Accurate numerical method for Liénard nonlinear differential equations

Figures & data

Figure 1. Numerical solution versus exact solution for Example 4.1 when $h=0.05$.

Figure 2. Pointwise absolute errors for Example 4.1 with different values of $h$.

Table 1. The pointwise absolute errors for Example 4.1 with $x\in \left[0,0.5\right]$.

		Present method
$x$	Magnus series [15] h = 0.001	$h=0.1$	$h=0.1$ Refinement	$h=0.01$	$h=0.001$
0		0	0	0	0
0.1	1.4409e−04	2.6794e−07	2.6794e−07	1.2975e−11	4.4409e−16
0.2	1.15476e−03	5.7177e−07	5.7177e−07	1.6940e−11	4.4409e−16
0.3	3.90185e−03	8.8764e−07	8.8764e−07	2.1276e−11	1.3323e−15
0.4	9.2132e−03	1.1954e−06	1.1954e−06	2.4793e−11	1.3323e−15
0.5	1.78199e−02	1.2849e−06	1.2747e−06	2.6595e−11	1.6653e−15

Table 2. The maximum absolute errors for different values of h and $x\in [0,1]$.

$h=1/50$	$h=1/\text{1}00$	$h=1/200$	$h=1/300$	$h=1/400$	$h=1/500$
6.6333e−10	2.6704e−11	1.1988e−12	2.0595e−13	7.2942e−14	2.9088e−14
Refinement
6.7708e−10	2.8566e−11	1.3498e−12	2.3725e−13	6.9500e−14	2.8089e−14

Figure 3. Numerical solution versus exact solution for Example 4.2 when $h=0.05$.

Figure 4. Pointwise absolute errors for Example 4.2 with different values of $h$.

Table 3. The pointwise absolute errors for Example 4.2 with $x\in \left[0,1\right]$ and different values of h.

	Differential transform [16]	Hybrid heuristic computation [17]		Present method
$x$	$h=0.1$	IPA	ASA	$h=0.1$	$h=0.1$ Refinement	$h=0.01$	$h=0.002$
0.1	2.2683e−06	6.24e−06	7.30e−06	6.7141e−08	6.7141e−08	3.1564e−12	1.8874e−15
0.2	2.5222e−06	5.60e−06	7.55e−06	1.1401e−07	1.1401e−07	4.0805e−12	2.9976e−15
0.3	1.4404e−05	5.15e−06	7.75e−06	1.4010e−07	1.4010e−07	5.5097e−12	4.5519e−15
0.4	6.1806e−05	5.08e−06	7.88e−06	1.4855e−07	1.4855e−07	7.2031e−12	6.1062e−15
0.5	2.2099e−04	5.00e−06	7.95e−06	1.4131e−07	1.3523e−07	8.8091e−12	6.9944e−15
0.6	6.5187e−04	4.60e−06	8.06e−06	2.6414e−07	9.8888e−08	9.9460e−12	7.6605e−15
0.7	1.6408e−03	3.96e−06	8.32e−06	4.3610e−07	5.4195e−08	1.0276e−11	7.1054e−15
0.8	3.6469e−03	3.39e−06	8.73e−06	5.8126e−07	1.3746e−08	9.5440e−12	5.1070e−15
0.9	7.3493e−03	3.15e−06	9.18e−06	6.4934e−07	1.4832e−08	7.5914e−12	1.3323e−15
1.0	1.3692e−02	3.18e−06	9.65e−06	6.2114e−07	2.9187e−08	4.3318e−12	5.2180e−15

Table 4. The maximum absolute errors for Example 4.2 for different values of h and $x\in [0,1]$.

$h=1/50$	$h=1/\text{1}00$	$h=1/200$	$h=1/300$	$h=1/400$	$h=1/500$
3.5866e−10	1.0288e−11	3.9369e−13	6.5170e−14	1.8652e−14	7.6605e−15
Refinement
1.5083e−10	7.0087e−12	3.4439e−13	6.1284e−14	1.8208e−14	7.5495e−15

Figure 5. Numerical solution versus exact solution for Example 4.3 when $h=0.05$.

Table 5. The pointwise absolute errors for Example 4.3 with $x\in \left[0,1\right]$ and different values of h.

	Differential transform [16]	Hybrid heuristic computation [17]		Present method
$x$	$h=0.1$	IPA	ASA	$h=0.1$	$h=0.1$ Refinement	$h=0.01$	$h=0.002$
0.1	4.0668e−06	4.33e−05	1.64e−05	2.5685e−08	2.5685e−08	8.5355e−12	8.5487e−15
0.2	3.7000e−06	5.31e−05	2.22e−05	5.1433e−07	5.1433e−07	2.3057e−11	3.2085e−14
0.3	7.1793e−06	5.84e−05	2.55e−05	1.3397e−06	1.3397e−06	4.2467e−11	6.3172e−14
0.4	4.5002e−05	6.37e−05	2.21e−05	2.3254e−06	2.3254e−06	6.3473e−11	9.6034e−14
0.5	2.4676e−04	6.49e−05	1.54e−05	3.3161e−06	3.2340e−06	8.2096e−11	1.2235e−13
0.6	9.9922e−04	6.01e−05	1.11e−05	5.0076e−06	3.8973e−06	9.4586e−11	1.3622e−13
0.7	3.2168e−03	5.27e−05	1.09e−05	7.9779e−06	4.2577e−06	9.8288e−11	1.3467e−13
0.8	8.7371e−03	4.67e−05	1.13e−05	1.2097e−05	4.3703e−06	9.2047e−11	1.1663e−13
0.9	2.0841e−02	4.28e−05	7.47e−06	1.6871e−05	4.3323e−06	7.6092e−11	8.2989e−14
1.0	4.4891e−02	3.83e−05	9.00e−07	2.1802e−05	4.2279e−06	5.1619e−11	3.5860e−14

Table 6. The maximum absolute errors for different values of h and $x\in [0,1]$.

$h=1/50$	$h=1/\text{1}00$	$h=1/200$	$h=1/300$	$h=1/400$	$h=1/500$
2.4198e−09	1.1685e−10	6.4769e−12	1.2325e−12	3.8236e−13	1.5532e−13
Refinement
2.0705e−09	9.8358e−11	5.6958e−12	1.0956e−12	3.3984e−13	1.3756e−13

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