An Efficient Semi-Analytical Solution of a One-Dimensional Curvature Equation that Describes the Human Corneal Shape
Abstract
:1. Introduction
2. Description of the Method
2.1. Construction of Green’s Function
2.2. Green-Picard Fixed Point Iteration
2.3. Green-Mann Fixed Point Iteration
3. Convergence Analysis
4. Results and Discussions
- The zero-order solution based on the hyperbolic cosine function [2]:
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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x | Proposed Method h5 | Taylor Equation (45) | Linearization Equation (40) | Zero-Order Equation (39) | Perturbation Equation (43) |
---|---|---|---|---|---|
0.0 | 0.00005 | 0.32318 | 3.23315 | 3.31177 | 9.93049 |
0.1 | 0.00002 | 0.32800 | 3.27178 | 3.36031 | 9.97343 |
0.2 | 0.00004 | 0.34291 | 3.35877 | 3.50557 | 10.10180 |
0.3 | 0.00005 | 0.36958 | 3.46262 | 3.74615 | 10.31389 |
0.4 | 0.00005 | 0.41066 | 3.56295 | 4.07985 | 10.60702 |
0.5 | 0.00005 | 0.46910 | 3.64695 | 4.50375 | 10.97761 |
0.6 | 0.00005 | 0.55066 | 3.70707 | 5.01426 | 11.42121 |
0.7 | 0.00004 | 0.65862 | 3.73951 | 5.60723 | 11.93270 |
0.8 | 0.00007 | 0.79862 | 3.74304 | 6.27816 | 12.50642 |
0.9 | 0.00018 | 0.97608 | 3.71836 | 7.02218 | 13.13624 |
x | Proposed Method h5 | Taylor Equation (45) | Linearization Equation (40) | Zero-Order Equation (39) | Perturbation Equation (43) |
---|---|---|---|---|---|
0.0 | 0.00023 | 1.36905 | 3.30851 | 5.91347 | 4.92646 |
0.1 | 0.00021 | 1.39359 | 3.35215 | 6.01828 | 4.95513 |
0.2 | 0.00016 | 1.46981 | 3.43570 | 6.33032 | 5.03704 |
0.3 | 0.00012 | 1.60514 | 3.51105 | 6.84277 | 5.16019 |
0.4 | 0.00007 | 1.81087 | 3.55219 | 7.54513 | 5.30551 |
0.5 | 0.00001 | 2.10070 | 3.54752 | 8.42437 | 5.44798 |
0.6 | 0.00007 | 2.48944 | 3.49474 | 9.46589 | 5.55769 |
0.7 | 0.00017 | 2.99174 | 3.39741 | 10.65448 | 5.60064 |
0.8 | 0.00027 | 3.62121 | 3.26263 | 11.97477 | 5.53921 |
0.9 | 0.00029 | 4.38984 | 3.09960 | 13.41166 | 5.33237 |
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Abukhaled, M.; Khuri, S. An Efficient Semi-Analytical Solution of a One-Dimensional Curvature Equation that Describes the Human Corneal Shape. Math. Comput. Appl. 2019, 24, 8. https://doi.org/10.3390/mca24010008
Abukhaled M, Khuri S. An Efficient Semi-Analytical Solution of a One-Dimensional Curvature Equation that Describes the Human Corneal Shape. Mathematical and Computational Applications. 2019; 24(1):8. https://doi.org/10.3390/mca24010008
Chicago/Turabian StyleAbukhaled, Marwan, and Suheil Khuri. 2019. "An Efficient Semi-Analytical Solution of a One-Dimensional Curvature Equation that Describes the Human Corneal Shape" Mathematical and Computational Applications 24, no. 1: 8. https://doi.org/10.3390/mca24010008
APA StyleAbukhaled, M., & Khuri, S. (2019). An Efficient Semi-Analytical Solution of a One-Dimensional Curvature Equation that Describes the Human Corneal Shape. Mathematical and Computational Applications, 24(1), 8. https://doi.org/10.3390/mca24010008