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Sachin Kaushal, Lovely Professional University Unit 22: Solution of Laplace Differential Equation

Unit 22: Solution of Laplace Differential Equation Notes

CONTENTS
Objectives
Introduction
22.1 Solution of Laplace Differential Equation Cylindrical Co-ordinates

22.2 Circular Harmonics
22.2.1 Solution of Laplace s Equation in Spherical Polar Co-ordinates
22.2.2 Steady Flow of Heat in Rectangular Plate
22.3 Summary
22.4 Keywords
22.5 Review Questions

22.6 Further Readings
Objectives

After studying this unit, you should be able to:
 Know that Laplace equation is a partial differential equation involving one dependent
variable and three independent variables.
 See that it has a vast number of applications in gravitational potential process in electrostatic
potential distributions, in the propagation of waves, in diffusion process or heat
conductions.
 Note that three major co-ordinate systems namely the Cartesian co-ordinate system the
spherical polar co-ordinate system or the cylindrical co-ordinate systems are used to
express Laplacian operator.

Introduction

This Laplace equation is seen to be written in such a way that the dependence of dependent
variable on three independent variables can be separated.
Both spherical polar co-ordinates and cylindrical co-ordinates are used to find the solution of
Laplace equation.

22.1 Solution of Laplace Differential Equation Cylindrical

Co-ordinates

The most important partial differential equation of applied mathematics is the differential
equation of Laplace i.e.
2
V = 0 ...(1)
,
,
The Laplace operator is expressed in general curvilinear co-ordinates u u u in the following
1
3
2
manner,
1 h h h h h h
2 = 2 3 3 1 1 2 ...(2)
h h h u 1 h 1 u 1 u 2 h 2 u 2 u 3 h 3 u 3
1 2 3
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