NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS PDF

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CHAPTER

NUMERICAL SOLUTION OF P

ARTIAL DIFFERENTIAL EQUATIONS

Chapter Objectives

Introduction

Classification of second order equations

Finite-difference approximations

Elliptic equations to partial derivatives

Solution of Laplace equation

Solution of Poisson"s equation

Solution of elliptic equations by relaxation

Parabolic equations method

Solution of one-dimensional heat equation

Solution of two-dimensional heat equation

Hyperbolic equations

Solution of wave equation

11.1 Introduction

Partial differential equations arise in the study of many branches of applied mathematics, e.g., in fluid dynamics, heat transfer, boundary layer flow, elasticity, quantum mechanics, and electro- magnetic theory. Only a few of these equations can be solved by analytical methods which are also complicated by requiring use of advanced mathematical techniques. In most of the cases, it is easier

492 NUMERICAL METHODS IN ENGINEERING AND SCIENCE

to develop approximate solutions by numerical methods. Of all the numeri- cal methods available for the solution of partial differential equations, the method of finite differences is most commonly used. In this method, the derivatives appearing in the equation and the boundary conditions are re- placed by their finite difference approximations. Then the given equation is changed to a system of linear equations which are solved by iterative pro- cedures. This process is slow but produces good results in many boundary value problems. An added advantage of this method is that the computation can be carried by electronic computers. To accelerate the solution, some- times the method of relaxation proves quite effective. Besides discussing the finite difference method, we shall briefly de- scribe the relaxation method also in this chapter.

11.2 Classification of Second Order Equations

The general linear partial differential equation of the second order in two independent variables is of the form 222
22
uuuuux,y x,y x,y x,y,u , 0xy x yxyAB Cd (1) Such a partial differential equation is said to be (i) elliptic if B 2 - 4AC < 0, (ii) parabolic if B 2 - 4AC = 0, and (iii) hyperbolic if B 2 - 4AC > 0. Obs. A partial equation is classified according to the region in which it is desired to be solved. For instance, the partial differential equation f xx f yy

0 is elliptic if y > 0, parabolic if y

0, and hyperbolic if y < 0.

EXAMPLE 11.1

Classify the following equations:

(i) 222
22

44 20uuuuu

xy x y xy (ii) 22
22
22

10,,1uuxy xyxy

NOTE NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 493 (iii) 222
222
22

15240.uuuxxxxtxt

Solution:

(i) Comparing this equation with (1) above, we find that t 2

A 1, B 4, C 4

B 2 - 4AC (4) 2 - 4 × 1 × 4 0

So the equation is parabolic.

(ii) Here A x 2 , B 0, C 1 - y 2 B 2 - 4AC 0 - 4x 2 (1 - y 2 ) 4x 2 (y 2 - 1)

For all x between - and , x

2 is positive

For all y between - 1 and 1, y

2 < 1 B 2 - 4AC < 0

Hence the equation is elliptic

(iii) Here A = 1 + x 2 , B = 5 + 2x 2 , C = 4 + x 2 B 2quotesdbs_dbs2.pdfusesText_2

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