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CHAPTER
11NUMERICAL 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 easier492 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 22222
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) 22222
44 20uuuuu
xy x y xy (ii) 2222
22
10,,1uuxy xyxy
NOTE NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 493 (iii) 222222
22
15240.uuuxxxxtxt
Solution:
(i) Comparing this equation with (1) above, we find that t 2A 1, B 4, C 4
B 2 - 4AC (4) 2 - 4 × 1 × 4 0So 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 positiveFor all y between - 1 and 1, y
2 < 1 B 2 - 4AC < 0Hence the equation is elliptic
(iii) Here A = 1 + x 2 , B = 5 + 2x 2 , C = 4 + x 2 B 2quotesdbs_dbs2.pdfusesText_2[PDF] 2004 ap computer science free response answers
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