent variables

Differential Equations

  • w.r.t. dependent variables

Differential Equations dependent variables

  • w.r.t. the presence of a Source Term

Differential Equations homogeneous non homogeneous

SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

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

Differential Equations equation

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

motion differential equation

motion differential equation

motion differential equation

motion differential equation

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

Sturm-Liouville Equation

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

S.L. Equation

S.L. Equation

#5 What is the Self-Adjoint Sturm-Liouville differential operator notation?

Self-Adjoint Sturm-Liouville differential operator notation

#6 Properties of Sturm-Liouville Equations

Properties of Sturm-Liouville Equations

#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

Regular Sturm Liouville System

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

Periodic Sturm Liouville System

#23 Wronskian of Two Functions

Wronskian of Two Functions

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