Question 11.3.2: Apply Richardson’s extrapolation to approximate the solution...
Apply Richardson’s extrapolation to approximate the solution to the boundary-value problem
y^{\prime \prime}=-\frac{2}{x} y^{\prime}+\frac{2}{x^{2}} y+\frac{\sin (\ln x)}{x^{2}}, \quad \text { for } 1 \leq x \leq 2, \text { with } y(1)=1 \text { and } y(2)=2,
using h = 0.1, 0.05, and 0.025.
The results are listed in Table 11.4. The first extrapolation is
\operatorname{Ext}_{1 i}=\frac{4 w_{i}(h=0.05)-w_{i}(h=0.1)}{3}
the second extrapolation is
\operatorname{Ext}_{2 i}=\frac{4 w_{i}(h=0.025)-w_{i}(h=0.05)}{3};
and the final extrapolation is
\operatorname{Ext}_{3 i}=\frac{16 \operatorname{Ext}_{2 i}- Ext _{1 i}}{15}
Table 11.4
| x_{i} | w_{i}(h=0.05) | w_{i}(h=0.025) | \operatorname{Ext}_{1 i} | Ext _{2 i} | EXt _{3 i} |
| 1.0 | 1.00000000 | 1.00000000 | 1.00000000 | 1.00000000 | 1.00000000 |
| 1.1 | 1.09262207 | 1.09262749 | 1.09262925 | 1.09262930 | 1.09262930 |
| 1.2 | 1.18707436 | 1.18708222 | 1.18708477 | 1.18708484 | 1.18708484 |
| 1.3 | 1.28337094 | 1.28337950 | 1.28338230 | 1.28338236 | 1.28338236 |
| 1.4 | 1.38143493 | 1.38144319 | 1.38144589 | 1.38144595 | 1.38144595 |
| 1.5 | 1.48114959 | 1.48115696 | 1.48115937 | 1.48115941 | 1.48115942 |
| 1.6 | 1.58238429 | 1.58239042 | 1.58239242 | 1.58239246 | 1.58239246 |
| 1.7 | 1.68500770 | 1.68501240 | 1.68501393 | 1.68501396 | 1.68501396 |
| 1.8 | 1.78889432 | 1.78889748 | 1.78889852 | 1.78889853 | 1.78889853 |
| 1.9 | 1.89392740 | 1.89392898 | 1.89392950 | 1.89392951 | 1.89392951 |
| 2.0 | 2.00000000 | 2.00000000 | 2.00000000 | 2.00000000 | 2.00000000 |
The values of w_{i}(h=0.1) are omitted from the table to save space, but they are listed in Table 11.3. The results for w_{i}(h=0.025) are accurate to approximately 3 \times 10^{-6} . However, the results of \operatorname{Ext}_{3 i} are correct to the decimal places listed. In fact, if sufficient digits had been used, this approximation would agree with the exact solution with maximum error of 6.3 \times 10^{-11} at the mesh points, an impressive improvement.
Table 11.3
\begin{array}{cccc}\hline x_{i} & w_{i} & y\left(x_{i}\right) & \left|w_{i}-y\left(x_{i}\right)\right| \\\hline 1.0 & 1.00000000 & 1.00000000 & \\1.1 & 1.09260052 & 1.09262930 & 2.88 \times 10^{-5} \\1.2 & 1.18704313 & 1.18708484 & 4.17 \times 10^{-5} \\1.3 & 1.28333687 & 1.28338236 & 4.55 \times 10^{-5} \\1.4 & 1.38140205 & 1.38144595 & 4.39 \times 10^{-5} \\1.5 & 1.48112026 & 1.48115942 & 3.92 \times 10^{-5} \\1.6 & 1.58235990 & 1.58239246 & 3.26 \times 10^{-5} \\1.7 & 1.68498902 & 1.68501396 & 2.49 \times 10^{-5} \\1.8 & 1.78888175 & 1.78889853 & 1.68 \times 10^{-5} \\1.9 & 1.89392110 & 1.89392951 & 8.41 \times 10^{-6} \\2.0 & 2.00000000 & 2.00000000 & \\\hline\end{array}