On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics
Abstract
1. Introduction
2. Formulation of the Crank–Nicolson Finite Difference Method (CN–FDM)
3. Formulation of the Explicit Decoupled Group Method (EDGM)
4. Stability Analysis
- (i)
- (ii)
- .
5. Convergence Analysis
6. Numerical Results and Discussion
7. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
References
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CN–FDM | EDGM | ||||||
---|---|---|---|---|---|---|---|
6 | 3.2214 | 24 | 2.4354 × 10−2 | 1.5446 | 11 | 2.5830 × 10−2 | |
12 | 57.0982 | 75 | 6.0448 × 10−3 | 14.1272 | 34 | 6.1778 × 10−3 | |
18 | 250.0227 | 145 | 2.5440 × 10−3 | 51.5105 | 66 | 2.6716 × 10−3 | |
24 | 623.4341 | 230 | 1.2371 × 10−3 | 148.0147 | 106 | 1.4167 × 10−3 | |
6 | 4.3391 | 24 | 2.4449 × 10−2 | 1.6299 | 12 | 2.5933 × 10−2 | |
12 | 63.6731 | 78 | 6.0859 × 10−3 | 17.5187 | 35 | 6.2206 × 10−3 | |
18 | 296.8552 | 150 | 2.5441 × 10−3 | 72.8325 | 68 | 2.6855 × 10−3 | |
24 | 765.8251 | 237 | 1.2493 × 10−3 | 212.8129 | 109 | 1.4199 × 10−3 | |
6 | 3.2722 | 24 | 2.4449 × 10−2 | 1.3767 | 12 | 2.5934 × 10−2 | |
12 | 58.3578 | 78 | 6.0862 × 10−3 | 16.7624 | 35 | 6.2144 × 10−3 | |
18 | 250.1333 | 150 | 2.5414 × 10−3 | 67.1791 | 68 | 2.6880 × 10−3 | |
24 | 723.0534 | 237 | 1.2615 × 10−3 | 169.6976 | 109 | 1.4235 × 10−3 | |
6 | 1.3651 | 9 | 2.3432 × 10−2 | 0.8389 | 5 | 2.4777 × 10−2 | |
12 | 14.9717 | 21 | 5.7997 × 10−3 | 4.4889 | 11 | 5.9192 × 10−3 | |
18 | 64.7063 | 39 | 2.4295 × 10−3 | 17.991 | 18 | 2.5349 × 10−3 | |
24 | 226.331 | 61 | 1.1482 × 10−3 | 63.1113 | 28 | 1.3300 × 10−3 | |
6 | 3.2977 | 25 | 2.4525 × 10−2 | 1.3007 | 12 | 2.6012 × 10−2 | |
12 | 44.7645 | 79 | 6.1073 × 10−3 | 12.388 | 36 | 6.2598 × 10−3 | |
18 | 215.872 | 152 | 2.5741 × 10−3 | 54.5599 | 69 | 2.7378 × 10−3 | |
24 | 671.5466 | 240 | 1.3141 × 10−3 | 157.9549 | 111 | 1.4666 × 10−3 | |
6 | 3.7105 | 24 | 2.4446 × 10−2 | 2.1392 | 12 | 2.5927 × 10−2 | |
12 | 48.9569 | 77 | 6.0763 × 10−3 | 14.094 | 35 | 6.2183 × 10−3 | |
18 | 247.4537 | 150 | 2.5423 × 10−3 | 51.0625 | 68 | 2.6972 × 10−3 | |
24 | 658.7823 | 236 | 1.2699 × 10−3 | 153.9475 | 109 | 1.4156 × 10−3 |
CN–FDM | EDGM | ||||||
---|---|---|---|---|---|---|---|
4 | 7.6451 | 493 | 3.3805 × 10−2 | 2.3588 | 211 | 3.3457 × 10−2 | |
8 | 22.1864 | 388 | 7.9937 × 10−3 | 5.0926 | 167 | 7.6691 × 10−3 | |
16 | 49.5047 | 283 | 2.0338 × 10−3 | 12.9866 | 123 | 1.7656 × 10−3 | |
32 | 123.0862 | 191 | 8.4239 × 10−4 | 31.4933 | 84 | 5.8782 × 10−4 | |
4 | 13.3397 | 599 | 4.1630 × 10−2 | 3.6625 | 257 | 4.1215 × 10−2 | |
8 | 33.5994 | 546 | 1.1022 × 10−2 | 7.8271 | 236 | 1.0530 × 10−2 | |
16 | 101.0359 | 491 | 3.3731 × 10−3 | 24.7500 | 216 | 2.8573 × 10−3 | |
32 | 307.0350 | 435 | 1.5878 × 10−3 | 77.1075 | 194 | 9.8333 × 10−4 | |
4 | 9.2405 | 483 | 3.2694 × 10−2 | 2.4086 | 206 | 3.2342 × 10−2 | |
8 | 20.9804 | 371 | 7.5679 × 10−3 | 5.6095 | 160 | 7.2467 × 10−3 | |
16 | 48.5232 | 260 | 1.9174 × 10−3 | 12.4909 | 113 | 1.6750 × 10−3 | |
32 | 111.5133 | 168 | 8.2150 × 10−4 | 28.1021 | 74 | 5.6681 × 10−4 |
CN–FDM | EDGM | ||||||
---|---|---|---|---|---|---|---|
6 | 1.1107 | 7 | 3.6260 × 10−3 | 0.8549 | 4 | 6.3340 × 10−3 | |
12 | 9.8215 | 14 | 9.1115 × 10−4 | 3.5592 | 8 | 1.4611 × 10−3 | |
18 | 41.471 | 26 | 3.1795 × 10−4 | 11.7159 | 13 | 6.2110 × 10−4 | |
24 | 109.8017 | 40 | 1.7590 × 10−4 | 32.605 | 19 | 2.9598 × 10−4 | |
6 | 2.8476 | 19 | 3.9242 × 10−3 | 1.2287 | 10 | 6.7795 × 10−3 | |
12 | 34.0943 | 58 | 1.0029 × 10−3 | 9.621 | 28 | 1.5797 × 10−3 | |
18 | 148.4631 | 109 | 3.1527 × 10−4 | 41.7843 | 53 | 6.7007 × 10−4 | |
24 | 398.0526 | 168 | 2.3551 × 10−4 | 110.8827 | 83 | 3.2448 × 10−4 | |
6 | 1.0247 | 6 | 3.6211 × 10−3 | 0.7978 | 4 | 6.3249 × 10−3 | |
12 | 8.1283 | 14 | 9.2666 × 10−4 | 2.9754 | 7 | 1.4593 × 10−3 | |
18 | 32.5322 | 24 | 3.2975 × 10−4 | 9.9754 | 12 | 6.2310 × 10−4 | |
24 | 90.0535 | 38 | 1.7051 × 10−4 | 24.3636 | 18 | 3.0412 × 10−4 | |
6 | 1.0972 | 7 | 3.6316 × 10−3 | 0.7846 | 4 | 6.3415 × 10−3 | |
12 | 8.8651 | 15 | 9.1747 × 10−4 | 3.3896 | 8 | 1.4587 × 10−3 | |
18 | 38.1758 | 27 | 3.2260 × 10−4 | 11.0987 | 13 | 6.1325 × 10−4 | |
24 | 107.0003 | 42 | 1.8170 × 10−4 | 28.3911 | 20 | 2.9483 × 10−4 | |
6 | 2.9888 | 19 | 3.9502 × 10−3 | 1.4219 | 10 | 6.8113 × 10−3 | |
12 | 41.7307 | 59 | 1.0089 × 10−3 | 10.3176 | 29 | 1.5737 × 10−3 | |
18 | 180.8514 | 111 | 3.2972 × 10−4 | 51.3228 | 54 | 6.6835 × 10−4 | |
24 | 496.4411 | 170 | 2.3834 × 10−4 | 134.1644 | 85 | 3.4322 × 10−4 | |
6 | 2.3473 | 19 | 3.9200 × 10−3 | 1.0168 | 9 | 6.7680 × 10−3 | |
12 | 37.1117 | 57 | 9.6831 × 10−4 | 10.5125 | 28 | 1.5614 × 10−3 | |
18 | 166.6839 | 108 | 3.2475 × 10−4 | 43.7882 | 52 | 6.7173 × 10−4 | |
24 | 462.7543 | 165 | 2.4535 × 10−4 | 130.107 | 82 | 3.1735 × 10−4 |
CPU Timing | Iteration Counting | |
---|---|---|
52.01–79.40% | 3.91–54.67% | |
62.44–75.47% | 50.00–55.13% | |
57.93–76.53% | 50.00–55.13% | |
38.55–72.20% | 44.44–54.10% | |
60.56–76.48% | 52.00–54.61% | |
42.35–79.36% | 50.00–54.67% |
CPU Timing | Iteration Counting | |
---|---|---|
23.03–71.75% | 42.86–52.50% | |
56.85–72.14% | 47.37–51.72% | |
22.14–72.95% | 33.33–52.63% | |
28.49–73.47% | 42.85–52.38% | |
49.41–75.28% | 47.36–51.35% | |
56.68–73.73% | 50.30–52.63% |
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Salama, F.M. On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics. Fractal Fract. 2024, 8, 282. https://doi.org/10.3390/fractalfract8050282
Salama FM. On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics. Fractal and Fractional. 2024; 8(5):282. https://doi.org/10.3390/fractalfract8050282
Chicago/Turabian StyleSalama, Fouad Mohammad. 2024. "On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics" Fractal and Fractional 8, no. 5: 282. https://doi.org/10.3390/fractalfract8050282
APA StyleSalama, F. M. (2024). On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics. Fractal and Fractional, 8(5), 282. https://doi.org/10.3390/fractalfract8050282