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Well-posedness of a nonlinear Hilfer fractional derivative model for the Antarctic circumpolar current

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Abstract

This article explores the Hilfer fractional derivative within the context of fractional differential equations and investigates a mathematical model formulated as a three-point boundary value problem (BVP). The primary focus is on the application of these models to analyze the jet flow of the Antarctic Circumpolar Current. The study establishes the existence of stream functions using Schaefer’s fixed point theorem under the assumption of the continuity of the vorticity function \(\Phi \). Furthermore, the article delves into the existence and uniqueness results of the stream functions by employing the Banach fixed point theorem. This analysis is conducted under the condition that the vorticity function \(\Phi \) is Lipschitz continuous with respect to the stream function. Additionally, the stability of the stream functions of the BVP is explored through Ulam–Hyers and generalized Ulam–Hyers stability analyses. In contrast to the foundational results presented for the three-point BVP, the article includes illustrative examples aimed at validating the findings.

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Acknowledgements

The third and fourth author expresses gratitude to the Council of Scientific and Industrial Research, Government of India, for their support through research project no-25(0321)/23/EMR-II. The authors extend their appreciation to the reviewers for offering valuable insights that have enhanced the quality of this paper.

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No funding was received for conducting this study.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada

    H. M. Srivastava

  2. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan, Republic of China

    H. M. Srivastava

  3. Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, 02447, Republic of Korea

    H. M. Srivastava

  4. Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007, Baku, Azerbaijan

    H. M. Srivastava

  5. Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taoyuan City, 320314, Taiwan

    H. M. Srivastava

  6. Section of Mathematics, International Telematic University Uninettuno, 00186, Rome, Italy

    H. M. Srivastava

  7. Yogananda School of AI, Computer, and Data Science, Shoolini University, Bajhol, Solan, Himachal Pradesh, 173229, India

    Kanika Dhawan

  8. Department of Mathematics and Scientific Computing, National Institute of Technology, Hamirpur, Himachal Pradesh, 177005, India

    Ramesh Kumar Vats

  9. Department of Mathematics and Scientific Computing, National Institute of Technology, Hamirpur, Himachal Pradesh, 177005, India

    Ankit Kumar Nain

Authors
  1. H. M. Srivastava
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  2. Kanika Dhawan
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  3. Ramesh Kumar Vats
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  4. Ankit Kumar Nain
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Contributions

KD and AKN conceived of the presented idea. KD and AKN developed the theory and performed the computations. RKV and HMS verified the analytical methods. HMS and RKV encouraged KD and AKN to investigate the stability aspect and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.

Corresponding author

Correspondence to Ramesh Kumar Vats.

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Srivastava, H.M., Dhawan, K., Vats, R.K. et al. Well-posedness of a nonlinear Hilfer fractional derivative model for the Antarctic circumpolar current. Z. Angew. Math. Phys. 75, 45 (2024). https://doi.org/10.1007/s00033-024-02192-0

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