Table 4.

Comparison between Caputo fractal fractional, Caputo-Fabrizio, ABC operator.

(x,t)	Exact	Caputo	$\mid Exact-Caputo\mid$	CF	$\mid Exact-CF\mid$	ABC	$\mid Exact-ABC\mid$
(− 3.0,0.05)	0.0187	0.0205	− 1.8  $\times {10}^{-03}$	0.0246	− 5.9  $\times {10}^{-03}$	0.0258	− 7.1 $\times {10}^{-4}$
(− 2.5,0.05)	0.0658	0.0674	− 1.6  $\times {10}^{-03}$	0.0713	− 5.5$\times {10}^{-03}$	0.0724	− 6.7 $\times {10}^{-04}$
(− 2.0,0.05)	0.1086	0.1100	− 1.5 $\times {10}^{-03}$	0.1134	− 4.9  $\times {10}^{-03}$	0.1144	− 5.9 $\times {10}^{-04}$
(− 1.5,0.05)	0.1448	0.1460	− 1.2 $\times {10}^{-03}$	0.1487	− 3.9 $\times {10}^{-03}$	0.1496	− 4.8 $\times {10}^{-04}$
(− 1.0,0.05)	0.1724	0.1732	− 8.3 $\times {10}^{-04}$	0.1752	− 2.8 $\times {10}^{-04}$	0.1757	− 3.4 $\times {10}^{-04}$
(− 0.5,0.05)	0.1896	0.1900	− 4.3 $\times {10}^{-04}$	0.1910	− 1.4 $\times {10}^{-04}$	0.1913	− 1.7 $\times {10}^{-04}$
(0.0,0.05)	0.1953	0.1953	− 2.7 $\times {10}^{-06}$	0.1953	− 2.7 $\times {10}^{-06}$	0.1953	− 2.7 $\times {10}^{-06}$
(0.5,0.03)	0.1892	0.1888	3.2 $\times {10}^{-04}$	0.1877	1.5 $\times {10}^{-04}$	0.1875	1.7 $\times {10}^{-04}$
(1.0,0.03)	0.1716	0.1710	6.2 $\times {10}^{-04}$	0.1688	2.8 $\times {10}^{-04}$	0.1683	3.3 $\times {10}^{-04}$
(1.5,0.03)	0.1437	0.1428	8.9 $\times {10}^{-04}$	0.1397	4.0 $\times {10}^{-04}$	0.1391	4.6 $\times {10}^{-04}$
(2.0,0.03)	0.1072	0.1061	1.1 $\times {10}^{-04}$	0.1022	5.0 $\times {10}^{-04}$	0.1015	5.7 $\times {10}^{-04}$
(2.5,0.03)	0.0642	0.0630	1.2 $\times {10}^{-03}$	0.0586	5.6 $\times {10}^{-03}$	0.0577	6.5 $\times {10}^{-04}$
(3.0,0.03)	0.0171	0.0157	1.3 $\times {10}^{-03}$	0.0111	6.0 $\times {10}^{-03}$	0.0101	6.9 $\times {10}^{-04}$