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Investigation of nonlinear fractional delay differential equation via singular fractional operator

  • Dildar Ahmad , Amjad Ali , Ibrahim Mahariq , Ghaus ur Rahman and Kamal Shah ORCID logo EMAIL logo
Published/Copyright: November 22, 2021
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 2

Abstract

The present research work is basically devoted to construction of a fractional order differential equation with time delay. Initially, integral representation is given to solution of the underline problem. Afterwards, operator form of solution is studied under some auxiliary hypothesis. Since uniqueness of solution is required, therefore we also provide results for exploring the uniqueness of solution for the underlying model. Using Lebesgue dominated convergence theorem and some other results from analysis, this work provides results devoted to existence of at least one solution. Also, for investigating the nature of solution for the proposed model, we study different kind of stability analysis. These stability related results show, how the solution behave with time. At the end of the article, we illustrate the obtained results via some examples.

Keywords: Banach theorem; existence theory; fractional delay DEs; Schaefer theorem; Ulam’s type stability
Mathematics subject classification: 26A33; 34A08; 93A30
Corresponding author: Kamal Shah, Department of Mathematics, University of Malakand, Chakdara Dir(L), 18000 Khyber Pakhtunkhwa, Pakistan; and Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia, E-mail:

Acknowledgments

The authors are agreed to submit this version.

  1. Author contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

  2. Research funding: No Funding is available.

  3. Conflict of interest statement: The authors declare that they have no competing interests.

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Received: 2021-04-06
Revised: 2021-08-04
Accepted: 2021-11-04
Published Online: 2021-11-22

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijnsns-2021-0150
Keywords for this article
Banach theorem; existence theory; fractional delay DEs; Schaefer theorem; Ulam’s type stability

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Numerical solutions for variable-order fractional Gross–Pitaevskii equation with two spectral collocation approaches
  4. Threshold dynamics of an HIV-1 model with both virus-to-cell and cell-to-cell transmissions, immune responses, and three delays
  5. Numerical scheme and analytical solutions to the stochastic nonlinear advection diffusion dynamical model
  6. On modeling column crystallizers and a Hermite predictor–corrector scheme for a system of integro-differential equations
  7. On a discrete fractional stochastic Grönwall inequality and its application in the numerical analysis of stochastic FDEs involving a martingale
  8. A new self adaptive Tseng’s extragradient method with double-projection for solving pseudomonotone variational inequality problems in Hilbert spaces
  9. Solitary waves of the RLW equation via least squares method
  10. Optical solitons and stability regions of the higher order nonlinear Schrödinger’s equation in an inhomogeneous fiber
  11. Weighted pseudo asymptotically Bloch periodic solutions to nonlocal Cauchy problems of integrodifferential equations in Banach spaces
  12. Modified inertial subgradient extragradient method for equilibrium problems
  13. Propagation of dark-bright soliton and kink wave solutions of fluidized granular matter model arising in industrial applications
  14. Existence and uniqueness of positive solutions for fractional relaxation equation in terms of ψ-Caputo fractional derivative
  15. Investigation of nonlinear fractional delay differential equation via singular fractional operator
  16. Travelling peakon and solitary wave solutions of modified Fornberg–Whitham equations with nonhomogeneous boundary conditions
  17. The (3 + 1)-dimensional Wazwaz–KdV equations: the conservation laws and exact solutions
  18. Stability and ψ-algebraic decay of the solution to ψ-fractional differential system
  19. Some new characterizations of a space curve due to a modified frame N ⃗ , C ⃗ , W ⃗ in Euclidean 3-space
  20. Numerical simulation of particulate matter propagation in an indoor environment with various types of heating
  21. Inertial accelerated algorithms for solving split feasibility with multiple output sets in Hilbert spaces
  22. Stability analysis and abundant closed-form wave solutions of the Date–Jimbo–Kashiwara–Miwa and combined sinh–cosh-Gordon equations arising in fluid mechanics
  23. Contrasting effects of prey refuge on biodiversity of species
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