Page 189 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Richa Nandra, Lovely Professional University

Notes Unit 11: Sturm–Liouville’s Boundary Value Problems

CONTENTS
Objectives

Introduction
11.1 Sturm-Liouville’s Equation
11.2 Boundary Conditions

11.3 Properties of the Eigenvalues and Eigenfunctions
11.4 Bessel’s Inequality, Approximation in the Mean and Completeness

11.5 Summary
11.6 Keywords
11.7 Review Questions

11.8 Further Readings

Objectives

After studying this unit, you should be able to:
 Understand the structure of self-adjoint equations. If we are dealing with only second
order differential equations, we see that under what conditions we can put them in self-
adjoint form.

 Know that Sturm-Liouville boundary value problem is a method of dealing with equations
which can be put into Sturm-Liouville form.

 Find the solutions for some values of the parameters. The solutions are known as
eigenfunctions and the values of the parameter are known as eigenvalues.
 Know that important examples of Sturm-Liouville boundary value problems are Legendre
equation, Bessel’s equations and many more.

Introduction

This method helps us in finding certain sets of functions which are orthogonal and we can
express any function in terms of these eigenfunctions on the interval a x b where a and b may
be finite or one of them finite and the other infinite or both a and b to be infinite.
These methods are known as Fourier Legendre expansion if we use Legendre polynomials and
so on.
11.1 Sturm-Liouville’s Equation

In the first four units we have studied linear second order differential equations. After examining
some solutions techniques that are applicable to such equations in general we studied the
particular cases of Legendre’s equation, Bessel’s equations, the Hermite equations and Laguerre’s
equations, as they frequently arise in models of physical systems in spherical, cylindrical
geometries and in Quantum mechanics. In each case we saw that we can construct a set of

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