Analysis of a class of fractional delay integro-differential equations with Riesz-Caputo derivative

Pratima Tiwari; Rajesh K. Pandey

doi:10.3934/dcdss.2024048

Article Contents

2025, Volume 18, Issue 5: 1267-1284. Doi: 10.3934/dcdss.2024048

This issue Previous Article Dynamics investigation and numerical simulation of fractional-order predator-prey model with Holling type $ II $ functional response Next Article Stock patterns in a class of delayed discrete-time population models

Analysis of a class of fractional delay integro-differential equations with Riesz-Caputo derivative

Pratima Tiwari^1, and
Rajesh K. Pandey^1, ,

1.
Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Uttar Pradesh, 221005, India

^*Corresponding author: Rajesh K. Pandey
^*Corresponding author: Rajesh K. Pandey

Received: January 2024

Revised: March 2024

Early access: April 2024

Published: May 2025

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

This paper investigates the existence and uniqueness of solutions for a particular category of fractional delay integro-differential (FDID) equations in the context of Banach space, incorporating the Riesz-Caputo fractional derivative. Employing fractional calculus techniques and multiple fixed-point theorems, we establish a few results regarding both existence and uniqueness. Further, by introducing a partial order in a Banach space of all continuous functions, we look into the existence of extremal solutions. To demonstrate the competence of the suggested outcomes, a few instances are presented at the conclusion.

Keywords:

Existence theorem,
uniqueness theorem,
Riesz-Caputo fractional derivative,
nonlinear fractional differential equation,
extremal solutions.

Mathematics Subject Classification: Primary: 34A08.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	N. Aboutabit, A new construction of an image edge detection mask based on Caputo-Fabrizio fractional derivative, The Visual Computer (Springer), 37 (2021), 1545-1557.
[2]	N. Adjimi and M. Benbachir, Existence results for langevin equation with Riesz-Caputo fractional derivative, Surveys in Mathematics and its Applications, 17 (2022), 225-239.
[3]	M. Alqhtani, K. M. Owolabi, K. M. Saad and E. Pindza, Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology, Chaos, Solitons & Fractals, 161 (2022), 112394. doi: 10.1016/j.chaos.2022.112394.
[4]	S. Balochian and H. Baloochian, Edge detection on noisy images using Prewitt operator and fractional order differentiation, Multimedia Tools and Applications (Springer), 81 (2022), 9759-9770.
[5]	A. Boutiara, N. Adjimi, M. Benbachir and A. Mohammed, Analysis of a fractional boundary value problem involving Riesz-Caputo fractional derivative, Advances in the Theory of Nonlinear Analysis and its Application, 6 (2022), 14-27.
[6]	F. Chen, D. Baleanu and G. C. Wu, Existence results of fractional differential equations with Riesz–Caputo derivative, The European Physical Journal Special Topics (Springer), 226 (2017), 3411-3425.
[7]	F. Chen, A. Chen and X. Wu, Anti-periodic boundary value problems with Riesz–Caputo derivative, Advances in Difference Equations (Springer), 2019 (2019), Paper No. 119, 13 pp. doi: 10.1186/s13662-019-2001-z.
[8]	B. C. Dhage and J. Henderson, Existence theory for nonlinear functional boundary value problems, Electronic Journal of Qualitative Theory of Differential Equations, 2004 (2004), 1-15. doi: 10.14232/ejqtde.2004.1.1.
[9]	T. Erneux, Applied Delay Differential Equations, Springer, Berlin, 2009.
[10]	G. S. F. Frederico and D. F. M. Torres, Fractional conservation laws in optimal control theory, Nonlinear Dynamics (Springer), 53 (2008), 215-222. doi: 10.1007/s11071-007-9309-z.
[11]	A. Granas and J. Dugundji, Fixed Point Theory, Springer, Berlin, Germany, 2003.
[12]	C.-Y. Gu, J. Zhang and G.-C. Wu, Positive solutions of fractional differential equations with the Riesz space derivative, Applied Mathematics Letters (Elsevier), 95 (2019), 59-64. doi: 10.1016/j.aml.2019.03.006.
[13]	J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer Science & Business Media, New York, 2013.
[14]	R. Isaacs, Differential Games: A mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, Courier Corporation, Mineola, New York, 1999.
[15]	H. Joshi and B. K. Jha, Fractional-order mathematical model for calcium distribution in nerve cells, Computational and Applied Mathematics (Springer), 39 (2020), Paper No. 56, 22 pp. doi: 10.1007/s40314-020-1082-3.
[16]	H. Joshi, M. Yavuz, S. Townley and B. K. Jha, Stability analysis of a non-singular fractional-order covid-19 model with nonlinear incidence and treatment rate, Physica Scripta (IOP Publishing), 98 (2023), 045216.
[17]	D. Kai, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, in: Lecture Notes in Mathematics, Springer, 2010.
[18]	K. Kaur, N. Jindal and K. Singh, QRFODD: Quaternion Riesz Fractional Order Directional Derivative for color image edge detection, Signal Processing (Elsevier), 212 (2023), 109170.
[19]	A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland Mathematical Studies, Amsterdam, 2006.
[20]	Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, San Diego, New York, 1993.
[21]	S. Kumar, R. Saxena and K. Singh, Fractional Fourier transform and fractional-order calculus-based image edge detection, Circuits, Systems, and Signal Processing (Springer), 36 (2017), 1493-1513.
[22]	A. E. Matouk, Advanced Applications of Fractional Differential Operators to Science and Technology, IGI Global, Hershey PA, USA, 2020.
[23]	R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports (Elsevier), 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.
[24]	O. Naifar and A. B. Makhlouf, Fractional Order Systems–Control Theory and Applications, Springer, Switzerland, 2022.
[25]	K. M. Owolabi, Analysis and numerical simulation of cross reaction-diffusion systems with the Caputo-Fabrizio and Riesz operators, Numerical Methods for Partial Differential Equations, 39 (2023), 1915-1937.
[26]	K. M. Owolabi, Numerical simulation of fractional-order reaction–diffusion equations with the Riesz and Caputo derivatives, Neural Computing and Applications, 32 (2020), 4093-4104.
[27]	K. M. Owolabi, J. F. G$\acute{o}$mez-Aguilar, Y. Karacha, Y. Li, B. Saleh and A. A. Ayman, Chaotic behavior in fractional Helmholtz and Kelvin-Helmholtz instability problems with Riesz operator, Fractals, 30 (2022), 2240182.
[28]	K. M. Owolabi and E. Pindza, Spatiotemporal chaos in diffusive systems with the Riesz fractional order operator, Chinese Journal of Physics, 77 (2022), 2258-2275. doi: 10.1016/j.cjph.2021.12.031.
[29]	I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Elsevier, San Diego, California, 1998.
[30]	K. Pyragas, Continuous control of chaos by self-controlling feedback, Physics Letters A (Elsevier), 170 (1992), 421-428.
[31]	W. Rahou, A. Salim, J. E. Lazreg and M. Benchohra, Existence and stability results for impulsive implicit fractional differential equations with delay and Riesz-Caputo derivative, Mediterranean Journal of Mathematics (Springer), 20 (2023), 143. doi: 10.1007/s00009-023-02356-8.
[32]	F. A. Rihan, Delay Differential Equations and Applications to Biology, Springer, Singapore, 2021.
[33]	J. Sabatier, P. Lanusse, P. Melchior and A. Oustaloup, Fractional Order Differentiation and Robust Control Design, Intelligent systems, Control and Automation: Science and Engineering, Springer, Dordrecht, 2015. doi: 10.1007/978-94-017-9807-5.
[34]	O. P. Sabatier and J. A. T. Machado, Advances in fractional calculus: The-oretical developments and applications in physics and engineering, edited by J., SIAM J. Appl. Math. (SIAM), 63 (2003), 612.
[35]	A. Salim, J. E. Lazreg and M. Benchohra, Existence, uniqueness and Ulam-Hyers-Rassias stability of differential coupled systems with Riesz-Caputo fractional derivative, Tatra Mountains Mathematical Publications, 84 (2023), 111-138. doi: 10.2478/tmmp-2023-0019.
[36]	A. K. Shukla, R. K. Pandey and R. B. Pachori, A fractional filter based efficient algorithm for retinal blood vessel segmentation, Biomedical Signal Processing and Control (Elsevier), 59 (2020), 101883.
[37]	A. K. Shukla, R. K. Pandey, S. Yadav and R. B. Pachori, Generalized fractional filter-based algorithm for image denoising, Circuits, Systems, and Signal Processing (Springer), 39 (2020), 363-390.
[38]	G.-C. Wu, D. Baleanu, Z.-G. Deng and S.-D. Zeng, Lattice fractional diffusion equation in terms of a Riesz-Caputo difference, Physica A: Statistical Mechanics and its Applications (Elsevier), 438 (2015), 335-339. doi: 10.1016/j.physa.2015.06.024.
[39]	G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Physics Reports (Elsevier), 371 (2002), 461-580. doi: 10.1016/S0370-1573(02)00331-9.
[40]	E. Zeidler, Nonlinear Functional Analysis: Part I, Springer, Verlag, 1985.
[41]	M. Zhou, Well-posedness for multi-point BVPs for fractional differential equations with Riesz-Caputo derivative, Authorea Preprints (Authorea), 2022.