×

zbMATH — the first resource for mathematics

On a nonlinear boundary value problem modeling corneal shape. (English) Zbl 1307.92129
Summary: We present some results concerning a boundary value problem for a nonlinear ordinary differential equation that was used before in modeling the topography of human cornea. These results generalize previously obtained theorems on existence and uniqueness. We show that our equation has a unique solution for all parameters and conditions that can arise in physical situation. In the second part of the article we derive some new estimates and approximate solutions. Numerical calculations verify that these approximations are very accurate.

MSC:
92C50 Medical applications (general)
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, K.; El-Sheikh, A.; Newson, T., Application of structural analysis to the mechanical behaviour of the cornea, J. Roy. Soc. Interface, 1, 3-15, (2004)
[2] Bielecki, A., Une remarque sur la method de Banach-cacciopoli-tikhnov dans la theorie des equations dierentilles ordinaries, Bull. Pol. Acad. Sci. Math. Phys. Astron., 4, 261-264, (1956) · Zbl 0070.08103
[3] Braun, R. J.; Fitt, A. D., Modelling drainage of the precorneal tear film after a blink, Math. Med. Biol., 20, 1-28, (2003) · Zbl 1042.92004
[4] Canning, C. R.; Dewynne, J. N.; Fitt, A. D.; Greaney, M. J., Fluid flow in the anterior chamber of a human eye, IMA J. Math. Appl. Med. Biol., 19, 31-60, (2002) · Zbl 1013.92014
[5] He, J.-H., Asymptotic methods for solitary solutions and compactons, Abstr. Appl. Anal., 1-130, (2012), art. no. 916793 · Zbl 1257.35158
[6] He, J.-H., A remark on “A nonlinear mathematical model of the corneal shape”, Nonlinear Anal. Real World Appl., 13, 2863-2865, (2012) · Zbl 1257.34010
[7] Iskander, D. R.; Collins, M. J.; Davis, B., Optimal modeling of corneal surfaces by Zernike polynomials, IEEE Trans. Biomed. Eng., 48, 87-95, (2001)
[8] Kasprzak, H.; Iskander, D. R., Approximating ocular surfaces by generalized conic curves, Ophthal. Physiol. Opt., 26, 602-609, (2006)
[9] Martínez-Finkelshtein, A.; López, D. R.; Castro, G. M.; Alió, J. L., Adaptive cornea modeling from keratometric data, Investig. Ophthalmol. Vis. Sci., 52, 4963-4970, (2011)
[10] Mejía-Barbosa, Y.; Malacara-Hernández, D., A review of methods for measuring corneal topography, Optom. Vis. Sci., 78, 240-253, (2001)
[11] Okrasiński, W.; Płociniczak, Ł., A nonlinear mathematical model of the corneal shape, Nonlinear Anal. Real World Appl., 13, 1498-1505, (2012) · Zbl 1239.34004
[12] Okrasiński, W.; Płociniczak, Ł., Bessel function model of corneal topography, Appl. Math. Comput., 223, 436-443, (2013) · Zbl 1329.92058
[13] Płociniczak, Ł.; Okrasiński, W., Regularization of an ill-posed problem in corneal topography, Inverse Probl. Sci. Eng., 21, 1090-1097, (2013) · Zbl 1308.35326
[14] Talu, S.; Talu, M., An overview on mathematical models of human corneal surface, (IFMBE Proceedings, vol. 26, (2009)), 291-294
[15] Trattler, W.; Majmudar, P.; Luchs, J. I.; Swartz, T., Cornea handbook, (2010), Slack Incorporated
[16] Zhang, X.; Feng, M., Exact number of solutions of a one-dimensional prescribed mean curvature equation with concave-convex nonlinearities, J. Math. Anal. Appl., 395, 393-402, (2012) · Zbl 1250.34020
[17] Zheng, S.; Ying, J.; Wang, B.; Xie, Z.; Huang, X.; Shi, M., Three-dimensional model for human anterior corneal surface, J. Biomedical Optics, 18, 1-5, (2013), art. no. 65002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.