1. Introduction
2. Fundamental ideas of fractional and fuzzy calculus
3. Some basic properties of the fuzzy generalized integral transform
4. Fuzzy formulation of generalized integral transform in connection with ADM
5. Application to the fuzzy fractional Boussineq equation
5.1. Fourth-order fuzzy fractional BSe in R
Fig. 1. Comparison of surface profile of Problem 5.1 established by (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when α=1 and uncertainty factor $\wp$∈[0,1]. |
Fig. 2. Comparison of multiple surface profiles of Problem 5.1 established by (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when different fractional orders correlate with uncertainty factor $\wp$∈[0,1]. |
Fig. 3. Comparison of multiple two dimensional profiles of Problem 5.1 established by (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when different fractional orders correlate with uncertainty factor $\wp$∈[0,1]. |
Fig. 4. Comparison of multiple two-dimensional profiles of the Problem 5.1 established by (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when fractional order α=0.7 correlate with the different uncertainty factors $\wp$∈[0,1]. |
Fig. 5. Comparison of surface profiles (b) Comparison of 2D profiles of Problem 5.1 established by considering fuzzy CFD and fuzzy ABC fractional derivative operators when α=1 and uncertainty factor $\wp$∈[0,1]. |
5.2. Fourth-order fuzzy fractional BSe in $\mathbb{R}^{n}$
Fig. 6. Comparison of 3D surface profile of Problem 5.2 established by (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when α=1 and uncertainty factor $\wp$∈[0,1]. |
Fig. 7. Comparison of multiple surface profiles of Problem 5.2 established by (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when different fractional orders correlate with uncertainty factor $\wp$∈[0,1]. |
Fig. 8. Comparison of multiple two dimensional profiles of Problem 5.2 established by (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when different fractional orders correlate with uncertainty factor $\wp$∈[0,1]. |
Fig. 9. Comparison of multiple two-dimensional profiles of the Problem 5.2 established by (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when fractional order α=0.7 correlate with the different uncertainty factors $\wp$∈[0,1]. |
Fig. 10. (a) Comparison of surface profiles (b) Comparison of 2D profiles of Problem 5.2 established by fuzzy CFD and fuzzy ABC fractional derivative operators when α=1 and uncertainty factor $\wp$∈[0,1]. |
5.3. (2ndth)-Order fuzzy fractional BSe in $\mathbb{R}$
Table 1. The exact, the lower and upper solutions via $G I A D M_{C F D}$ and the lower and upper solutions via $G I A D M_{A B C}$ solutions of Example 5.3 with varying values of y and ℘ in comparison with the solution profile of [94] obtained by iterative method. |
y | ω | $\underline{\mathbf{y}}_{\text {CFD }}$ | $\overline{\mathbf{y}}_{C F D}$ | $\overline{\mathbf{y}}_{A B C}$ | $\overline{\mathbf{y}}_{A B C}$ | y [94] | Exact | ||
---|---|---|---|---|---|---|---|---|---|
0.1 | -1.094119307 | 1.094119307 | -1.094119307 | 0.8958123558 | 1.105171018 | 1.105170918 | |||
0.2 | -0.9900031196 | 0.9900031196 | -0.9900031196 | 0.9900031196 | 1.000003152 | 1.000000000 | |||
0.2 | 0.3 | -0.8958123558 | 0.8958123558 | -0.8958123558 | 0.8958123558 | 0.9048609657 | 0.9048374180 | ||
0.4 | -0.8106401246 | 0.8106401246 | -0.8106401246 | 0.8106401246 | 0.8188284089 | 0.8187307531 | |||
0.5 | -0.7337004534 | 0.7337004534 | -0.7337004534 | 0.7337004534 | 0.7411115693 | 0.7408182207 | |||
0.1 | -1.336360341 | 1.336360341 | -1.336360341 | 1.336360341 | 1.349858930 | 1.349858808 | |||
0.2 | -1.209192542 | 1.209192542 | -1.209192542 | 1.209192542 | 1.221406607 | 1.221402758 | |||
0.4 | 0.3 | -1.094147683 | 1.094147683 | -1.094147683 | 1.094147683 | 1.105199680 | 1.105170918 | ||
0.4 | -0.9901180848 | 0.9901180848 | -0.9901180848 | 0.9901180848 | 1.000119278 | 1.000000000 | |||
0.5 | -0.8961437581 | 0.8961437581 | -0.8961437581 | 0.8961437581 | 0.9051957152 | 0.9048374180 | |||
0.1 | -1.632234205 | 1.632234205 | -1.632234205 | 1.632234205 | 1.648721420 | 1.648721271 | |||
0.2 | -1.476911105 | 1.476911105 | -1.476911105 | 1.476911105 | 1.491829399 | 1.491824698 | |||
0.6 | 0.3 | -1.336394997 | 1.336394997 | -1.336394997 | 1.336394997 | 1.349893936 | 1.349858808 | ||
0.4 | -1.209332959 | 1.209332959 | -1.209332959 | 1.209332959 | 1.221548443 | 1.221402758 | |||
0.5 | -1.094552457 | 1.094552457 | -1.094552457 | 1.094552457 | 1.105608543 | 1.105170918 | |||
0.1 | -1.993615361 | 1.993615361 | -1.993615361 | -1.993615361 | 2.013752890 | 2.013752707 | |||
0.2 | -1.803903297 | 1.803903297 | -1.803903297 | -1.803903297 | 1.822124542 | 1.822118800 | |||
0.8 | 0.3 | -1.632276536 | 1.632276536 | -1.632276536 | 1.632276536 | 1.648764178 | 1.648721271 | ||
0.4 | -1.477082612 | 1.477082612 | -1.477082612 | 1.477082612 | 1.492002638 | 1.491824698 | |||
0.5 | -1.336889391 | 1.336889391 | -1.336889391 | 1.336889391 | 1.350393324 | 1.349858808 | |||
0.1 | -2.4389324561 | 2.4389324561 | -2.4389324561 | 2.4389324561 | 2.449603111 | 2.459603111 | |||
0.2 | -2.203292463 | 2.203292463 | -2.203292463 | 2.203292463 | 2.225547944 | 2.225540928 | |||
1.0 | 0.3 | -1.993667063 | 1.993667063 | -1.993667063 | 1.993667063 | 2.013805114 | 2.013752707 | ||
0.4 | -1.804112776 | 1.804112776 | -1.804112776 | 1.804112776 | 1.822336137 | 1.822118800 | |||
0.5 | -1.632880389 | 1.632880389 | -1.632880389 | 1.632880389 | 1.649374132 | 1.648721271 |
Table 2. The comparison analysis via the absolute error among iterative method (IM) [94], the absolute error of the lower and upper solutions of CFD and the lower and upper solutions of GIADMABC of Example 5.3 for approximated results of $\Theta(\mathbf{y}, \omega ; \wp) $ at α=1 with varying values of y and ω. |
y | ω | EIM | $E_{\mathbf{\underline{y} }} \text { for } C F D$ | $E_{\mathbf{\overline{y} }} \text { for } C F D$ | $E_{\mathbf{\overline{y} }} \text { for } A B C$ | $E_{\mathbf{\ underline{y} }} \text { for } A B C$ | ||
---|---|---|---|---|---|---|---|---|
0.6 | 1.99998E−7 | 1.99999E−7 | 2.00001E−7 | 2.00011E−7 | 2.00567E−7 | |||
0.7 | 2.27800E−6 | 1.00990E−1 | 1.00889E−3 | 1.99990E−1 | 1.00998E−3 | |||
0.1 | 0.8 | 2.55678E−5 | 1.77809E−1 | 1.89023E−3 | 1.99560E−1 | 1.78894E−3 | ||
0.9 | 3.76890E−5 | 3.89767E−4 | 3.67789E−3 | 3.37456E−3 | 89769E−3 | |||
1.0 | 2.67800E−4 | 2.65478E−4 | 2.24478E−3 | 2.67891E−3 | 2.66551E−3 | |||
0.6 | 1.00986E−7 | 1.62342E−3 | 1.44578E−2 | 2.33440E−2 | 1.98178E−2 | |||
0.7 | 3.08725E−6 | 2.40369E−2 | 1.22991E−2 | 2.62860E−2 | 1.55202E−2 | |||
0.2 | 0.8 | 2.87620E−5 | 2.19932E−4 | 1.10232E−2 | 2.19932E−4 | 1.10232E−2 | ||
0.9 | 1.18111E−4 | 1.21112E−3 | 9.83332E−3 | 1.11442E−2 | 9.21122E−3 | |||
1.0 | 3.57661E−4 | 1.81238E0 | 8.65556E−3 | 1.89998E−3 | 8.61236E−3 | |||
0.6 | 1.49666E−7 | 3.82346E−3 | 1.87271E−2 | 3.98123E−4 | 1.77171E−2 | |||
0.7 | 4.72340E−6 | 2.96001E−5 | 1.81136E−2 | 2.97712E−4 | 1.41236E−2 | |||
0.3 | 0.8 | 3.51280E−5 | 2.68625E−5 | 1.34638E−2 | 2.68625E−2 | 1.34638E−2 | ||
0.9 | 1.47775E−4 | 2.43904E−3 | 1.26668E−2 | 2.40014E−3 | 1.299998E−2 | |||
1.0 | 4.38925E−4 | 2.19345E−4 | 1.09125E−2 | 2.19991E−2 | 1.09995E−2 | |||
0.6 | 1.00891E−7 | 4.70037E−5 | 2.30073E−2 | 4.30097E−6 | 2.09813E−2 | |||
0.7 | 5.00891E−6 | 3.60289E−3 | 1.55901E−2 | 3.02555E−1 | 1.82991E−2 | |||
0.4 | 0.8 | 4.09823E−5 | 3.10022E−3 | 1.47982E−2 | 3.00091E−2 | 1.63345E−2 | ||
0.9 | 1.00981E−4 | 2.00000E−2 | 1.34567E−2 | 2.23411E−1 | 1.23401E−2 | |||
1.0 | 5.77899E−4 | 2.89932E−1 | 1.782201E−2 | 2.789345E−1 | 1.98230E−2 | |||
0.6 | 2.00256E−7 | 4.98671E−1 | 2.245789E−2 | 4.235671E−3 | 2.98564E−2 | |||
0.7 | 6.00087E−6 | 5.43423E−3 | 3.23344E−2 | 5.23678E−1 | 2.44567E−2 | |||
0.5 | 0.8 | 6.070894E−5 | 3.23400E−2 | 2.99123E−2 | 3.23456E−3 | 2.44344E−2 | ||
0.9 | 1.99882E−4 | 2.65781E−2 | 3.89967E−2 | 2.23451E−2 | 4.00982E−2 | |||
1.0 | 6.43567E−4 | 4.00991E−6 | 2.90023E−2 | 1.66689E−3 | 2.00099E−2 |
Fig. 11. Comparison of surface profile of Problem 5.3 established by considering (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when α=1 and uncertainty factor $\wp$∈[0,1]. |
Fig. 12. Comparison of multiple surface profiles of Problem 5.3 established by considering (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when different fractional orders correlate with uncertainty factor $\wp$∈[0,1]. |
Fig. 13. Comparison of multiple two dimensional profiles of Problem 5.3 established by considering (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when different fractional orders correlate with uncertainty factor $\wp$∈[0,1]. |
Fig. 14. Comparison of multiple two-dimensional profiles of the Problem 5.3 established by considering (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when fractional order α=0.7 correlate with the different uncertainty factors $\wp$∈[0,1]. |
Fig. 15. Three dimensional comparison of Problem 5.3 established by considering (a) fuzzy CFD (b) fuzzy ABC fractional derivative operators when α=1 and uncertainty factor $\wp$∈[0,1]. |