Existence and Uniqueness Theorems for Fractional Differential Equations with Proportional Delay

Document Type : Research Paper

Authors

1 Department of Mathematics, Shivaji University, Kolhapur - 416004, India.

2 Department of Mathematics, Jaysingpur College, Jaysingpur (Affiliated to Shivaji University, Kolhapur) - 416101, India.

Abstract

In this paper, we applied successive approximation method (SAM) to deal with the solution of non-linear differential equations (DEs) with proportional delay. Utilizing SAM we derived the results about existence and uniqueness. The differential equations (DEs) with proportional delay are a particular case of the time-dependent delay differential equations (DDEs). In this sense, we demonstrated that the equilibrium solution of time-dependent DDEs is asymptotically stable on finite time intervals. We obtained a series solution of pantograph and Ambartsumian equations and proved its convergence. Further, we proved that the zero solution of pantograph and Ambartsumian equations are asymptotically stable. The outcomes of integer order obtained for DEs with proportional delay and time-dependent DDEs have been extended to initial value problem (IVP) for fractional DDEs and a system of fractional DDEs involving Caputo fractional derivative. Finally, we illustrate the efficacy of the SAM by considering particular non-linear DEs with proportional delay. The results obtained for non-linear DEs with proportional delay by SAM are compared with exact solutions and other iterative methods. It is noted that SAM is easier to use than other techniques and the solutions obtained using SAM are consistent with the exact solution.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 08 July 2024

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History

  • Receive Date: 22 July 2023
  • Revise Date: 10 January 2024
  • Accept Date: 06 May 2024

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