Page 165 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Sachin Kaushal, Lovely Professional University

Notes Unit 9: Adjoint and Self-Adjoint Equations

CONTENTS
Objectives
Introduction
9.1 Adjoint and Self-adjoint Operators

9.2 Boundary Conditions
9.3 Eigenvalues and Eigenfunctions of Hermitian Linear Operators
9.4 Eigenfunction Expansions
9.5 Summary
9.6 Keywords

9.7 Review Question
9.8 Further Readings

Objectives

After studying this unit, you should be able to:
 See that adjoint and self-adjoint operators play an important part in the solution of certain
types of equations.
 Observe that the properties of the solutions as well as the values of certain parameter are
obtained in a systematic manner.
 Notice that the self-adjoint equations when solved under certain boundary conditions
yield values of the solutions known as eigenfunctions corresponding to certain eigenvalues.

Introduction

In this unit the method of putting an equation into a self-adjoint form is dealt with. This method
and the Sturm Liouville’s method leads us to the solutions of the differential equations which
are orthogonal.
The solutions form a set of eigenfunctions which are complete and so any function on the given
interval can be expanded in terms of these eigenfunctions.

9.1 Adjoint and Self-adjoint Operators

In this unit we are interested in solving inhomogeneous boundary value problems for linear,
second order differential equations. We will now develop an approach that is based upon the
idea of linear algebra. We shall work with the simplest possible type of linear differential
2
operator L, C [a, b} C{a, b} being in self-adjoint form:
d d
L = p ( ) q ( ) ...(1)
x
x
dx dx
1
where p(x) C [a, b] and is strictly non-zero for all x (a, b), and q(x) C [a, b]. The reasons for
referring to such an operator as self-adjoint will become clear later in this unit.

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