Solving singularly perturbed delay differential equations via fitted mesh and exact difference method

Figures & data

Figure 1. Effect of the delay parameter $(\delta )$ on solution: (a) Example 4.1 and (b) Example 4.2 for $\epsilon =2^{-4}$.

Figure 2. Formation of the boundary layer as $\epsilon$ goes small in (a) Example 4.1 in (b) Example 4.2.

Figure 3. Example 4.1, Layer resolving property of the schemes for $\epsilon =2^{-10}$: on (a) Scheme I, on (b) Scheme III for $N=2^6$

Figure 4. Layer resolving property of the schemes for $\epsilon =2^{-10}$ Example 4.2: in (a) Scheme I, in (b) Scheme III for $N=2^6$

Figure 5. Example 4.1, absolute error of the schemes for different mesh numbers on (a) Scheme I, on (b) Scheme II and on (c) Scheme III for $\epsilon =2^{-20}$

Figure 6. Example 4.2, absolute error of the schemes for different mesh numbers on (a) Scheme I, on (b) Scheme II and on (c) Scheme III for $\epsilon =2^{-20}$

Table 1. Example 4.1, maximum absolute error of Scheme II in (27) for $\delta =0.5\epsilon$

$\epsilon \downarrow$	N $\to$ $2^5$	$2^6$	$2^7$	$2^8$	$2^9$	$2^{10}$
$2^{-1}$	4.4251e-03	2.2784e-03	1.1565e-03	5.8257e-04	2.9238e-04	1.4646e-04
$2^{-4}$	3.2208e-02	2.0620e-02	1.2640e-02	7.1665e-03	3.6525e-03	1.8441e-03
$2^{-8}$	3.1420e-02	1.9795e-02	1.2021e-02	7.0567e-03	4.0454e-03	2.2756e-03
$2^{-12}$	3.8756e-02	2.4129e-02	1.4568e-02	8.5257e-03	4.8728e-03	2.7339e-03
$2^{-16}$	3.8722e-02	2.4105e-02	1.4552e-02	8.5148e-03	4.8664e-03	2.7306e-03
$2^{-20}$	3.8720e-02	2.4103e-02	1.4551e-02	8.5140e-03	4.8658e-03	2.7302e-03
$2^{-24}$	3.8720e-02	2.4103e-02	1.4551e-02	8.5139e-03	4.8657e-03	2.7302e-03
$2^{-30}$	3.8720e-02	2.4103e-02	1.4551e-02	8.5139e-03	4.8657e-03	2.7302e-03
$E^N$	3.8756e-02	2.4129e-02	1.4568e-02	8.5257e-03	4.8728e-03	2.7339e-03
$r^N$	0.6837	0.7279	0.7729	0.8071	0.8338	-

Table 2. Example 4.1, maximum absolute error of Scheme III in (30) for $\delta =0.5\epsilon$

$\epsilon \downarrow$	N $\to$ $2^5$	$2^6$	$2^7$	$2^8$	$2^9$	$2^{10}$
$2^{-1}$	5.4777e-04	2.8329e-04	1.4398e-04	7.2573e-05	3.6432e-05	1.8252e-05
$2^{-4}$	5.4013e-04	3.0846e-04	1.1991e-04	8.7367e-05	4.3864e-05	1.9777e-05
$2^{-8}$	6.5860e-04	1.7524e-04	1.8706e-04	5.4203e-05	5.3422e-05	2.1310e-05
$2^{-12}$	9.9983e-04	5.0405e-04	2.5259e-04	1.2650e-04	6.3418e-05	3.2934e-05
$2^{-16}$	9.8946e-04	4.9744e-04	2.4859e-04	1.2417e-04	6.2046e-05	3.1014e-05
$2^{-20}$	9.8881e-04	4.9702e-04	2.4834e-04	1.2402e-04	6.1963e-05	3.0968e-05
$2^{-24}$	9.8877e-04	4.9700e-04	2.4832e-04	1.2402e-04	6.1958e-05	3.0965e-05
$2^{-30}$	9.8876e-04	4.9700e-04	2.4832e-04	1.2401e-04	6.1958e-05	3.0965e-05
$E^N$	9.9983e-04	5.0405e-04	2.5259e-04	1.2650e-04	6.3418e-05	3.2934e-05
$r^N$	0.9881	0.9968	0.9977	0.9962	0.9453	-

Table 3. Example 4.1, maximum absolute error of Scheme III in (30) for different values of $\delta$ with $\epsilon =2^{-20}$

$\delta \downarrow ,$	$N\to 2^5$	$2^6$	$2^7$	$2^8$	$2^9$	$2^{10}$
$0$	9.8881e-04	4.9702e-04	2.4834e-04	1.2402e-04	6.1963e-05	3.0968e-05
$0.1\epsilon$	9.8468e-04	4.9622e-04	2.4820e-04	1.2400e-04	6.1960e-05	3.0968e-05
$0.3\epsilon$	9.8305e-04	4.9553e-04	2.4803e-04	1.2397e-04	6.1954e-05	3.0967e-05
$0.5\epsilon$	9.8848e-04	4.9616e-04	2.4809e-04	1.2397e-04	6.1954e-05	3.0967e-05
$0.7\epsilon$	9.9791e-04	4.9776e-04	2.4836e-04	1.2402e-04	6.1963e-05	3.0969e-05

Table 4. Example 4.1, comparison of maximum absolute error of Scheme III in (30) with results in (Kadalbajoo & Ramesh, 2007; Kumar & Kadalbajoo, 2012) and (Woldaregay & Duressa, 2021a)

$\epsilon \downarrow$	N $\to$ $2^5$	$2^6$	$2^7$	$2^8$	$2^9$	$2^{10}$
	Scheme III in (30)
$2^{-16}$	9.8946e-04	4.9744e-04	2.4859e-04	1.2417e-04	6.2046e-05	3.1014e-05
$2^{-20}$	9.8881e-04	4.9702e-04	2.4834e-04	1.2402e-04	6.1963e-05	3.0968e-05
$2^{-24}$	9.8877e-04	4.9700e-04	2.4832e-04	1.2402e-04	6.1958e-05	3.0965e-05
	Result in (Kumar & Kadalbajoo, 2012)
$2^{-16}$	3.6701e-02	1.2463e-02	4.1666e-03	1.4086e-03	4.7513e-04	1.5946e-04
$2^{-20}$	3.6709e-02	1.2478e-02	4.1763e-03	1.3898e-03	4.5369e-04	1.5379e-04
$2^{-24}$	3.6710e-02	1.2479e-02	4.1785e-03	1.3936e-03	4.5637e-04	1.4895e-04
	Method 1 in (Kadalbajoo & Ramesh, 2007)
$2^{-16}$	3.19e-2	1.90e-2	1.10e-2	6.21e-3	3.47Ee-3	1.93e-3
$2^{-20}$	3.19e-2	1.90e-2	1.09e-2	6.19e-3	3.46e-3	1.91e-3
$2^{-24}$	3.19e-2	1.90e-2	1.09e-2	6.19e-3	3.46e-3	1.91e-3
	Method 2 in (Kadalbajoo & Ramesh, 2007)
$2^{-16}$	3.39e-2	1.99e-2	1.14e-2	6.41e-3	3.57e-3	1.97e-3
$2^{-20}$	3.39e-2	1.99e-2	1.14e-2	6.39e-3	3.56e-3	1.96e-3
$2^{-24}$	3.39e-2	1.99e-2	1.14e-2	6.39e-3	3.56e-3	1.96e-3
	Method 3 in (Kadalbajoo & Ramesh, 2007)
$2^{-16}$	9.11e-3	3.84e-3	1.68e-3	7.71e-4	3.72e-4	1.89e-4
$2^{-20}$	9.09e-3	3.82e-3	1.66e-3	7.51e-4	3.52e-4	1.69e-4
$2^{-24}$	9.08e-3	3.82e-3	1.66e-3	7.50e-4	3.50e-4	1.68e-4
	Result in (Woldaregay & Duressa, 2021a)
$2^{-16}$	1.0745e-03	5.5307e-04	2.8038e-04	1.4117e-04	7.0825e-05	3.5473e-05
$2^{-20}$	1.0745e-03	5.5307e-04	2.8039e-04	1.4117e-04	7.0826e-05	3.5473e-05
$2^{-24}$	1.0745e-03	5.5307e-04	2.8039e-04	1.4117e-04	7.0826e-05	3.5473e-05

Table 5. Example 4.2, maximum absolute error of Scheme II in (27) for $\delta =0.5\epsilon ,$ for $\sigma =1$

$\epsilon \downarrow$	N $\to 2^5$	$2^6$	$2^7$	$2^8$	$2^9$	$2^{10}$
$2^{-1}$	4.9803e-03	2.5564e-03	1.2951e-03	6.5187e-04	3.2703e-04	1.6379e-04
$2^{-4}$	2.7963e-02	1.8022e-02	1.1082e-02	6.2982e-03	3.2111e-03	1.6215e-03
$2^{-8}$	3.0497e-02	1.9057e-02	1.1510e-02	6.7282e-03	3.8424e-03	2.1557e-03
$2^{-12}$	3.0453e-02	1.9034e-02	1.1513e-02	6.7419e-03	3.8552e-03	2.1638e-03
$2^{-16}$	3.0438e-02	1.9021e-02	1.1502e-02	6.7348e-03	3.8509e-03	2.1616e-03
$2^{-20}$	3.0437e-02	1.9020e-02	1.1502e-02	6.7342e-03	3.8504e-03	2.1613e-03
$2^{-24}$	3.0437e-02	1.9020e-02	1.1502e-02	6.7342e-03	3.8504e-03	2.1612e-03
$2^{-30}$	3.0437e-02	1.9020e-02	1.1502e-02	6.7342e-03	3.8504e-03	2.1612e-03
$E^N$	3.0453e-02	1.9034e-02	1.1513e-02	6.7419e-03	3.8552e-03	2.1638e-03
$r^N$	0.6780	0.7253	0.7720	0.8064	0.8333	-

Table 6. Example 4.2, maximum absolute error of Scheme III in (30) for $\delta =0.5\epsilon$

$\epsilon \downarrow$	N $\to 2^5$	$2^6$	$2^7$	$2^8$	$2^9$	$2^{10}$
$2^{-1}$	7.5977e-04	3.7960e-04	1.8981e-04	9.4937e-05	4.7478e-05	2.3742e-05
$2^{-4}$	1.3153e-03	4.9853e-04	1.9598e-04	1.0356e-04	5.3303e-05	2.6682e-05
$2^{-8}$	1.1163e-03	5.1315e-04	1.4435e-04	1.2709e-04	5.0211e-05	2.4331e-05
$2^{-12}$	1.3840e-03	6.0117e-04	2.7968e-04	1.3440e-04	6.3466e-05	2.4170e-05
$2^{-16}$	1.3861e-03	6.0216e-04	2.8020e-04	1.3511e-04	6.6747e-05	3.3305e-05
$2^{-20}$	1.3862e-03	6.0223e-04	2.8023e-04	1.3513e-04	6.6596e-05	3.3220e-05
$2^{-24}$	1.3862e-03	6.0223e-04	2.8023e-04	1.3513e-04	6.6586e-05	3.3215e-05
$2^{-30}$	1.3862e-03	6.0223e-04	2.8023e-04	1.3513e-04	6.6586e-05	3.3215e-05
$E^N$	1.3862e-03	6.0223e-04	2.8023e-04	1.3513e-04	6.6586e-05	3.3215e-05
$r^N$	1.2027	1.1037	1.0523	1.0211	1.0034	-

Table 7. Example 4.2, maximum absolute error of Scheme III in (30) for different values of $\delta$ with $\epsilon =2^{-20}$

$\delta \downarrow ,$	$N=2^5$	$2^6$	$2^7$	$2^8$	$2^9$	$2^{10}$
$0$	1.3862e-03	6.0223e-04	2.8023e-04	1.3513e-04	6.6545e-05	3.3210e-05
$0.1\epsilon$	1.3862e-03	6.0223e-04	2.8023e-04	1.3513e-04	6.6545e-05	3.3210e-05
$0.3\epsilon$	1.3862e-03	6.0223e-04	2.8023e-04	1.3513e-04	6.6545e-05	3.3210e-05
$0.5\epsilon$	1.3862e-03	6.0223e-04	2.8023e-04	1.3513e-04	6.6545e-05	3.3210e-05
$0.7\epsilon$	1.3862e-03	6.0223e-04	2.8023e-04	1.3513e-04	6.6545e-05	3.3210e-05

Kadalbajoo, M. K., & Ramesh, V. P. (2007). Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy. Applied Mathematics and Computation, 188(2), 1816–1831. https://doi.org/10.1016/j.amc.2006.11.046

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Kumar, D., & Kadalbajoo, M. K. (2012). Numerical treatment of singularly perturbed delay differential equations using B-Spline collocation method on Shishkin mesh. Jnaiam, 7(3–4), 73–90.

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Woldaregay, M. M., & Duressa, G. F. (2021a). Robust mid-point upwind scheme for singularly perturbed delay differential equations. Computational and Applied Mathematics, 40(5), 1–12. https://doi.org/10.1007/s40314-021-01569-5

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