1.2 Forcing Function 

Let us consider another example with the external forcing function is a concentrated loading, according to section (2), we use Dirac delta function to represent the loading.

According to (3.2) and (3.3), we decompose our problem into the following problems,

and

The same as the first example, we only have the particular solution  . And also we have the same Green's function as (1.1.11), and according to (2.9), then we can have

Therefore, the exact solution of the problem (1.2.1) is

Using the same notations as (1.1.13) and (1.1.14), the finite difference method for the problem (1.2.1) can be done in the following procedure. Consider

By using the property of (2.5), we have

We can use the method of finite difference to approximate the derivative and trapezoidal rule to approximate the integral in (1.2.6). Then, we could have

Since we can use the average of  and  and the average of  and  to approximate  and  respectively. Then, we have the following difference equation

Imposing the boundary conditions, we can rewrite (1.2.8) to

In matrix form, we have

where A is an order (n-1) square matrix, u and b are  column matrices, such that

TABLE 2.1

h=1/4	.10324725x10-2	-.2.1182702x10-5	.13191358x10-8	-.22654101x10-12
h=1/8	.25652942x10-3	-.1.3115520x10-6	.20388496x10-10	-.17763568x10-14
h=1/16	.64033988x10-4	-.8.1780855x10-8	.31682990x10-12	.41355808x10-14
h=1/32	.16002363x10-4	-.51083332x10-9	.90205621x10-14	-.82989171x10-14
h=1/64	.40002077x10-5	-.31918634x10-10	-.80213614x10-14
h=1/128	.10000280x10-5	-.20024260x10-11
h=1/256	.25000550x10-6

 
 

TABLE 2.2

r(1)	.24846126	.061916179	.015455950	.0078412154
r(2)	.24961655	.062354262	.015539640
r(3)	.24990421	.062463681	.028471310
r(4)	.24997606	.062483462
r(5)	.24999402	.062735328
r(6)	.24999850

 

For the problem (1.2.1), the forcing function is concentrated at the nodal point x = 1/2 where leads to a singular point, the Table 2.1 and Table 2.2 display the numerical results at this point. The same as problem (1.1.1), we see from Table 2.1 and Table 2.2 that we have a similar result at this singular point. The numerical experimental solutions show that at any nodal point, we have the same asymptotic error expansion as (1.1.19) [See Appendix {B.2a-B.2b]. In addition, when h = 1/256, the finite difference solution gives the error at x = 1/2  which is larger than the error of the extrapolated solution obtained from the three solutions corresponding to h = 1/4, 1/8 and 1/16. Although the forcing function can be concentrated at any other nodal points, the numerical solutions also show that we can obtain the same asymptotic error expansion [See Appendix B.2c-B.2h].

Wed Sep 15 10:03:39 HKT 1999