Non-polynomial septic spline method for singularly perturbed two point boundary value problems of order three

Figures & data

Table 1. Maximum absolute errors for Example 5.1 with different values of $\epsilon$ and $h$.

$\epsilon \downarrow$	$N=10$	$N=20$	$N=40$
Present method
$1/16$	2.8930e−04	5.3006e−06	2.6033e−08
$1/32$	1.0962e−04	1.9394e−06	1.3221e−08
$1/64$	3.8007e−05	6.8026e−07	6.2298e−09
Reference [5]
$1/16$	6.2854e−03	–	–
$1/32$	1.9707e−03	–	–
$1/64$	3.9065e−04	–	–
Reference [3]
$1/16$	1.3e−02	1.1e−03	7.8e−05
$1/32$	3.2e−03	2.7e−04	1.8e−05
$1/64$	3.4e−04	2.2e−05	1.1e−06

Table 2. Maximum absolute errors for Example 5.1 when $\epsilon <<h$.

$\epsilon \downarrow$	$N=10$	$N=50$	$N=100$	$N=150$	$N=200$
Present method
${10}^{-1}$	5.3363e−04	5.9813e−08	9.1841e−09	2.1617e−09	7.3808e−10
${10}^{-2}$	1.8773e−05	2.2337e−09	2.5972e−10	6.0042e−11	2.0403e−11
${10}^{-3}$	1.5441e−06	5.5630e−11	3.8697e−12	8.5417e−13	2.8902e−13
${10}^{-4}$	1.8248e−08	3.2291e−12	5.6471e−14	1.0585e−14	3.3903e−15
${10}^{-5}$	1.1853e−10	3.3754e−13	3.6503e−15	2.7729e−16	4.4615e−17
${10}^{-6}$	1.1362e−12	4.5181e−15	2.9874e−16	1.8873e−17	3.6840e−18
${10}^{-7}$	1.1314e−14	2.5989e−17	2.5748e−18	2.4307e−18	2.7110e−19
${10}^{-8}$	1.1309e−16	2.4638e−19	1.6575e−20	3.7102e−21	1.4808e−21
${10}^{-9}$	1.1309e−18	2.4508e−21	1.5883e−22	3.2062e−23	1.0424e−23
${10}^{-10}$	1.1309e−20	2.4495e−23	1.5815e−24	3.1605e−25	1.0074e−25
${10}^{-11}$	1.1309e−22	2.4490e−25	1.5805e−26	3.1550e−27	1.0035e−27
${10}^{-12}$	1.1307e−24	2.4469e−27	1.5764e−28	3.6037e−29	3.1729e−29
Reference [5]
${10}^{-1}$	1.6190e−02	7.3371e−04	6.4463e−04	6.3671e−04	6.3496e−04
${10}^{-2}$	5.4777e−04	3.5302e−05	3.2708e−05	3.3005e−05	3.3331e−05
${10}^{-3}$	4.3814e−05	2.4150e−06	1.3966e−06	1.1544e−06	1.2348e−06
${10}^{-4}$	7.5623e−06	2.4329e−07	1.1223e−07	7.6323e−08	6.1521e−08

Table 3. Maximum absolute errors for Example 5.2 with different values of $\epsilon$ and $h$.

$\epsilon \downarrow$	$N=10$	$N=20$	$N=40$
Present method
$1/16$	9.4405e−06	5.4886e−07	2.5658e−08
$1/32$	3.1645e−06	1.9215e−07	9.1282e−09
$1/64$	9.9920e−07	6.1969e−08	2.9364e−09
Reference [13]
$1/16$	1.02e−02	1.40e−03	1.73e−04
$1/32$	3.80e−03	4.84e−04	6.15e−05
$1/64$	1.40e−03	1.00e−04	2.00e−05
Reference [3]
$1/16$	2.5e−03	1.9e−04	1.4e−05
$1/32$	6.8e−04	5.7e−05	5.0e−06
$1/64$	1.2e−04	1.3e−05	1.6e−06

Table 4. Maximum absolute errors for Example 5.2 when $\epsilon <<h$.

$\epsilon \downarrow$	$N=10$	$N=50$	$N=100$	$N=150$	$N=200$
Present method
${10}^{-3}$	4.5809e−08	8.2990e−12	4.1392e−13	7.3211e−14	2.6225e−14
${10}^{-4}$	9.3573e−10	4.6893e−14	5.1063e−15	9.5009e−16	3.4922e−16
${10}^{-5}$	6.1195e−12	1.5171e−14	9.3190e−17	6.0495e−18	7.5145e−18
${10}^{-6}$	5.8655e−14	2.0745e−16	1.3348e−17	8.2310e−19	1.5064e−19
${10}^{-7}$	5.8409e−16	1.1931e−18	1.1543e−19	1.0764e−19	1.1465e−20
${10}^{-8}$	5.8386e−18	1.1336e−20	7.5708e−22	1.7702e−22	7.8479e−23
${10}^{-9}$	5.8398e−20	1.1575e−22	8.7888e−24	3.3087e−24	3.3087e−24
${10}^{-10}$	5.8564e−22	1.5574e−24	4.1359e−25	3.6189e−25	1.6529e−25
${10}^{-11}$	6.0132e−24	4.8872e−26	3.9081e−26	3.8774e−26	3.8774e−26
${10}^{-12}$	7.2701e−26	3.2817e−27	3.2312e−27	2.8273e−27	2.8273e−27
Reference [5]
${10}^{-3}$	2.5276e−03	1.7508e−04	7.0100e−05	3.6747e−05	2.0355e−05
${10}^{-4}$	1.9938e−03	2.5243e−05	1.1440e−05	7.3469e−06	5.3722e−06
${10}^{-5}$	…	2.0415e−05	1.3333e−06	8.3294e−07	6.0941e−07

Figure 1. Numerical solution versus exact solution of Examples 5.1 and 5.2 respectively for $N=30$ and $\epsilon ={10}^{-5}$.

Figure 2. Absolute errors of Example 5.1 for different values of $\epsilon$ and $h$.

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