1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

402 Chapter 7 Numerical Methods

Thus, the values ofuatx 0 ,x 1 ,...,xn, which satisfy Eq. (19) exactly, will nearly
satisfy Eq. (20); vice versa, the numbersu 0 ,u 1 ,...,un, which satisfy the re-
placement equations (20), nearly satisfy Eq. (19). It can be proved that the
calculated numbersu 0 ,u 1 ,...,undo indeed approach the appropriate values
ofu(xi)as xapproaches 0 (under continuity and other conditions onk(x),
p(x),f(x)).

EXERCISES

1.Set up and solve replacement equations withn=4 for the problem

d^2 u

dx^2 =−^1 ,^0 <x<^1 ,

u( 0 )= 0 , u( 1 )= 1.

2.Solve the problem of Exercise 1 analytically. On the basis of Eqs. (15)

and (16), explain why the numerical solution agrees exactly with the ana-

lytical solution.

3.Set up and solve replacement equations withn=4 for the problem

d^2 u

dx^2 −u=−^2 x,^0 <x<^1 ,

u( 0 )= 0 , u( 1 )= 1.

4.Solve the problem in Exercise 3 analytically, and compare the numerical

results with the true solution.

5.Set up and solve replacement equations withn=4 for the problem

d^2 u

dx^2 =x,^0 <x<^1 ,

u( 0 )−du

dx

( 0 )= 1 , u( 1 )= 0.

6.Solve the problem in Exercise 5 analytically, and compare the numerical

results with the true solution.

7.Set up and solve replacement equations for the problem

d^2 u

dx^2 +^10 u=^0 ,^0 <x<^1 ,

u( 0 )= 0 , u( 1 )=− 1.