Existence and stability of nonlinear discrete fractional initial value problems with application to vibrating eardrum

George Maria Selvam; Jehad Alzabut; Vignesh Dhakshinamoorthy; Jagan Mohan Jonnalagadda; Kamaleldin Abodayeh; George Maria Selvam; Jehad Alzabut; Vignesh Dhakshinamoorthy; Jagan Mohan Jonnalagadda; Kamaleldin Abodayeh

doi:10.3934/mbe.2021195

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 4: 3907-3921. doi: 10.3934/mbe.2021195

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Existence and stability of nonlinear discrete fractional initial value problems with application to vibrating eardrum

1.
Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur-635601, Tamil Nadu, India
2.
Department of Mathematics and General Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia
3.
Group of Mathematics, Faculty of Engineering, Ostim Technical University, 06374 Ankara, Turkey
4.
Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad-500078, Telangana, India

Academic Editor: Juan Luis García Guirao

Received: 26 February 2021 Accepted: 20 April 2021 Published: 06 May 2021

It is well known that Newton's second law can be applied in various biological processes including the behavior of vibrating eardrums. In this work, we consider a nonlinear discrete fractional initial value problem as a model describing the dynamic of vibrating eardrum. We establish sufficient conditions for the existence, uniqueness, and Hyers-Ulam stability for the solutions of the proposed model. To examine the validity of our findings, a concrete example of forced eardrum equation along with numerical simulation is analyzed.
- nonlinear discrete fractional problem,
- existence and uniqueness,
- Hyers–Ulam stability,
- vibrating eardrum
Citation: George Maria Selvam, Jehad Alzabut, Vignesh Dhakshinamoorthy, Jagan Mohan Jonnalagadda, Kamaleldin Abodayeh. Existence and stability of nonlinear discrete fractional initial value problems with application to vibrating eardrum[J]. Mathematical Biosciences and Engineering, 2021, 18(4): 3907-3921. doi: 10.3934/mbe.2021195

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Abstract

It is well known that Newton's second law can be applied in various biological processes including the behavior of vibrating eardrums. In this work, we consider a nonlinear discrete fractional initial value problem as a model describing the dynamic of vibrating eardrum. We establish sufficient conditions for the existence, uniqueness, and Hyers-Ulam stability for the solutions of the proposed model. To examine the validity of our findings, a concrete example of forced eardrum equation along with numerical simulation is analyzed.

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