1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

398 Chapter 7 Numerical Methods

x:0. 00. 20. 40. 60. 81. 0

u(x):1. 00. 643 0. 302 − 0. 026 − 0. 406 − 1. 0

Table 1 Approximate solution of Eqs. (1) and (2)

Differential equation Boundary condition

u(x)→ui u( 0 )→u 0

d^2 u

dx^2 (x)→

ui+ 1 − 2 ui+ui− 1

(   x)^2

du

dx(^0 )→

u 1 −u− 1

2    x

du

dx(x)→

ui+ 1 −ui− 1

2    x u(^1 )→un

f(x)→f(xi) dudx( 1 )→un+^12 − xun−^1

Table 2 Constructing replacement equations

First, the values ofxfor the table will be uniformly spaced across the interval
0 ≤x≤1, which we assume to be the interval of the boundary value problem

xi=i x, x=

1

n.

These are calledmeshpoints. The numbers approximating the values ofuare

ui∼=u(xi), i= 0 , 1 ,...,n.

These numbers are required to satisfy a set of equations obtained from the
boundary value problem by making the replacements shown in Table 2. The
entryf(x)refers to any coefficient or inhomogeneity in the differential equa-
tion.

Example.
The boundary value problem in Eqs. (1) and (2) would be replaced by the
algebraic equations

ui+ 1 − 2 ui+ui− 1

(   x)^2

− 12 xiui=− 1 , i= 1 , 2 ,...,n− 1 , (3)

u 0 = 1 , un=− 1. (4)

Equation (3) holds fori= 1 ,...,n−1, so the unknownsu 1 ,...,un− 1 would
be determined by this set of equations. The equations become specific when we
choosen.Letustaken=5, so x= 1 /5, and the four(i= 1 , 2 , 3 , 4 )versions