Tri-prong scheme for regularized long wave equation

Peer review under responsibility of University of Bahrain.

Figures & data

Figure 1d 4th- order approximation solution by present technique when h = 0.4476 in Example 4.1.

Figure 1e 4th-order approximation solution by standard VIM in Example 4.1.

Figure 4 Error analysis of solution of Eq. (13)(13) by finite difference method.

Figure 5a Absolute error for the 3rd-order approximation by standard VIM for u₃(x,t) in Example 4.3.

Figure 5b Absolute error for the 3rd-order approximation by present technique when h = 0.2712 in Example 4.3.

Figure 6 Error analysis of solution of Example 4.3 by finite difference method.

Figure 1a Absolute error for the 4th-order approximation by standard VIM in Example 4.1.

Figure 1b Absolute error for the 4th-order approximation by present technique when h = 0.4476 in Example 4.1.

Figure 1c Exact solution for Example 4.1.

Figure 1f The plot of numerical results for u₄(x,t) by standard VIM and present technique in comparison with the exact solution at t = 3 s, in Example 4.1.

Table

x	Exact solution	Approx. sol. k = 0.6	Approx. sol. k = 0.1	Approx. sol. k = 0.01
0.0	0.600000	0.600000	0.600000	0.600000
2.0	0.510146	0.511240	0.510120	0.510146
4.0	0.328115	0.310154	0.328784	0.328114
6.0	0.175583	0.174125	0.175142	0.175581
8.0	0.084973	0.084875	0.084453	0.084970
10.0	0.039146	0.034125	0.039145	0.039141

Figure 2 Error analysis of solution of Eq. (7)(7) by finite difference method.

Table

x	Exact solution	Approx. sol. k = 0.6	Approx. sol. k = 0.1	Approx. sol. k = 0.01
0.0	0.090000	0.090000	0.090000	0.090000
5.0	0.075418	0.075210	0.075124	0.075411
10.0	0.046803	0.046421	0.046812	0.046812
15.0	0.023980	0.023142	0.023921	0.023940
20.0	0.011112	0.011242	0.011231	0.011121
25.0	0.004912	0.001420	0.004221	0.004899

Figure 3a Absolute error for the 3rd-order approximation by standard VIM for u₃(x,t) in Example 4.2.

Figure 3b Absolute error for the 3rd-order approximation by present technique when h = 0.025 in Example 4.2.

Table

x	Exact solution	Approx. sol. k = 0.6	Approx. sol. k = 0.1	Approx. sol. k = 0.01
0.0	0.300000	0.300000	0.300000	0.300000
5.0	0.178117	0.176312	0.178112	0.178117
10.0	0.053475	0.052568	0.053445	0.053474
15.0	0.012753	0.011245	0.012714	0.012752
20.0	0.002872	0.005483	0.002845	0.002871
25.0	0.000638	0.001254	0.000636	0.000629
30.0	0.000141	0.002133	0.000147	0.000148

Figure 7 Travelling wave solution of Eq. (1)(1) for different values of parameters.