Figure 1.
Graphs of the proposed rational transformations , , with respect to in (a) and graphs of with respect to in (b) for the case of .
Figure 1.
Graphs of the proposed rational transformations , , with respect to in (a) and graphs of with respect to in (b) for the case of .
Figure 2.
Graphs of the proposed rational transformations , , compared with (: dotted line) and (: solid line).
Figure 2.
Graphs of the proposed rational transformations , , compared with (: dotted line) and (: solid line).
Figure 3.
Relative errors of , , for the end-point weakly singular integral , with and , in Example 2. Dotted horizontal lines indicate relative errors of for from top to bottom.
Figure 3.
Relative errors of , , for the end-point weakly singular integral , with and , in Example 2. Dotted horizontal lines indicate relative errors of for from top to bottom.
Figure 4.
Relative errors of , , for the end-point logarithmic singular integral , with , in Example 2. Dotted horizontal lines indicate relative errors of for from top to bottom.
Figure 4.
Relative errors of , , for the end-point logarithmic singular integral , with , in Example 2. Dotted horizontal lines indicate relative errors of for from top to bottom.
Figure 5.
Graphs of the proposed rational transformation
, with
, for
in (
a) and graphs of
, with
and
m chosen by (
39), for
in (
b) compared with the composite transformation
and the Doblaré-Gracia transformation
.
Figure 5.
Graphs of the proposed rational transformation
, with
, for
in (
a) and graphs of
, with
and
m chosen by (
39), for
in (
b) compared with the composite transformation
and the Doblaré-Gracia transformation
.
Figure 6.
Relative errors of , , for the CPV integral , for , , in Example 3. Dotted horizontal lines indicate relative errors of corresponding to the chosen values of .
Figure 6.
Relative errors of , , for the CPV integral , for , , in Example 3. Dotted horizontal lines indicate relative errors of corresponding to the chosen values of .
Figure 7.
Relative errors of , , for the CPV integral , for , , in Example 3. Dotted horizontal lines indicate relative errors of corresponding to the chosen values of .
Figure 7.
Relative errors of , , for the CPV integral , for , , in Example 3. Dotted horizontal lines indicate relative errors of corresponding to the chosen values of .
Table 1.
Relative errors of the N-point Gauss-Legendre quadrature rule corresponding to the existing transformations , and the presented transformation for the interior-point weakly singular integral , with and , in Example 1.
Table 1.
Relative errors of the N-point Gauss-Legendre quadrature rule corresponding to the existing transformations , and the presented transformation for the interior-point weakly singular integral , with and , in Example 1.
| | Existing Transformations | | Presented Transformation |
---|
| | | | | | |
---|
7 | 10 | | | | | |
20 | | | | | |
30 | | | | | |
40 | | | | | |
11 | 10 | | | | | |
20 | | | | | |
30 | | | | | |
40 | | | | | |
15 | 10 | | | | | |
20 | | | | | |
30 | | | | | |
40 | | | | | |
Table 2.
Relative errors of the N-point Gauss-Legendre quadrature rule corresponding to the existing transformations , and the presented transformation for the end-point weakly singular integral , with and , in Example 2.
Table 2.
Relative errors of the N-point Gauss-Legendre quadrature rule corresponding to the existing transformations , and the presented transformation for the end-point weakly singular integral , with and , in Example 2.
| | Existing Transformations | | Presented Transformation |
---|
| | | | | | |
---|
3 | 10 | | | | | |
20 | | | | | |
30 | | | | | |
5 | 10 | | | | | |
20 | | | | | |
30 | | | | | |
7 | 10 | | | | | |
20 | | | | | |
30 | | | | | |
9 | 10 | | | | | |
20 | | | | | |
30 | | | | | |
Table 3.
Relative errors of the N-point Gauss-Legendre quadrature rule corresponding to the existing transformations , and the presented transformation for the end-point logarithmic singular integral , with , in Example 2.
Table 3.
Relative errors of the N-point Gauss-Legendre quadrature rule corresponding to the existing transformations , and the presented transformation for the end-point logarithmic singular integral , with , in Example 2.
| | Existing Transformations | | Presented Transformation |
---|
| | | | | | |
---|
3 | 10 | | | | | |
20 | | | | | |
30 | | | | | |
5 | 10 | | | | | |
20 | | | | | |
30 | | | | | |
7 | 10 | | | | | |
20 | | | | | |
30 | | | | | |
9 | 10 | | | | | |
20 | | | | | |
30 | | | | | |
Table 4.
Relative errors of the N-point Gauss-Legendre quadrature rule corresponding to the Doblaré-Gracia transformation and the presented transformation for the CPV integral , , in Example 3.
Table 4.
Relative errors of the N-point Gauss-Legendre quadrature rule corresponding to the Doblaré-Gracia transformation and the presented transformation for the CPV integral , , in Example 3.
| | Existing Transformation | | Presented Transformation |
---|
| | | | | | |
---|
0.2 | 4 | | | | | |
8 | | | | | |
12 | | | | | |
16 | | | | | |
20 | | | | | |
0.4 | 4 | | | | | |
8 | | | | | |
12 | | | | | |
16 | | | | | |
20 | | | | | |
0.6 | 4 | | | | | |
8 | | | | | |
12 | | | | | |
16 | | | | | |
20 | | | | | |
Table 5.
Relative errors of the N-point Gauss-Legendre quadrature rule corresponding to the composite transformation and the presented transformation for the CPV integral , with ’s near the end-point, in Example 3.
Table 5.
Relative errors of the N-point Gauss-Legendre quadrature rule corresponding to the composite transformation and the presented transformation for the CPV integral , with ’s near the end-point, in Example 3.
| | Existing Transformation | | Presented Transformation |
---|
| | | | | | |
---|
0.90 | 16 | | | | | |
24 | | | | | |
32 | | | | | |
40 | | | | | |
48 | | | | | |
0.95 | 16 | | | | | |
24 | | | | | |
32 | | | | | |
40 | | | | | |
48 | | | | | |
0.99 | 16 | | | | | |
24 | | | | | |
32 | | | | | |
40 | | | | | |
48 | | | | | |
Table 6.
Values of
m evaluated by Equation (
39) for each
and
selected in
Table 5.
Table 6.
Values of
m evaluated by Equation (
39) for each
and
selected in
Table 5.
| | m |
---|
| | | | |
---|
0.90 | | 5 | 5 | 7 |
0.95 | | 7 | 11 | 13 |
0.99 | | 33 | 51 | 61 |