Rational Biparameter Homotopy Perturbation Method and Laplace-Padé Coupled Version

Table 2

Comparison numerical solution (RKF45) for (7.1) and its first order approximate solutions given by RBHPM (7.8), LPRBHPM (7.11), HPM (7.15), and HPM (7.16).

Order		[2,2]	[2,2]	2	4
𝑡	RKF45 for (7.1)	RBHPM (7.8)	LPRBHPM (7.11)	HPM (7.15)	HPM (7.16)
0.00	0.000000000000	0.000000000000	−2.04249011 (−13)	0.000000000000	0.000000000000
0.10	0.094996749389	0.0963305722762	0.0963305680033	0.095001281746	0.0949968616125
0.20	0.179964338949	0.181413957989	0.181413692649	0.180042107937	0.179964612584
0.30	0.254886117960	0.255755918036	0.255752978888	0.255328628571	0.254889069792
0.40	0.319824133934	0.319864038194	0.319847953401	0.321424660317	0.319850515572
0.50	0.374962243912	0.374261968751	0.374202151797	0.379476686508	0.3751195795
0.60	0.420620714410	0.419505461842	0.419331298484	0.431478857143	0.421336349604
0.70	0.457249830610	0.456198011306	0.455769940137	0.480583988889	0.459927483684
0.80	0.485408608033	0.485004250862	0.484075378004	0.531466565079	0.49405125938
0.90	0.505735525317	0.506659764874	0.504828319804	0.590743735714	0.53057647436
1.00	0.518917385708	0.521976574206	0.518630236646	0.66746031746	0.583931084348
1.10	0.525660073662	0.531844221386	0.526097880422	0.773643793651	0.683143422756
1.20	0.526664476070	0.537227026265	0.527855737463	0.924935314286	0.884097382101
1.30	0.522608206722	0.539158647966	0.524527343795	1.14130269603	1.2900013214
1.40	0.514133103782	0.538735520529	0.516726377800	1.44784142222	2.08441286596
1.50	0.501837942107	0.537111006964	0.505048313285	1.87566964286	3.58297069111
1.60	0.486275223293	0.535492254682	0.490063208382	2.4629231746	6.31238329453
1.70	0.467950980699	0.535141785803	0.472309972142	3.25585650079	11.128360908
1.80	0.447326548414	0.537385900716	0.452292231487	4.31005577143	19.3882241197
1.90	0.424821677896	0.54363212026	0.430475743238	5.69176980317	33.1990855168
2.00	0.400817960941	0.555398274173	0.407287171967	7.47936507937	55.769016169
2.10	0.375662771298	0.574356631824	0.383113985298	9.76491075	91.8967515218
2.20	0.349672525187	0.602397906631	0.358305196653	12.6558996317	148.645576503
2.30	0.323136129463	0.641722425466	0.333172699524	16.2771112079	236.259417439
2.40	0.296317411714	0.694969880802	0.307992974314	20.7726226286	369.394267712
2.50	0.269457558120	0.765406030734	0.28300899685	26.3079737103	568.756347393
2.60	0.242776634418	0.857196643743	0.258432227399	33.0724919365	863.260364544
2.70	0.216475028642	0.975820118585	0.234444603854	41.2817834571	1292.84749061
2.80	0.190734203918	1.12870912808	0.211200498849	51.1803960889	1912.13383476
2.90	0.165717437177	1.32628670185	0.188828626412	63.0446603151	2795.09702573
3.00	0.141570064350	1.58371504978	0.167433899647	77.1857142857	4041.05178634
	A.A.E.	0.231	0.00641	12.1	401