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Existence and stability of nonlinear discrete fractional initial value problems with application to vibrating eardrum
George Maria Selvam1, Jehad Alzabut2,3, Vignesh Dhakshinamoorthy1
1Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur-635601, Tamil Nadu, India.
Mathematical Biosciences and Engineering : MBE
|July 2, 2021
Summary
This study models vibrating eardrum dynamics using a nonlinear discrete fractional model. It proves the existence, uniqueness, and stability of solutions, validating findings with a forced eardrum equation example.
Area of Science:
- Biophysics
- Mathematical Biology
- Nonlinear Dynamics
Background:
- Newton's second law applies to biological systems like vibrating eardrums.
- Modeling eardrum dynamics requires understanding complex mechanical behaviors.
- Fractional calculus offers advanced tools for modeling such phenomena.
Purpose of the Study:
- To develop a nonlinear discrete fractional model for vibrating eardrum dynamics.
- To establish conditions for the existence, uniqueness, and Hyers-Ulam stability of the model's solutions.
- To analyze the model's validity through a specific eardrum equation and numerical simulation.
Main Methods:
- Utilizing nonlinear discrete fractional calculus for dynamic modeling.
- Applying mathematical analysis to prove existence and uniqueness theorems.
- Establishing Hyers-Ulam stability criteria for the fractional model.
- Conducting numerical simulations for a forced eardrum equation.
Main Results:
- Sufficient conditions for the existence and uniqueness of solutions were established.
- Hyers-Ulam stability of the solutions for the proposed model was proven.
- Numerical simulations confirmed the theoretical findings for a forced eardrum equation.
Conclusions:
- The nonlinear discrete fractional model effectively describes vibrating eardrum dynamics.
- The established conditions ensure reliable and stable solutions for eardrum vibration modeling.
- This work provides a robust mathematical framework for analyzing eardrum mechanics.
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