Figure 1.
The ℏ-curve corresponding to the 15th-order approximate series solution at and .
Figure 1.
The ℏ-curve corresponding to the 15th-order approximate series solution at and .
Figure 2.
Truncated series solution of Example 1 using several values of m.
Figure 2.
Truncated series solution of Example 1 using several values of m.
Figure 3.
The ℏ-curve corresponding to the 16th-order approximate series solution at and .
Figure 3.
The ℏ-curve corresponding to the 16th-order approximate series solution at and .
Figure 4.
Truncated series solution of Example 2 using several values of m.
Figure 4.
Truncated series solution of Example 2 using several values of m.
Figure 5.
The ℏ-curve corresponding to the 14th-order approximate series solution at and .
Figure 5.
The ℏ-curve corresponding to the 14th-order approximate series solution at and .
Figure 6.
Truncated series solution of Example 2 using several values of m.
Figure 6.
Truncated series solution of Example 2 using several values of m.
Table 1.
Comparative numerical results between q-HATM and LDM at , , and different values of k, x, and t.
Table 1.
Comparative numerical results between q-HATM and LDM at , , and different values of k, x, and t.
| | | | | t | | | |
---|
| | | 0.1 | | 1 | | 5 | |
---|
| | | | | | | | |
---|
0.1 | 1 | | −0.200720 | −0.226554 | 3.59117 | 2.2964 | 227.281 | 207.308 |
| | | 5.28214 | 4.91095 | 0.831902 | 3.58913 | −96.6100 | −38.8371 |
| | | | | | | | |
| 2 | | −0.226554 | −0.226554 | 2.29640 | 2.29640 | 207.308 | 207.308 |
| | | 4.91095 | 4.91095 | 3.58913 | 3.58913 | −38.8371 | −38.8371 |
| | | | | | | | |
| 20 | | −0.226554 | −0.226554 | 2.29640 | 2.29640 | 207.308 | 207.308 |
| | | 4.91095 | 4.91095 | 3.58913 | 3.58913 | −38.8371 | −38.8371 |
| | | | | | | | |
0.5 | 1 | | −0.0578117 | −0.0836456 | 1.60200 | 0.307238 | 89.5254 | 69.5526 |
| | | 1.89810 | 1.52691 | −0.368544 | 2.38868 | −94.5321 | −36.7592 |
| | | | | | | | |
| 2 | | −0.0836456 | −0.0836456 | 0.307238 | 0.307238 | 69.5526 | 69.5526 |
| | | 1.52691 | 1.52691 | 2.38868 | 2.38868 | −36.7592 | −36.7592 |
| | | | | | | | |
| 20 | | −0.0836456 | −0.0836456 | 0.307238 | 0.307238 | 69.5526 | 69.5526 |
| | | 1.52691 | 1.52691 | 2.38868 | 2.38868 | −36.7592 | −36.7592 |
| | | | | | | | |
0.9 | 1 | | −0.00561994 | −0.0314538 | 0.875532 | −0.419230 | 39.2154 | 19.2426 |
| | | 0.662211 | 0.291022 | −0.806961 | 1.95026 | −93.7732 | −36.0004 |
| | | | | | | | |
| 2 | | −0.0314538 | −0.0314538 | −0.419230 | −0.419230 | 19.2426 | 19.2426 |
| | | 0.291022 | 0.291022 | 1.95026 | 1.95026 | −36.0004 | −36.0004 |
| | | | | | | | |
| 20 | | −0.0314538 | −0.0314538 | −0.419230 | −0.419230 | 19.2426 | 19.2426 |
| | | 0.291022 | 0.291022 | 1.95026 | 1.95026 | −36.0004 | −36.0004 |
Table 2.
Comparative numerical results between q-HATM and LDM at , , and different values of k, x, and t.
Table 2.
Comparative numerical results between q-HATM and LDM at , , and different values of k, x, and t.
| | | | | t | | | |
---|
| | | 0.1 | | 0.5 | | 5 | |
---|
x | | | | | | | | |
---|
0.1 | 1 | | −0.0668385 | −0.226571 | 1.27317 | 0.602298 | 287.393 | 255.466 |
| | | 1.97995 | 5.76698 | 1.37805 | 2.27969 | −112.984 | −20.3642 |
| | | | | | | | |
| 2 | | −0.0695093 | −0.226571 | 0.602298 | 0.602298 | 255.466 | 255.466 |
| | | 1.73925 | 5.76698 | 2.27969 | 2.27969 | −20.3642 | −20.3642 |
| | | | | | | | |
| 20 | | −0.0695093 | −0.226571 | 0.602298 | 0.602298 | 255.466 | 255.466 |
| | | 1.73925 | 5.76698 | 2.27969 | 2.27969 | −20.3642 | −20.3642 |
| | | | | | | | |
0.5 | 1 | | −0.0668385 | −0.0695093 | 0.699324 | 0.0284545 | 112.934 | 81.0069 |
| | | 1.97995 | 1.73925 | 0.125798 | 1.02744 | −107.633 | −15.0128 |
| | | | | | | | |
| 2 | | −0.0695093 | −0.0695093 | 0.0284545 | 0.0284545 | 81.0069 | 81.0069 |
| | | 1.73925 | 1.73925 | 1.02744 | 1.02744 | −15.0128 | −15.0128 |
| | | | | | | | |
| 20 | | −0.0695093 | −0.0695093 | 0.0284545 | 0.0284545 | 81.0069 | 81.0069 |
| | | 1.73925 | 1.73925 | 1.02744 | 1.02744 | −15.0128 | −15.0128 |
| | | | | | | | |
0.9 | 1 | | −0.00947778 | −0.0121486 | 0.489749 | −0.181120 | 49.2199 | 17.2923 |
| | | 0.508974 | 0.268272 | −0.331539 | 0.570103 | −105.679 | −13.0584 |
| | | | | | | | |
| 2 | | −0.0121486 | −0.0121486 | −0.181120 | −0.181120 | 17.2923 | 17.2923 |
| | | 0.268272 | 0.268272 | 0.570103 | 0.570103 | −13.0584 | −13.0584 |
| | | | | | | | |
| 20 | | −0.0121486 | −0.0121486 | −0.181120 | −0.181120 | 17.2923 | 17.2923 |
| | | 0.268272 | 0.268272 | 0.570103 | 0.570103 | −13.0584 | −13.0584 |
Table 3.
Comparative numerical results between q-HATM and LDM at , , and different values of k, x, and t.
Table 3.
Comparative numerical results between q-HATM and LDM at , , and different values of k, x, and t.
| | | | | t | | | |
---|
| | | 0.1 | | 1 | | 5 | |
---|
x | | | | | | | | |
---|
0.1 | 1 | | −2.49107 | −2.55566 | −4.38644 | −7.62334 | 41.2062 | −8.72583 |
| | | −3.01524 | −5.21112 | 3.15269 | −7.85278 | 69.3833 | 35.4296 |
| | | | | | | | |
| 2 | | −2.55566 | −2.55566 | −7.62334 | −7.62334 | −8.72583 | −8.72583 |
| | | −5.21112 | −5.21112 | −7.85278 | −7.85278 | 35.4296 | 35.4296 |
| | | | | | | | |
| 20 | | −2.55566 | −2.55566 | −7.62334 | −7.62334 | −8.72583 | −8.72583 |
| | | −5.21112 | −5.21112 | −7.85278 | −7.85278 | 35.4296 | 35.4296 |
| | | | | | | | |
0.5 | 1 | | −0.743360 | −0.807945 | −0.0180716 | −3.25498 | 65.1720 | 15.2400 |
| | | −0.683663 | −2.87954 | 6.57607 | -4.42941 | 74.4459 | 40.4921 |
| | | | | | | | |
| 2 | | −0.807945 | −0.807945 | −3.25498 | −3.25498 | 15.2400 | 15.2400 |
| | | −2.87954 | −2.87954 | −4.42941 | −4.42941 | 40.4921 | 40.4921 |
| | | | | | | | |
| 20 | | −0.807945 | −0.807945 | −3.25498 | −3.25498 | 15.2400 | 15.2400 |
| | | −2.87954 | −2.87954 | −4.42941 | −4.42941 | 40.4921 | 40.4921 |
| | | | | | | | |
0.9 | 1 | | −0.105074 | −0.169659 | 1.57731 | −1.65959 | 73.9246 | 23.9926 |
| | | 0.167859 | −2.02802 | 7.82632 | −3.17915 | 76.2948 | 42.3410 |
| | | | | | | | |
| 2 | | −0.169659 | −0.169659 | −1.65959 | −1.65959 | 23.9926 | 23.9926 |
| | | −2.02802 | −2.02802 | −3.17915 | −3.17915 | 42.3410 | 42.3410 |
| | | | | | | | |
| 20 | | −0.169659 | −0.169659 | −1.65959 | −1.65959 | 23.9926 | 23.9926 |
| | | −2.02802 | −2.02802 | −3.17915 | −3.17915 | 42.3410 | 42.3410 |
Table 4.
Comparative numerical results between q-HATM and LDM at , , and different values of k, x, and t.
Table 4.
Comparative numerical results between q-HATM and LDM at , , and different values of k, x, and t.
| | | | | t | | | |
---|
| | | 0.1 | | 1 | | 5 | |
---|
x | | | | | | | | |
---|
0.1 | 1 | | −2.35149 | −2.35817 | −4.03540 | −5.71257 | 44.3256 | −35.4934 |
| | | −2.72454 | −3.04503 | 1.87245 | −6.17798 | 86.2490 | 9.62286 |
| | | | | | | | |
| 2 | | −2.35817 | −2.35817 | −5.71257 | −5.71257 | −35.4934 | −35.4934 |
| | | −3.04503 | −3.04503 | 1.87245 | −6.17798 | 9.62286 | 9.62286 |
| | | | | | | | |
| 20 | | −2.35817 | −2.35817 | −5.71257 | −5.71257 | −35.4934 | −35.4934 |
| | | −3.04503 | −3.04503 | 1.87245 | −6.17798 | 9.62286 | 9.62286 |
| | | | | | | | |
0.5 | 1 | | −0.706464 | −0.713141 | −0.424620 | −2.10179 | 79.3945 | −0.424480 |
| | | −0.714807 | −1.03530 | 5.26992 | −2.78051 | 92.9483 | 16.3222 |
| | | | | | | | |
| 2 | | −0.713141 | −0.713141 | −2.10179 | −2.10179 | −0.424480 | −0.424480 |
| | | −1.03530 | −1.03530 | −2.78051 | −2.78051 | 16.3222 | 16.3222 |
| | | | | | | | |
| 20 | | −0.713141 | −0.713141 | −2.10179 | −2.10179 | −0.424480 | −0.424480 |
| | | −1.03530 | −1.03530 | −2.78051 | −2.78051 | 16.3222 | 16.3222 |
| | | | | | | | |
0.9 | 1 | | −0.105680 | −0.112357 | 0.894081 | −0.783093 | 92.2020 | 12.3831 |
| | | 0.0191709 | −0.301323 | 6.51072 | −1.53971 | 95.3950 | |
| | | | | | | | |
| 2 | | −0.112357 | −0.112357 | −0.783093 | −0.783093 | 12.3831 | 12.3831 |
| | | −0.301323 | −0.301323 | −1.53971 | −1.53971 | 18.7688 | |
| | | | | | | | |
| 20 | | −0.112357 | −0.112357 | −0.783093 | −0.783093 | 12.3831 | 12.3831 |
| | | −0.301323 | −0.301323 | −1.53971 | −1.53971 | 18.7688 | 18.7688 |
Table 5.
Comparative numerical results between q-HATM and LDM at , , and different values of k, x, and t.
Table 5.
Comparative numerical results between q-HATM and LDM at , , and different values of k, x, and t.
| | | | | t | | | |
---|
| | | 0.1 | | 1 | | 5 | |
---|
x | | | | | | | | |
---|
0.1 | 1 | | 7.13659 | −5.70225 | 8.15776 | −225.788 | 95.1809 | 9327.71 |
| | | −3.14325 | −14.2973 | −6.90776 | −1000.92 | 567.526 | 104,323 |
| | | | | | | | |
| 2 | | 7.13571 | −48.9027 | 6.90776 | −19793.9 | −269.402 | |
| | | −2.64825 | −906.041 | −6.90776 | −57,543.6 | −32.4740 | |
| | | | | | | | |
| 4 | | 7.13571 | −23,084.0 | 6.90776 | | −269.402 | |
| | | −2.64825 | −2.83707 | −6.90776 | | −32.4740 | |
| | | | | | | | |
0.5 | 1 | | 2.14894 | 1.85859 | 3.32944 | 5.00487 | 283.485 | −2324.10 |
| | | −1.29220 | 1.57557 | −2.07944 | −25.9033 | 590.224 | 4778.26 |
| | | | | | | | |
| 2 | | 2.14806 | 1.73289 | 2.07944 | −89.7380 | −81.0982 | 42115.8 |
| | | −0.797202 | −25.4133 | −2.07944 | −192.244 | −9.77565 | 4281.22 |
| | | | | | | | |
| 4 | | 2.14806 | −8.60542 | 2.07944 | −25,471.4 | −81.0982 | |
| | | −0.797202 | −610.033 | −2.07944 | −9.77565 | −175,454 | |
| | | | | | | | |
0.9 | 1 | | 0.327387 | 0.401310 | 1.56608 | 8.69824 | 352.256 | −2985.78 |
| | | −0.616177 | 1.77678 | −0.316082 | −14.8761 | 598.514 | 1485.35 |
| | | | | | | | |
| 2 | | 0.326512 | 0.380032 | 0.316082 | −8.58753 | −12.3272 | 12334.8 |
| | | −0.121177 | −6.34458 | −0.316082 | −22.0275 | −1.48593 | 4887.13 |
| | | | | | | | |
| 4 | | 0.326512 | −1.94451 | 0.316082 | −2128.81 | −12.3272 | 319,519 |
| | | −0.121177 | −133.995 | −0.316082 | −7827.00 | −1.48593 | −10,299.8 |
Table 6.
Comparative numerical results between q-HATM and LDM at , , and different values of k, x, and t.
Table 6.
Comparative numerical results between q-HATM and LDM at , , and different values of k, x, and t.
| | | | | t | | | |
---|
| | | 0.1 | | 1 | | 5 | |
---|
x | | | | | | | | |
---|
0.1 | 1 | | 7.13458 | 3.83500 | 7.82014 | −181.595 | 103.806 | 9818.09 |
| | | −4.17120 | −1.82618 | −5.89382 | −641.931 | 472.924 | 98,809.2 |
| | | | | | | | |
| 2 | | 7.12751 | 2.12439 | 5.15639 | −6433.88 | −210.775 | 698,279 |
| | | −2.96769 | −116.019 | −10.4020 | −24,560.5 | 9.82302 | |
| | | | | | | | |
| 4 | | 7.12751 | −52.1407 | 5.15639 | | −210.775 | |
| | | −2.96769 | −265,147 | −10.4020 | | 9.82302 | |
| | | | | | | | |
0.5 | 1 | | 2.14920 | 2.07148 | 3.56567 | 1.60583 | 273.436 | −1819.880 |
| | | −2.11295 | 1.1827 | −0.329041 | −13.5076 | 510.471 | 4669.25 |
| | | | | | | | |
| 2 | | 2.14213 | 2.06707 | 0.901923 | −25.8340 | −41.1445 | 28,486.1 |
| | | −0.909437 | −2.73552 | −4.83725 | −138.091 | 47.3700 | 3870.46 |
| | | | | | | | |
| 4 | | 2.14213 | 1.93882 | 0.901923 | −2014.43 | −41.1445 | |
| | | −0.909437 | −8.46116 | −4.83725 | −27079.7 | 47.3700 | |
| | | | | | | | |
0.9 | 1 | | 0.328481 | 0.343344 | 2.01188 | 4.67504 | 335.387 | −2514.04 |
| | | −1.36125 | 1.19248 | 1.70329 | −5.84939 | 524.184 | 1366.79 |
| | | | | | | | |
| 2 | | 0.321411 | 0.342650 | −0.651863 | −0.372439 | 20.8067 | 8949.48 |
| | | −0.157739 | −0.00607621 | −2.80492 | −28.9225 | 61.0826 | 5635.91 |
| | | | | | | | |
| 4 | | 0.321411 | 0.305315 | −0.651863 | −202.622 | 20.8067 | −323,568 |
| | | −0.157739 | −1.33735 | −2.80492 | −1529.67 | 61.0826 | −587,610 |