Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay
Abstract
:1. Introduction
2. Preliminaries
3. Idea of Fractional Natural Transform Decomposition Method
4. Results
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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NTDM () | HPM [8] | Exact | AE | |||
---|---|---|---|---|---|---|
0.25 | 0.25 | 0.390443474 | 0.321004230 | 0.321004232 | 0.3210063542 | 2.12240 × 10 |
0.50 | 0.529812605 | 0.412109374 | 0.412109375 | 0.4121803178 | 7.09428 × 10 | |
0.75 | 0.695464609 | 0.528686521 | 0.528686522 | 0.5292500042 | 5.63480 × 10 | |
1.00 | 0.893414574 | 0.677083333 | 0.677083335 | 0.6795704570 | 2.48712 × 10 | |
0.50 | 0.25 | 0.780886948 | 0.642008463 | 0.642008465 | 0.6420127085 | 4.24500 × 10 |
0.50 | 1.059625210 | 0.824218749 | 0.824218750 | 0.8243606355 | 1.41885 × 10 | |
0.75 | 1.390929217 | 1.057373039 | 1.057373040 | 1.0585000080 | 1.12696 × 10 | |
1.00 | 1.786829148 | 1.354166666 | 1.354166670 | 1.3591409140 | 4.97424 × 10 | |
0.75 | 0.25 | 1.171330424 | 0.963012695 | 0.963012698 | 0.9630190628 | 6.36750 × 10 |
0.50 | 1.589437815 | 1.236328118 | 1.236328120 | 1.2365409530 | 2.12828 × 10 | |
0.75 | 2.086393828 | 1.586059568 | 1.586059570 | 1.5877500130 | 1.69044 × 10 | |
1.00 | 2.680243723 | 2.031248999 | 2.031250000 | 2.0387113710 | 7.46137 × 10 |
Exact | Absolute Error | ||||
---|---|---|---|---|---|
0.25 | 0.25 | 0.0976593036 | 0.08025158853 | 0.08025158856 | 3.00000 × 10 |
0.50 | 0.1330574978 | 0.10304507290 | 0.10304507940 | 6.50000 × 10 | |
0.75 | 0.1765973237 | 0.13231233190 | 0.13231250110 | 1.69200 × 10 | |
1.00 | 0.2314929791 | 0.16989087310 | 0.16989261420 | 1.74110 × 10 | |
0.50 | 0.25 | 0.3906372147 | 0.32100635410 | 0.32100635420 | 1.00000 × 10 |
0.50 | 0.5322299913 | 0.41218029200 | 0.41218031780 | 2.58000 × 10 | |
0.75 | 0.7063892947 | 0.52924932750 | 0.52925000420 | 6.76700 × 10 | |
1.00 | 0.9259719163 | 0.67956349210 | 0.67957045700 | 6.96490 × 10 | |
1 | 0.25 | 1.5625488590 | 1.28402541600 | 1.28402541700 | 1.00000 × 10 |
0.50 | 2.1289199640 | 1.64872116800 | 1.64872127100 | 1.03000 × 10 | |
0.75 | 2.8255571770 | 2.11699731000 | 2.11700001700 | 2.70700 × 10 | |
1.00 | 3.7038876660 | 2.71825396900 | 2.71828182800 | 2.78590 × 10 |
x | t | Exact | Absolute Error | ||
---|---|---|---|---|---|
0.25 | 0.25 | 0.0976593036 | 0.04867504891 | 0.04867504894 | 3.00000 × 10 |
0.50 | 0.1330574978 | 0.03790816051 | 0.03790816623 | 5.72000 × 10 | |
0.75 | 0.1765973237 | 0.02952276638 | 0.02952290954 | 1.43160 × 10 | |
1.00 | 0.2314929791 | 0.02299107144 | 0.02299246508 | 1.39364 × 10 | |
0.50 | 0.25 | 0.3906372147 | 0.19470019570 | 0.19470019580 | 1.00000 × 10 |
0.50 | 0.5322299913 | 0.15163264200 | 0.15163266490 | 2.29000 × 10 | |
0.75 | 0.7063892947 | 0.11809106550 | 0.11809163820 | 5.72700 × 10 | |
1.00 | 0.9259719163 | 0.09196428572 | 0.09196986030 | 5.57458 × 10 | |
1 | 0.25 | 1.5625488590 | 0.77880078270 | 0.77880078310 | 4.00000 × 10 |
0.50 | 2.1289199640 | 0.60653056800 | 0.60653065970 | 9.17000 × 10 | |
0.75 | 2.8255571770 | 0.47236426220 | 0.47236655270 | 2.29050 × 10 | |
1.00 | 3.7038876660 | 0.36785714290 | 0.36787944120 | 2.22983 × 10 |
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Shah, R.; Khan, H.; Kumam, P.; Arif, M.; Baleanu, D. Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay. Mathematics 2019, 7, 532. https://doi.org/10.3390/math7060532
Shah R, Khan H, Kumam P, Arif M, Baleanu D. Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay. Mathematics. 2019; 7(6):532. https://doi.org/10.3390/math7060532
Chicago/Turabian StyleShah, Rasool, Hassan Khan, Poom Kumam, Muhammad Arif, and Dumitru Baleanu. 2019. "Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay" Mathematics 7, no. 6: 532. https://doi.org/10.3390/math7060532
APA StyleShah, R., Khan, H., Kumam, P., Arif, M., & Baleanu, D. (2019). Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay. Mathematics, 7(6), 532. https://doi.org/10.3390/math7060532