An Artificial Neural Network Approach for Solving Space Fractional Differential Equations
Abstract
:1. Introduction
2. Preliminaries and Notation
Problem Description
3. Implement ANNs
4. Numerical Simulations
4.1. Example 1
4.2. Example 2
4.3. Example 3
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Iter | n = 3 | n = 5 | ||||
---|---|---|---|---|---|---|
s = 10 | s = 15 | s = 20 | s = 10 | s = 15 | s = 20 | |
500 | 2.6523 | 2.9910 | 6.0083 | 3.3160 | 2.8045 | 2.3854 |
1000 | 1.6914 | 1.8374 | 3.5842 | 5.4319 | 9.5441 | 9.3812 |
1500 | 8.6155 | 8.7060 | 1.6233 | 1.3877 | 2.1126 | 2.3634 |
2000 | 3.3664 | 2.9976 | 5.1971 | 1.2209 | 5.3879 | 3.4650 |
Iter | n = 7 | n = 9 | ||||
s = 10 | s = 15 | s = 20 | s = 10 | s = 15 | s = 20 | |
500 | 1.0702 | 1.2441 | 1.6539 | 1.1291 | 4.0604 | 8.8801 |
1000 | 3.4174 | 9.9941 | 1.1063 | 5.1723 | 1.2310 | 3.6049 |
1500 | 2.4528 | 7.0533 | 9.2353 | 1.8850 | 4.4209 | 5.0940 |
2000 | 1.7723 | 4.4926 | 6.5423 | 8.3826 | 1.7819 | 7.4516 |
Iter | n = 3 | n = 5 | ||||
---|---|---|---|---|---|---|
s = 10 | s = 15 | s = 20 | s = 10 | s = 15 | s = 20 | |
500 | 1.0996 | 8.6408 | 9.0756 | 6.3823 | 5.1589 | 1.0696 |
1000 | 9.0466 | 7.0190 | 7.9873 | 4.4332 | 4.0132 | 5.3003 |
1500 | 6.9259 | 5.3601 | 6.6735 | 3.7657 | 2.6475 | 2.8372 |
2000 | 5.0009 | 3.9275 | 5.3449 | 3.1037 | 9.7093 | 2.3107 |
Iter | n = 7 | n = 9 | ||||
s = 10 | s = 15 | s = 20 | s = 10 | s = 15 | s = 20 | |
500 | 6.6813 | 8.7033 | 1.6382 | 1.6684 | 1.4755 | 5.0869 |
1000 | 1.9484 | 4.5613 | 3.4877 | 5.8310 | 2.3891 | 1.8448 |
1500 | 8.2505 | 2.5685 | 1.4557 | 4.1485 | 9.6104 | 7.5542 |
2000 | 3.6007 | 1.4610 | 6.3923 | 3.6057 | 4.4318 | 5.2341 |
Iter | n = 3 | n = 5 | ||||
---|---|---|---|---|---|---|
s = 10 | s = 15 | s = 20 | s = 10 | s = 15 | s = 20 | |
500 | 8.2966 | 5.2626 | 1.0117 | 5.1567 | 1.4999 | 9.1941 |
1000 | 4.4255 | 4.6605 | 1.8228 | 3.4654 | 1.3074 | 1.3050 |
1500 | 2.8333 | 1.5791 | 1.8212 | 4.6556 | 1.2690 | 2.4591 |
2000 | 4.6642 | 1.9708 | 1.8203 | 4.4605 | 1.2715 | 2.2832 |
Iter | n = 7 | n = 9 | ||||
s = 10 | s = 15 | s = 20 | s = 10 | s = 15 | s = 20 | |
500 | 1.0564 | 5.2482 | 1.3604 | 9.5831 | 1.2413 | 1.4771 |
1000 | 4.6845 | 1.9055 | 6.8806 | 2.4573 | 4.7626 | 5.2961 |
1500 | 3.6713 | 1.3278 | 5.2282 | 8.5461 | 1.2382 | 1.9865 |
2000 | 3.1830 | 9.2861 | 4.1084 | 6.6666 | 3.2416 | 9.2339 |
Iter | n = 3 | n = 5 | ||||
---|---|---|---|---|---|---|
s = 10 | s = 15 | s = 20 | s = 10 | s = 15 | s = 20 | |
500 | 1.3920 | 1.3119 | 1.1351 | 1.8116 | 4.5366 | 8.4585 |
1000 | 8.8657 | 9.2285 | 4.4556 | 1.8071 | 1.2556 | 7.2918 |
1500 | 5.4655 | 2.9104 | 1.2161 | 1.8011 | 6.6137 | 6.1525 |
2000 | 5.2418 | 9.2827 | 1.1887 | 1.7939 | 5.9639 | 5.0514 |
Iter | n = 7 | n = 9 | ||||
s = 10 | s = 15 | s = 20 | s = 10 | s = 15 | s = 20 | |
500 | 4.4860 | 5.8645 | 4.1541 | 1.5837 | 1.4489 | 1.6768 |
1000 | 6.3946 | 3.6488 | 9.1928 | 2.5737 | 3.1810 | 4.2928 |
1500 | 2.6374 | 9.1063 | 7.7642 | 5.1589 | 1.9570 | 1.0630 |
2000 | 9.3822 | 6.7459 | 6.1965 | 7.4717 | 1.7564 | 7.4550 |
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Dai, P.; Yu, X. An Artificial Neural Network Approach for Solving Space Fractional Differential Equations. Symmetry 2022, 14, 535. https://doi.org/10.3390/sym14030535
Dai P, Yu X. An Artificial Neural Network Approach for Solving Space Fractional Differential Equations. Symmetry. 2022; 14(3):535. https://doi.org/10.3390/sym14030535
Chicago/Turabian StyleDai, Pingfei, and Xiangyu Yu. 2022. "An Artificial Neural Network Approach for Solving Space Fractional Differential Equations" Symmetry 14, no. 3: 535. https://doi.org/10.3390/sym14030535
APA StyleDai, P., & Yu, X. (2022). An Artificial Neural Network Approach for Solving Space Fractional Differential Equations. Symmetry, 14(3), 535. https://doi.org/10.3390/sym14030535