Finite Element Approximations to Caputo–Hadamard Time-Fractional Diffusion Equation with Application in Parameter Identification
Abstract
:1. Introduction
2. Preliminaries
3. Analysis of a Finite Element Scheme
3.1. Analysis of Truncation Errors
3.2. Analysis of the Finite Element Method
3.3. Numerical Experiments of the Finite Element Method
4. An Inversion Algorithm to Evaluate the Fractional Order
4.1. L–M Regularization Method
Algorithm 1:(A Levenberg–Marquardt Algorithm): Given the initial data, the boundary information and the observation data . |
|
4.2. Numerical Experiment of the Finite Element Levenberg–Marquardt Method
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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32 | 4.61 | 2.12 | 8.64 | |||
64 | 2.27 | 1.02 | 1.11 | 0.94 | 4.19 | 1.04 |
128 | 1.13 | 1.01 | 5.79 | 0.94 | 2.07 | 1.02 |
256 | 5.64 | 1.00 | 2.97 | 0.97 | 1.03 | 1.01 |
8 | 7.54 | 1.55 | 2.00 | |||
16 | 1.93 | 1.97 | 2.69 | 1.97 | 5.11 | 1.97 |
32 | 4.85 | 1.99 | 6.77 | 1.99 | 1.28 | 2.00 |
64 | 1.22 | 2.00 | 1.69 | 2.00 | 3.10 | 2.04 |
Iter. | ||||
---|---|---|---|---|
2.9997 | 3.4048 | 11 | ||
2.9997 | 3.3971 | 11 | ||
2.9997 | 3.3984 | 12 | ||
5.9995 | 5.1359 | 12 | ||
5.9995 | 5.1364 | 11 | ||
5.9995 | 5.1287 | 11 | ||
8.9994 | 5.8136 | 12 | ||
8.9994 | 5.8134 | 12 | ||
8.9994 | 5.8135 | 11 |
Iter. | ||||
---|---|---|---|---|
7.4999 | 1.1803 | 12 | ||
7.4992 | 7.6152 | 12 | ||
7.4967 | 3.3354 | 12 | ||
7.5078 | 7.8329 | 12 | ||
7.5104 | 1.0446 | 12 | ||
7.5257 | 2.5709 | 12 | ||
1 | 7.3639 | 1.3607 | 12 | |
7.4047 | 9.5338 | 12 | ||
7.4961 | 3.9271 | 12 |
Iter. | ||||
---|---|---|---|---|
3.008 | 8.4911 | 11 | ||
3.008 | 8.4911 | 11 | ||
3.008 | 8.4911 | 12 | ||
6.008 | 8.3445 | 12 | ||
6.008 | 8.3445 | 11 | ||
6.008 | 8.3445 | 11 | ||
9.008 | 8.3297 | 12 | ||
9.008 | 8.3297 | 12 | ||
9.008 | 8.3297 | 11 |
Iter. | ||||
---|---|---|---|---|
7.5085 | 8.5595 | 12 | ||
7.5082 | 8.2265 | 12 | ||
7.5081 | 8.1000 | 12 | ||
7.5089 | 8.9797 | 12 | ||
7.5076 | 7.6190 | 12 | ||
7.5102 | 1.0166 | 12 | ||
1 | 7.5415 | 4.1538 | 12 | |
7.4991 | 8.5762 | 12 | ||
7.5305 | 3.0522 | 12 |
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Cheng, S.; Du, N.; Wang, H.; Yang, Z. Finite Element Approximations to Caputo–Hadamard Time-Fractional Diffusion Equation with Application in Parameter Identification. Fractal Fract. 2022, 6, 525. https://doi.org/10.3390/fractalfract6090525
Cheng S, Du N, Wang H, Yang Z. Finite Element Approximations to Caputo–Hadamard Time-Fractional Diffusion Equation with Application in Parameter Identification. Fractal and Fractional. 2022; 6(9):525. https://doi.org/10.3390/fractalfract6090525
Chicago/Turabian StyleCheng, Shijing, Ning Du, Hong Wang, and Zhiwei Yang. 2022. "Finite Element Approximations to Caputo–Hadamard Time-Fractional Diffusion Equation with Application in Parameter Identification" Fractal and Fractional 6, no. 9: 525. https://doi.org/10.3390/fractalfract6090525
APA StyleCheng, S., Du, N., Wang, H., & Yang, Z. (2022). Finite Element Approximations to Caputo–Hadamard Time-Fractional Diffusion Equation with Application in Parameter Identification. Fractal and Fractional, 6(9), 525. https://doi.org/10.3390/fractalfract6090525