Theory and Modern Applications
Table 2 Solution and residual error for Example 3 when \(\alpha =0.7\) and \(\beta =0.5\) at several values of x and t using 60 terms of the series (\(\hbar =-0.85\))
From: Homotopy-Sumudu transforms for solving system of fractional partial differential equations
x | t | ||||||
|---|---|---|---|---|---|---|---|
0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 | ||
0 | u | 1.0000 | 1.4041 | 1.8025 | 2.3377 | 3.1298 | 4.4451 |
Eu | 0 | 2.220 10−16 | 1.776 10−15 | 3.183 10−12 | 3.231 10−9 | 7.500 10−6 | |
0.25 | u | 1.2840 | 1.8145 | 2.3303 | 3.0091 | 3.9948 | 5.6054 |
Eu | 1.110 10−16 | 1.110 10−16 | 1.887 10−15 | 3.581 10−12 | 3.783 10−9 | 1.576 10−5 | |
0.5 | u | 1.6487 | 2.3416 | 3.0081 | 3.8712 | 5.1055 | 7.0953 |
Eu | 0 | 3.331 10−16 | 3.109 10−15 | 5.343 10−12 | 8.557 10−9 | 2.179 10−5 | |
0.75 | u | 2.1170 | 3.0183 | 3.8783 | 4.9782 | 6.5317 | 9.0084 |
Eu | 0 | 4.4415 10−16 | 4.774 10−15 | 3.638 10−12 | 1.973 10−8 | 3.162 10−5 | |
1 | u | 2.7183 | 3.8872 | 4.9956 | 6.3995 | 8.3629 | 11.465 |
Eu | 0 | 1.110 10−16 | 6.661 10−15 | 2.615 10−12 | 3.097 10−8 | 4.801 10−5 | |
0 | v | 1.0000 | 0.60517 | 0.52571 | 0.51761 | 0.57958 | 0.76370 |
Ev | 2.220 10−16 | 0 | 5.551 10−16 | 2.785 10−12 | 5.413 10−9 | 1.654 10−6 | |
0.25 | v | 0.77880 | 0.46439 | 0.40597 | 0.40975 | 0.47746 | 0.65966 |
Ev | 3.331 10−16 | 2.220 10−16 | 2.220 10−16 | 3.979 10−13 | 1.164 10−10 | 3.025 10−6 | |
0.5 | v | 0.60653 | 0.35474 | 0.31271 | 0.32574 | 0.39793 | 0.57863 |
Ev | 2.220 10−16 | 3.331 10−16 | 4.441 10−16 | 7.390 10−13 | 1.397 10−9 | 7.093 10−6 | |
0.75 | v | 0.47237 | 0.26935 | 0.24008 | 0.26032 | 0.33599 | 0.51552 |
Ev | 0 | 5.551 10−16 | 0 | 1.137 10−13 | 9.662 10−9 | 1.100 10−5 | |
1 | v | 0.36788 | 0.20285 | 0.18352 | 0.20937 | 0.28775 | 0.46638 |
Ev | 0 | 6.661 10−16 | 2.220 10−16 | 2.274 10−13 | 7.916 10−9 | 1.256 10−5 | |