Modified homotopy perturbation method and its application to analytical solitons of fractional-order Korteweg–de Vries equation

Beni-Suef University Journal of Basic and Applied Sciences

Table 3 Numerical results of problem three at an exact value \(\alpha = 1\)

\(t\)	Value of (x)	Numerical solution	Exact solution	Abs error
\(t = 0\)	− 0.04	2.668682966	2.668682966	0
	− 0.02	1.334341483	1.334341483	0
	0	0	0	0
	0.02	− 1.334341483	− 1.334341483	0
	0.04	− 2.668682966	− 2.668682966	0
\(t = 0.02\)	− 0.04	3.031965441	3.032594279	0.000628838
	− 0.02	1.515982720	1.516297139	0.000314419
	0	0	0	0
	0.02	− 1.515982720	− 1.516297139	0.000314419
	0.04	− 3.031965441	− 3.032594279	0.000628838
\(t = 0.04\)	− 0.04	3.499774890	3.511424956	0.011650066
	− 0.02	1.749887445	1.755712478	0.005825033
	0	0	0	0
	0.02	− 1.749887445	− 1.755712478	0.005825033
	0.04	− 3.499774890	− 3.511424956	0.011650066
\(t = 0.06\)	− 0.04	4.099780218	4.169817134	0.070036916
	− 0.02	2.049890109	2.084908567	0.035018458
	0	0	0	0
	0.02	− 2.049890109	− 2.084908567	0.035018458
	0.04	− 4.099780218	− 4.169817134	0.070036916
\(t = 0.08\)	− 0.04	4.859650332	5.132082627	0.272432295
	− 0.02	2.429825166	2.566041313	0.136216147
	0	0	0	0
	0.02	− 2.429825166	− 2.566041313	0.136216147
	0.04	− 4.859650332	− 5.132082627	0.272432295
\(t = 0.10\)	− 0.04	5.807054134	6.671707415	0.864653281
	− 0.02	2.903527067	3.335853708	0.432326641
	0	0	0	0
	0.02	− 2.903527067	− 3.335853708	0.432326641
	0.04	− 5.807054134	− 6.671707415	0.864653281

As occurred in the first two problems, the data in Table 3 indicate that the absolute error progressively becomes worse as the value of t increases. It was discovered that the error initially increases from 0 to an order that significantly increases from \(\times 10^{ - 2}\) to \(\times 10^{ - 1}\).