TARU PUBLICATIONS
Journal of Interdisciplinary Mathematics cover
Hybrid ·Peer-reviewed·ISSN (Online): 2169-012X·ISSN (Print): 0972-0502

Monthly Journal: Publishes the methodological and theoretical role of mathematics and mathematical applications underpinning scientific research.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

Fractional differential equation with movable boundary conditions

* , , ,

* Corresponding author · click or hover a name for details

pp. 233–243Vol. 27Issue 2March 2024DOI: 10.47974/JIM-1817XML
Published Online:
19 Mar 2024
Article type:
Research Article
Language:
EN
Article no.:
JIM-1817
Pages:
233–243

Abstract

In this research paper, we discuss the complex-valued solutions for the nonlinear fractional boundary value problem (FBVP) of complex order (δ = τ + ιa; 1 < τ ≤ 2, a ∈ R+) with movable boundary conditions. The fractional operators are taken in the sense of Riemann-Liouville (R-L) with complex order. By using the concept of Green’s function, the existence and uniqueness of solutions are established in this article. Also, we prove that the FBVP of complex order with movable boundary conditions is Ulam-Hyers Stable. Using illustrative examples, the results for this nonlinear FBVP have been shown.

Keywords

Subject Classifications

Primary 26A33Secondary 34A0835R11

References

[1] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations. Wiley (1993).
[2] R.Alchikh, and S. Khuri, On the solutions of the fractional bratu’s problem,”  Journal of Interdisciplinary Mathematics vol. 25,no.6, pp. 1093-1107 (2020).
[3] Z. Baitiche, and C. Derbazi, “Implicit fractional differential equations with φ–caputo fractional derivative in banach spaces,” Journal of Interdisciplinary Mathematics, vol.25, no. 5, pp. 1237–1252 (2022).
[4] B. Ross, Fractional calculus and its applications: Proceedings of the international conference held at the University of New Haven, June 1974, vol. 457. Springer (2006).
[5] R. Metzler, W. Schick, H.-G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: A fractional calculus approach,” The Journal of Chemical Physics, vol. 103, no. 16, pp. 7180-7186 (1995).
[6] R. Magin, “Fractional calculus in bioengineering, part 1,” Critical Reviews in Biomedical Engineering, vol. 32, no. 1 (2004).
[7] Y. Shen, J. Hua, W. Fan, Y. Liu, X. Yang, and L. Chen, “Optimal design and dynamic performance analysis of a fractional-order electrical network-based vehicle mechatronic ISD suspension,” Mechanical Systems and Signal Processing, vol. 184, p. 109718 (2023).
[8] S. Arora, T. Mathur, S. Agarwal, K. Tiwari, and P. Gupta, “Applications of fractional calculus in computer vision: A survey,” Neurocomputing, vol. 489, pp. 407-428 (2022).
[9] N. Singh, S. Arora, T. Mathur, S. Agarwal, and K. Tiwari, “Stock price prediction using fractional gradient-based long short term memory,” in Journal of Physics: Conference series, vol. 1969, p. 012038 (2021).
[10] J. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57-68 (1998).
[11] T. M. Atanacković, S. Konjik, S. Pilipović, and D. Zorica, “Complex order fractional derivatives in viscoelasticity,” Mechanics of Time-Dependent Materials, vol. 20, pp. 175-195 (2016).
[12] S.P. Goyal, and T. Mathur, “On generalized fractional diffusion equations,” South East Asian Journal of Mathematics and mathematical Sciences, vol. 4, pp. 53-61 (2006).
[13] T. Mathur, “On generalized fractional diffusion equation-II,” Journal of Rajasthan Academy of Physical Sciences, vol. 3, pp. 183-190 (2004).
[14] A. Yadav, T. Mathur, S. Agarwal, and B. Yadav, “Fractional boundary value problem in complex domain,” Journal of Mathematical Analysis and Applications, vol. 526, no. 1 (2023).
[15] W. Xie, J. Xiao, and Z. Luo, “Existence of extremal solutions for nonlinear fractional differential equation with nonlinear boundary conditions,” Applied Mathematics Letters, vol. 41, pp. 46-51 (2015).
[16] A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, vol. 204. Elsevier (2006).
[17] K. Hussain, A. Hamoud, and N. Mohammed, “Some new uniqueness results for fractional integro-differential equations,” Nonlinear Functional Analysis and Applications, vol. 24, no. 4, pp. 827-836 (2019).
[18] M. Benchohra and S. Bouriah, “Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order,” Moroccan Journal of Pure and Applied Analysis, vol. 1, pp. 1-16 (2015).
[19] Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 916-924 (2010).
[20] K. Ma, X. Li, and S. Sun, “Boundary value problems of fractional q-difference equations on the half-line,” Boundary Value Problems, vol. 2019, no. 1, pp. 1-16 (2019).
[21] Kumar, Anil , Gupta, Amit Kumar, Panwar, Deepak , Chaurasia, Sandeep & Goyal, Dinesh. Operating system security with discrete mathematical structure for secure round robin scheduling method with intelligent time quantum, Journal of Discrete Mathematical Sciences and Cryptography, 26:5, 1519–1533 (2023).
[22] Kushwaha, Satpal Singh, Joshi, Sandeep & Gupta, Amit Kumar. An efficient approach to secure smart contract of Ethereum blockchain using hybrid security analysis approach, Journal of Discrete Mathematical Sciences and Cryptography, 26:5, 1499–1517 (2023), DOI: https://doi.org/10.47974/JDMSC-1815.
[23] Joshi, Ruchi, Mathur, Priya, Gupta, Amit Kumar, Singh, Suyesha, Paliwal, Vismita & Nayar, Sejal. Mathematical modeling of intelligent system for predicting effectiveness of premenstrual syndrome, Journal of Interdisciplinary Mathematics, 26:3, 551-562 (2023). 

Views: 324Downloads: 79Citations: 1