Electronic Journal of Differential Equations 15th annual Conference on Applied Mathematics, Univ. of Central Oklahoma,
Electron. J. Diff. Eqns., Conf. 02, 1999, pp. 115-124.

### Mathematical model for the basilar membrane as a two dimensional plate H. Y. Alkahby, M. A. Mahrous, & B. Mamo

Abstract:
In this paper we present two mathematical models for the basilar membrane. In the first model the membrane is represented as an annular region. In the second model the basilar membrane is treated as a rectangular region. Comparison of the two models allows us to study the effect of the curvature of the basilar membrane on the range of the frequencies of hearing. The differential equation of both models is a fourth order partial differential equation derived from the classical plate theory. Boundary conditions are defined as a region with four sides. The conditions are different on each side and together form an interesting physiological combination, relative to standard engineering problems. Eigenvalues of the differential equations of the two models are obtained numerically. A comparison of the eigenvalues of the two models clearly shows that the range of the hearing frequencies of the first model is larger than that of the second model. The results indicate strongly that the curvature of the basilar membrane plays an important role in the hearing process. Curvature and measurement of curvature should be allowed in future models and experiments of the inner ear.

Published January 21, 2000.
Subject classfications: 92C05, 92610, 35G15, 34B10.
Key words: Basilar membrane, eigenvalue, hearing frequencies.

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Hadi Y. Alkahby & B. Mamo
Division of the Natural Sciences, Dillard University
New Orleans, LA 70122 USA
email: halkahby@aol.com   Tel.: 504-286-4731

M. A. Mahrous
Department of Mathematics, University of New Orleans
New Orleans, LA 70148, USA