Qualitative analysis of the fractional Basset problem with boundary conditions via the Caputo–Fabrizio derivative

Shayma Adil Murad; Ava Sh. Rafeeq; Alan M. Omar; Mohammed O. Mohammed; Thabet Abdeljawad; Manar A. Alqudah; Shayma Adil Murad; Ava Sh. Rafeeq; Alan M. Omar; Mohammed O. Mohammed; Thabet Abdeljawad; Manar A. Alqudah

doi:10.3934/math.2025945

AIMS Mathematics

2025, Volume 10, Issue 9: 21159-21192. doi: 10.3934/math.2025945

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Qualitative analysis of the fractional Basset problem with boundary conditions via the Caputo–Fabrizio derivative

1.
Department of Mathematics, College of Science, University of Duhok, Duhok 42001, Iraq
2.
Department of Mathematics, College of Science, University of Zakho, Duhok, Iraq
3.
Department of Fundamental Sciences, Faculty of Engineering and Architecture, Istanbul Gelisim University, Avcılar-Istanbul 34310, Turkiye
4.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
5.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
6.
Department of Mathematical Sciences, College of Sciences, Princess Nourah bint Abdulrahman University P. O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 05 June 2025 Revised: 25 August 2025 Accepted: 02 September 2025 Published: 15 September 2025
MSC : 34A08, 26A33, 34K20, 34A12, 97M50

The concept of fixed points serves as an effective and essential tool in analyzing nonlinear phenomena. This study investigates the existence and uniqueness of solutions for a class of Basset-type fractional differential equations with boundary conditions involving the Caputo–Fabrizio fractional derivative. These equations emerge from the generalized Basset force describing the motion of a sphere settling in a viscous fluid. Darbo's fixed point theorem, combined with the measure of noncompactness, is applied to establish the existence of solutions. Uniqueness is ensured via Banach's fixed point theorem. Additionally, stability analysis is performed using Ulam–Hyers and Ulam–Hyers–Rassias concepts. An illustrative example, supported by tables and figures, demonstrates the applicability of the theoretical results.
Citation: Shayma Adil Murad, Ava Sh. Rafeeq, Alan M. Omar, Mohammed O. Mohammed, Thabet Abdeljawad, Manar A. Alqudah. Qualitative analysis of the fractional Basset problem with boundary conditions via the Caputo–Fabrizio derivative[J]. AIMS Mathematics, 2025, 10(9): 21159-21192. doi: 10.3934/math.2025945

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Abstract

The concept of fixed points serves as an effective and essential tool in analyzing nonlinear phenomena. This study investigates the existence and uniqueness of solutions for a class of Basset-type fractional differential equations with boundary conditions involving the Caputo–Fabrizio fractional derivative. These equations emerge from the generalized Basset force describing the motion of a sphere settling in a viscous fluid. Darbo's fixed point theorem, combined with the measure of noncompactness, is applied to establish the existence of solutions. Uniqueness is ensured via Banach's fixed point theorem. Additionally, stability analysis is performed using Ulam–Hyers and Ulam–Hyers–Rassias concepts. An illustrative example, supported by tables and figures, demonstrates the applicability of the theoretical results.

References

[1]	R. L. Bagley, On the fractional order initial value problem and its engineering applications, In: Fractional calculus and its applications, Tokyo: College of Engineering, Nihon University, 1990, 12–20.
[2]	C. Ingo, R. L. Magin, T. B. Parrish, New insights into the fractional order diffusion equation using entropy and kurtosis, Entropy, 16 (2014), 5838–5852. https://doi.org/10.3390/e16115838 doi: 10.3390/e16115838
[3]	R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl., 59 (2010), 1586–1593. https://doi.org/10.1016/j.camwa.2009.08.039 doi: 10.1016/j.camwa.2009.08.039
[4]	F. Mainardi, Fractional calculus and waves in linear viscoelasticity: An introduction to mathematical models, Singapore: World Scientific, 2010. https://doi.org/10.1142/p614
[5]	M. M. Bhatti, M. Marin, R. Ellahi, I. M. Fudulu, Insight into the dynamics of EMHD hybrid nanofluid (ZnO/CuO-SA) flow through a pipe for geothermal energy applications, J. Therm. Anal. Calorim., 148 (2023), 14261–14273. https://doi.org/10.1007/s10973-023-12565-8 doi: 10.1007/s10973-023-12565-8
[6]	A. Hobiny, I. Abbas, M. Marin, The influences of the hyperbolic two-temperatures theory on waves propagation in a semiconductor material containing spherical cavity, Mathematics, 10 (2022), 121. https://doi.org/10.3390/math10010121 doi: 10.3390/math10010121
[7]	A. K. Yadav, E. Carrera, M. Marin, M. I. Othman, Reflection of hygrothermal waves in a Nonlocal Theory of coupled thermo-elasticity, Mech. Adv. Mater. Struct., 31 (2024), 1083–1096. https://doi.org/10.1080/15376494.2022.2130484 doi: 10.1080/15376494.2022.2130484
[8]	J. X. Zhang, Y. Q. Liu, T. Chai, Singularity-free low-complexity fault-tolerant prescribed performance control for spacecraft attitude stabilization, IEEE Trans. Autom. Sci. Eng., 22 (2025), 15408–15419. https://doi.org/10.1109/TASE.2025.3569566 doi: 10.1109/TASE.2025.3569566
[9]	X. Wang, X. Zhang, W. Pedrycz, S. H. Yang, D. Boutat, Consensus of TS fuzzy fractional-order, singular perturbation, multi-agent systems, Fractal Fract., 8 (2024), 523. https://doi.org/10.3390/fractalfract8090523 doi: 10.3390/fractalfract8090523
[10]	A. B. Basset, A treatise on hydrodynamics: With numerous examples, Cambridge: Deighton, Bell and Company, 1888.
[11]	A. B. Basset, The descent of a sphere in a viscous liquid, Nature, 83 (1910), 521.
[12]	F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fract., 7 (1996), 1461–1477. https://doi.org/10.1016/0960-0779(95)00125-5 doi: 10.1016/0960-0779(95)00125-5
[13]	F. Mainardi, P. Pironi, F. Tampieri, On a generalization of the Basset problem via fractional calculus, In: Proceedings of the 15th Canadian Congress of Applied Mechanics, CANCAM'95, Victoria: University of Victoria, 1995,836–837.
[14]	S. Salahshour, A. Ahmadian, N. Senu, D. Baleanu, P. Agarwal, On analytical solutions of the fractional differential equation with uncertainty: Application to the Basset problem, Entropy, 17 (2015), 885–902. https://doi.org/10.3390/e17020885 doi: 10.3390/e17020885
[15]	V. Govindaraj, K. Balachandran, Stability of Basset equation, J. Fract. Calc. Appl., 5 (2014), 1–15. https://doi.org/10.13140/2.1.1869.6962
[16]	K. Nouri, S. Elahi-Mehr, L. Torkzadeh, Investigation of the behavior of the fractional Bagley–Torvik and Basset equations via numerical inverse Laplace transform, Rom. Rep. Phys., 68 (2016), 503–514.
[17]	J. G. Abulahad, S. A. Murad, Existence, uniqueness and stability theorems for certain functional fractional initial value problem, Al-Rafidain J. Comput. Sci. Math., 8 (2011), 59–70. https://doi.org/10.33899/csmj.2011.163608 doi: 10.33899/csmj.2011.163608
[18]	Q. Dai, R. Gao, Z. Li, C. Wang, Stability of Ulam–Hyers and Ulam–Hyers–Rassias for a class of fractional differential equations, Adv. Differ. Equ., 2020 (2020), 103. https://doi.org/10.1186/s13662-020-02558-4 doi: 10.1186/s13662-020-02558-4
[19]	Z. Cui, Z. Zhou, Existence of solutions for Caputo fractional delay differential equations with nonlocal and integral boundary conditions, Fixed Point Theory Algorithms Sci. Eng., 2023 (2023), 1. https://doi.org/10.1186/s13663-022-00738-3 doi: 10.1186/s13663-022-00738-3
[20]	S. A. Murad, Certain analysis of solution for the nonlinear Two-point boundary value problem with Caputo fractional derivative, J. Funct. Space., 2022 (2022), 1385355. https://doi.org/10.1155/2022/1385355 doi: 10.1155/2022/1385355
[21]	S. A. Murad, Z. A. Ameen, Existence and Ulam stability for fractional differential equations of mixed Caputo-Riemann derivatives. AIMS Math., 7 (2022), 6404–6419. https://doi.org/10.3934/math.2022357
[22]	S. A. Murad, A. S. Rafeeq, T. Abdeljawad, Caputo-Hadamard fractional boundary-value problems in $\mathfrak{L}^\mathfrak{p}$-spaces, AIMS Math., 9 (2024), 17464–17488. https://doi.org/10.3934/math.2024849 doi: 10.3934/math.2024849
[23]	P. Castro, A. S. Silva, On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem, Math. Biosci. Eng., 19 (2022), 10809–10825. https://doi.org/10.3934/mbe.2022505 doi: 10.3934/mbe.2022505
[24]	H. Fazli, F. Bahrami, J. J. Nieto, General Basset-Boussinesq-Oseen equation: Existence, uniqueness, approximation and regularity of solutions, Int. J. Comput. Math., 97 (2019), 1792–1805. https://doi.org/10.1080/00207160.2019.1658870 doi: 10.1080/00207160.2019.1658870
[25]	H. Baghani, J. Alzabut, J. J. Nieto, Further results on the existence of solutions for generalized fractional Basset–Boussinesq–Oseen equation, Iran. J. Sci. Technol., Trans. A Sci., 44 (2020), 1461–1467. https://doi.org/10.1007/s40995-020-00942-z doi: 10.1007/s40995-020-00942-z
[26]	A. Ashyralyev, Well-posedness of the Basset problem in spaces of smooth functions, Appl. Math. Lett., 24 (2011), 1176–1180. https://doi.org/10.1016/j.aml.2011.02.002 doi: 10.1016/j.aml.2011.02.002
[27]	L. Cona, Fixed point approach to Basset problem, New Trends Math. Sci., 5 (2017), 175–181. http://doi.org/10.20852/ntmsci.2017.195 doi: 10.20852/ntmsci.2017.195
[28]	D. Bahuguna, J. Anjali, Application of Rothe's method to fractional differential equations, Malaya J. Matem., 7 (2019), 399–407. https://doi.org/10.26637/MJM0703/0006 doi: 10.26637/MJM0703/0006
[29]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[30]	S. Ullah, M. A. Khan, M. Farooq, Z. Hammouch, D. Baleanu, A fractional model for the dynamics of tuberculosis infection using Caputo–Fabrizio derivative, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 975–993. https://doi.org/10.3934/dcdss.2020057 doi: 10.3934/dcdss.2020057
[31]	A. Boudaoui, Y. El Hadj Moussa, Z. Hammouch, S. Ullah, A fractional-order model describing the dynamics of the novel coronavirus (COVID-19) with nonsingular kernel, Chaos Solitons Fract., 146 (2021), 110859. https://doi.org/10.1016/j.chaos.2021.110859 doi: 10.1016/j.chaos.2021.110859
[32]	F. Mansal, N. Sene, Analysis of fractional fishery model with reserve area in the context of time-fractional order derivative, Chaos Solitons Fract., 140 (2020), 110200. https://doi.org/10.1016/j.chaos.2020.110200 doi: 10.1016/j.chaos.2020.110200
[33]	K. Maazouz, R. Rodríguez-López, Differential equations of arbitrary order under Caputo–Fabrizio derivative: Some existence results and study of stability, Math. Biosci. Eng., 19 (2022), 6234–6251. https://doi.org/10.3934/mbe.2022291 doi: 10.3934/mbe.2022291
[34]	K. K. Ali, K. R. Raslan, A. Abd-Elall Ibrahim, M. S. Mohamed, On study the fractional Caputo–Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type, AIMS Math., 8 (2023), 18206–18222. https://doi.org/10.3934/math.2023925 doi: 10.3934/math.2023925
[35]	J. Bravo, C. Lizama, The abstract Cauchy problem with Caputo–Fabrizio fractional derivative, Mathematics, 10 (2022), 3540. https://doi.org/10.3390/math10193540 doi: 10.3390/math10193540
[36]	T. M. Atanacković, S. Pilipović, D. Zorica, Properties of the Caputo–Fabrizio fractional derivative and its distributional settings, Fract. Calc. Appl. Anal., 21 (2018), 29–44. https://doi.org/10.1515/fca-2018-0003 doi: 10.1515/fca-2018-0003
[37]	M. O. Mohammed, A. S. Rafeeq, New results for existence, uniqueness, and Ulam stable theorem to Caputo–Fabrizio fractional differential equations with periodic boundary conditions, Int. J. Appl. Comput. Math., 10 (2024), 109. https://doi.org/10.1007/s40819-024-01741-5 doi: 10.1007/s40819-024-01741-5
[38]	A. Shaikh, A. Tassaddiq, K. S. Nisar, D. Baleanu, Analysis of differential equations involving Caputo–Fabrizio fractional operator and its applications to reaction–diffusion equations, Adv. Differ. Equ., 2019 (2019), 178. https://doi.org/10.1186/s13662-019-2115-3 doi: 10.1186/s13662-019-2115-3
[39]	X. Wu, F. Chen, S. Deng, Hyers-Ulam stability and existence of solutions for weighted Caputo–Fabrizio fractional differential equations, Chaos Solitons Fract., 5 (2020), 100040. https://doi.org/10.1016/j.csfx.2020.100040 doi: 10.1016/j.csfx.2020.100040
[40]	J. Banaś, On measures of noncompactness in Banach spaces, Comment. Math. Univ. Carolin., 21 (1980), 131–143.
[41]	B. Chandra Deuri, M. V. Paunoviá, A. Das, V. Parvaneh, Solution of a fractional integral equation using the Darbo fixed point theorem, J. Math., 2022 (2022), 8415616. https://doi.org/10.1155/2022/8415616 doi: 10.1155/2022/8415616
[42]	H. Işik, S. Banaei, F. Golkarmanesh, V. Parvaneh, C. Park, M. Khorshidi, On new extensions of Darbo's fixed point theorem with applications, Symmetry, 12 (2020), 424. https://doi.org/10.3390/sym12030424 doi: 10.3390/sym12030424
[43]	H. Yépez-Martínez, J. F. Gómez-Aguilar, A new modified definition of Caputo–Fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method (MHAM), J. Comput. Appl. Math., 346 (2019), 247–260. https://doi.org/10.1016/j.cam.2018.07.023 doi: 10.1016/j.cam.2018.07.023
[44]	M. Al-Refai, Proper inverse operators of fractional derivatives with nonsingular kernels, Rend. Circ. Mat. Palermo, Ser. 2, 71 (2022), 525–535. https://doi.org/10.1007/s12215-021-00638-2 doi: 10.1007/s12215-021-00638-2
[45]	H. P. Heinz, On the behavior of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983), 1351–1371. https://doi.org/10.1016/0362-546X(83)90006-8 doi: 10.1016/0362-546X(83)90006-8
[46]	D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Isr. J. Math., 108 (1998), 109–138. https://doi.org/10.1007/BF02783044 doi: 10.1007/BF02783044
[47]	E. Zeidler, Nonlinear functional analysis and its applications: II/B: Nonlinear monotone operators, Berlin: Springer, 2013.
[48]	T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Differ. Equ., 2017 (2017), 313. https://doi.org/10.1186/s13662-017-1285-0 doi: 10.1186/s13662-017-1285-0

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