Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation

Mostafa M A Khater,S K Elagan,M A El-Shorbagy,S H Alfalqi,J F Alzaidi,Nawal A Alshehri

Table 2. Absolute errors between the analysis obtained using the MKud method and the constructed semi-analytical solution using the VI scheme for different values of x and the following values of the abovementioned parameters a0 = 1, k = e, r1 = −1, and r2 = −2.

Value of x t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 t = 8 t = 9 t = 10 0 0.000 761 9 0.004 856 691 0.035 343 152 0.236 875 694 0.997 209 591 1.757 184 901 1.957 641 682 1.986 335 208 1.987 919 89 1.985 454 506 1 0.000 280 247 0.001 789 003 0.013 152 735 0.094 285 709 0.536 850 64 1.460 485 799 1.902 784 858 1.983 386 09 1.993 952 21 1.994 397 482 2 0.000 103 091 0.000 658 453 0.004 859 343 0.035 763 763 0.238 025 35 0.999 398 599 1.760 722 794 1.962 837 217 1.993 496 363 1.997 353 871 3 3.792 45E-05 0.000 242 274 0.001 790 473 0.013 308 884 0.094 711 666 0.537 661 43 1.461 796 342 1.904 709 968 1.986 040 478 1.997 450 479 4 1.395 15E-05 8.913 34E-05 0.000 659 061 0.004 916 979 0.035 920 873 0.238 324 368 0.999 881 943 1.761 432 869 1.963 816 413 1.994 787 057 5 5.132 47E-06 3.279 11E-05 0.000 242 507 0.001 811 702 0.013 366 737 0.094 821 77 0.537 839 407 1.462 057 816 1.905 070 56 1.986 515 805 6 1.888 13E-06 1.206 33E-05 8.922 03E-05 0.000 666 874 0.004 938 269 0.035 961 392 0.238 389 864 0.999 978 169 1.761 565 573 1.963 991 345 7 6.946 04E-07 4.437 85E-06 3.282 33E-05 0.000 245 382 0.001 819 536 0.013 381 645 0.094 845 868 0.537 874 811 1.462 106 642 1.905 134 923 8 2.5553E-07 1.6326E-06 1.207 51E-05 9.027 79E-05 0.000 669 756 0.004 943 754 0.035 970 257 0.238 402 889 0.999 996 132 1.761 589 252 9 9.400 44E-08 6.005 99E-07 4.442 21E-06 3.321 23E-05 0.000 246 442 0.001 821 553 0.013 384 906 0.094 850 659 0.537 881 42 1.462 115 353 10 3.458 23E-08 2.209 48E-07 1.6342E-06 1.221 83E-05 9.066 79E-05 0.000 670 498 0.004 944 954 0.035 972 02 0.238 405 321 0.999 999 336 11 1.272 21E-08 8.128 22E-08 6.011 89E-07 4.494 87E-06 3.335 58E-05 0.000 246 715 0.001 821 995 0.013 385 555 0.094 851 554 0.537 882 599 12 4.6802E-09 2.990 21E-08 2.211 65E-07 1.653 57E-06 1.227 11E-05 9.076 84E-05 0.000 670 661 0.004 945 192 0.035 972 349 0.238 405 754 13 1.721 75E-09 1.100 04E-08 8.136 21E-08 6.083 15E-07 4.514 29E-06 3.339 28E-05 0.000 246 775 0.001 822 082 0.013 385 676 0.094 851 713 14 6.333 96E-10 4.0468E-09 2.993 14E-08 2.237 87E-07 1.660 72E-06 1.228 46E-05 9.079 04E-05 0.000 670 693 0.004 945 237 0.035 972 408 15 2.330 11E-10 1.488 73E-09 1.101 12E-08 8.232 66E-08 6.109 43E-07 4.519 29E-06 3.340 09E-05 0.000 246 786 0.001 822 099 0.013 385 697 16 8.572 03E-11 5.476 75E-10 4.050 78E-09 3.028 62E-08 2.247 54E-07 1.662 56E-06 1.228 76E-05 9.079 47E-05 0.000 670 699 0.004 945 245 17 3.1537E-11 2.0148E-10 1.4902E-09 1.114 17E-08 8.268 22E-08 6.1162E-07 4.520 38E-06 3.340 25E-05 0.000 246 789 0.001 822 102 18 1.160 18E-11 7.412 03E-11 5.482 14E-10 4.0988E-09 3.041 71E-08 2.250 03E-07 1.662 96E-06 1.228 82E-05 9.079 56E-05 0.000 670 7 19 4.269 03E-12 2.7268E-11 2.016 77E-10 1.507 86E-09 1.118 98E-08 8.277 38E-08 6.117 68E-07 4.5206E-06 3.340 28E-05 0.000 246 789 20 1.569 41E-12 1.003 02E-11 7.419 18E-11 5.547 12E-10 4.1165E-09 3.045 08E-08 2.250 57E-07 1.663 04E-06 1.228 83E-05 9.079 57E-05 21 5.768 72E-13 3.690 38E-12 2.729 33E-11 2.040 67E-10 1.514 38E-09 1.120 22E-08 8.279 39E-08 6.117 98E-07 4.520 64E-06 3.340 28E-05 22 2.144 95E-13 1.359 36E-12 1.004 26E-11 7.507 42E-11 5.5711E-10 4.121 07E-09 3.045 82E-08 2.250 68E-07 1.663 05E-06 1.228 83E-05 23 7.815 97E-14 5.004 89E-13 3.694 82E-12 2.761 79E-11 2.049 49E-10 1.516 06E-09 1.120 49E-08 8.279 78E-08 6.118 03E-07 4.520 65E-06 24 2.930 99E-14 1.838 53E-13 1.3598E-12 1.016 03E-11 7.539 75E-11 5.577 27E-10 4.122 06E-09 3.045 96E-08 2.2507E-07 1.663 06E-06 25 1.065 81E-14 6.750 16E-14 4.996E-13 3.737 45E-12 2.773 69E-11 2.051 76E-10 1.516 42E-09 1.120 55E-08 8.279 86E-08 6.118 04E-07 26 5.773 16E-15 2.620 13E-14 1.856 29E-13 1.376 23E-12 1.020 52E-11 7.548 18E-11 5.578 63E-10 4.122 26E-09 3.045 99E-08 2.2507E-07 27 3.552 71E-15 1.154 63E-14 6.9722E-14 5.084 82E-13 3.756 11E-12 2.776 93E-11 2.052 28E-10 1.5165E-09 1.120 56E-08 8.279 87E-08 28 8.881 78E-16 3.108 62E-15 2.4869E-14 1.860 73E-13 1.380 67E-12 1.021 49E-11 7.549 83E-11 5.578 88E-10 4.1223E-09 3.045 99E-08 29 1.776 36E-15 3.108 62E-15 1.065 81E-14 7.016 61E-14 5.0937E-13 3.759 66E-12 2.7776E-11 2.052 37E-10 1.516 51E-09 1.120 56E-08 30 −1.77636E-15 −1.33227E-15 1.776 36E-15 2.353 67E-14 1.851 85E-13 1.381 12E-12 1.021 63E-11 7.550 09E-11 5.578 91E-10 4.1223E-09