Page 173 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Sachin Kaushal, Lovely Professional University

Notes Unit 10: Green’s Function Method

CONTENTS
Objectives
Introduction

10.1 Boundary Value Problem of Sturm Liouville Type
10.2 Green’s Function for one dimensional problem
10.3 Periodic Solutions. Generalized Green’s Function

10.3.1 Construction of Green’s Function
10.4 Green’s Function for Two independent Variables
10.5 Green’s Function for Two Dimensional Problem
10.6 Summary
10.7 Keywords

10.8 Review Questions
10.9 Further Readings

Objectives

After studying this unit, you should be able to see that:
 Green’s function plays an important part in the solution of the differential equations.

 It finds its applications in most of the boundary value problems.
 Green’s function is quite helpful in converting a differential equation into an integral
equation.

Introduction

Green’s function method helps in solving most of the boundary value problems. It is quite
useful in reducing a differential equation to an integral equation. With the help of the Green’s
function method the problem of solution of differential equations becomes simpler.
10.1 Boundary Value Problem of Sturm Liouville Type

We consider a differential equation of the second order

2
d y dy
x
p 1 ( ) p 2 ( )y 0 ...(1)
x
dx 2 dx
where p (x), p (x) are real-valued continuous function on a closed interval a x b. The equation
1 2
(1) can be put into the form
d dy
p ( ) x q ( ) x y ...(2)
dx dx

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