Study of complex-valued differential equations with the Hilfer fractional derivative

Keerthana Nagaraj; Vivek Devaraj; Abdulah A. Alghamdi; Mohamed M. El-Dessoky; Elsayed Mohamed Elsayed; Keerthana Nagaraj; Vivek Devaraj; Abdulah A. Alghamdi; Mohamed M. El-Dessoky; Elsayed Mohamed Elsayed

doi:10.3934/math.2026739

AIMS Mathematics

2026, Volume 11, Issue 6: 18171-18201. doi: 10.3934/math.2026739

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Research article

Study of complex-valued differential equations with the Hilfer fractional derivative

1.
Department of Mathematics, PSG College of Arts & Science, Coimbatore 641014, India
2.
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 14 April 2026 Revised: 01 June 2026 Accepted: 03 June 2026 Published: 22 June 2026
MSC : 26A33, 34A37, 34K05, 47H10

In this paper, a class of nonlinear impulsive pantograph-type Hilfer complex-valued systems (IPH-CVSes) is investigated. Two cases corresponding to the fractional orders $ \rho\in(0, 1) $ and $ \rho\in(1, 2) $ are studied. Explicit solution representations for the considered models are derived. Moreover, existence and uniqueness results are established using Krasnoselskii's fixed-point theorem and the Banach contraction principle. Finally, an application arising from an aerodynamic flow model is provided to demonstrate the applicability of the obtained theoretical results.
- Hilfer fractional derivative,
- impulsive pantograph equations,
- complex-valued systems,
- Banach contraction principle,
- Krasnoselskii's fixed-point theorem
Citation: Keerthana Nagaraj, Vivek Devaraj, Abdulah A. Alghamdi, Mohamed M. El-Dessoky, Elsayed Mohamed Elsayed. Study of complex-valued differential equations with the Hilfer fractional derivative[J]. AIMS Mathematics, 2026, 11(6): 18171-18201. doi: 10.3934/math.2026739

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Abstract

In this paper, a class of nonlinear impulsive pantograph-type Hilfer complex-valued systems (IPH-CVSes) is investigated. Two cases corresponding to the fractional orders $ \rho\in(0, 1) $ and $ \rho\in(1, 2) $ are studied. Explicit solution representations for the considered models are derived. Moreover, existence and uniqueness results are established using Krasnoselskii's fixed-point theorem and the Banach contraction principle. Finally, an application arising from an aerodynamic flow model is provided to demonstrate the applicability of the obtained theoretical results.

References

[1]	B. Ross, The development of fractional calculus 1695–1900, Hist. Math., 4 (1977), 75–89. https://doi.org/10.1016/0315-0860(77)90039-8 doi: 10.1016/0315-0860(77)90039-8
[2]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000.
[3]	T. F. Nonnenmacher, R. Metzler, On the Riemann-Liouville fractional calculus and some recent applications, Fractals, 3 (1995), 557–566. https://doi.org/10.1142/S0218348X95000497 doi: 10.1142/S0218348X95000497
[4]	M. Caputo, Linear models of dissipation whose Q is almost frequency independent-Ⅱ, Geophys. J. Int., 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
[5]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier Science, 2006. https://doi.org/10.1016/S0304-0208(06)80001-0
[6]	J. J. Nieto, A. Ouahab, V. Venktesh, Implicit fractional differential equations via the Liouville-Caputo derivative, Mathematics, 3 (2015), 398–411. https://doi.org/10.3390/math3020398 doi: 10.3390/math3020398
[7]	K. M. Furati, M. D. Kassim, N. E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616–1626. https://doi.org/10.1016/j.camwa.2012.01.009 doi: 10.1016/j.camwa.2012.01.009
[8]	T. Zahra, H. M. Fahad, M. Rehman, On weighted fractional calculus with respect to functions, Math. Methods Appl. Sci., 48 (2025), 5642–5659. https://doi.org/10.1002/mma.10626 doi: 10.1002/mma.10626
[9]	A. Dawuken, G. Maimaitiaili, A. Rozi, A. Abdurahman, Application of fractional Laplacian partial differential equation in mural image inpainting, Complex Syst. Stab. Control, 2 (2026), 100005. https://doi.org/10.53941/cssc.2026.100005 doi: 10.53941/cssc.2026.100005
[10]	Q. Liao, D. Luo, Exponential stability criteria for fractional order switched system based on multiple discontinuous Lyapunov function method, Complex Syst. Stab. Control, 2 (2026), 100003. https://doi.org/10.53941/cssc.2026.100003 doi: 10.53941/cssc.2026.100003
[11]	T. Zhang, Y. Li, Exponential Euler scheme of multi-delay Caputo-Fabrizio fractional-order differential equations, Appl. Math. Lett., 124 (2022), 107709. https://doi.org/10.1016/j.aml.2021.107709 doi: 10.1016/j.aml.2021.107709
[12]	T. Zhang, Y. Li, Global exponential stability of discrete-time almost automorphic Caputo-Fabrizio BAM fuzzy neural networks via exponential Euler technique, Knowl.-Based Syst., 246 (2022), 108675. https://doi.org/10.1016/j.knosys.2022.108675 doi: 10.1016/j.knosys.2022.108675
[13]	J. Singh, A. Gupta, J. J. Nieto, M. Rutkoski, Computational analysis of fractional Drinfeld-Sokolov-Wilson equation associated with regularized form of Hilfer-Prabhakar derivative, Netw. Heterog. Media, 21 (2026), 1–21. https://doi.org/10.3934/nhm.2026001 doi: 10.3934/nhm.2026001
[14]	W. Wang, R. Metzler, Ž. Tomovski, Distributed-order fractional diffusion equation with Hilfer-Prabhakar fractional derivative, Phys. A, 689 (2026), 131370. https://doi.org/10.1016/j.physa.2026.131370 doi: 10.1016/j.physa.2026.131370
[15]	J. J. Nieto, A. Petruşel, H. K. Xu, New perspectives in fixed point theory and its applications, Fixed Point Theory Algorithm. Sci. Eng., 2026 (2026), 19. https://doi.org/10.1186/s13663-025-00820-6 doi: 10.1186/s13663-025-00820-6
[16]	M. Kirane, Y. V. Rogovchenko, Comparison results for systems of impulse parabolic equations with applications to population dynamics, Nonlinear Anal., 28 (1997), 263–276. https://doi.org/10.1016/0362-546X(95)00159-S doi: 10.1016/0362-546X(95)00159-S
[17]	G. Stamov, I. Stamova, C. Spirova, Impulsive reaction-diffusion delayed models in biology: integral manifolds approach, Entropy, 23 (2021), 1631. https://doi.org/10.3390/e23121631 doi: 10.3390/e23121631
[18]	H. Runvik, A. Medvedev, M. C. Kjellsson, Impulsive feedback modeling of levodopa pharmacokinetics subject to intermittently interrupted gastric emptying, 2020 American Control Conference, 2020, 1323–1328. https://doi.org/10.23919/ACC45564.2020.9147668
[19]	M. Li, Z. Liu, L. Hua, O. M. Kwon, K. Shi, Sliding mode control for robust trajectory tracking of robotic manipulators under impulsive and external disturbances, Int. J. Control Autom. Syst., 23 (2025), 1909–1918. https://doi.org/10.1007/s12555-024-1128-1 doi: 10.1007/s12555-024-1128-1
[20]	S. Ansari, M. Malik, R. Dhayal, Optimal controls for multi-term fractional stochastic integro-differential equations with impulses and infinite delay, Int. J. Syst. Sci., 56 (2025), 1862–1883. https://doi.org/10.1080/00207721.2024.2435568 doi: 10.1080/00207721.2024.2435568
[21]	A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, Singapore: World Scientific, 1995.
[22]	R. Agarwal, S. Hristova, D. O'Regan, Non-instantaneous impulses in differential equations, Cham: Springer International Publishing, 2017. https://doi.org/10.1007/978-3-319-66384-5_1
[23]	H. Xu, Q. Zhu, Stability of discrete-time impulsive stochastic systems with hybrid non-deterministic delays, IEEE Trans. Autom. Control, 2026. https://doi.org/10.1109/tac.2026.3666702
[24]	Y. Lu, D. Ruan, Q. Zhu, Symmetric analysis of stability criteria for nonlinear systems with multi-delayed periodic impulses: intensity periodicity and averaged delay, Symmetry, 17 (2025), 1481. https://doi.org/10.3390/sym17091481 doi: 10.3390/sym17091481
[25]	K. Lan, Existence and uniqueness of solutions of nonlinear Cauchy-type problems for first-order fractional differential equations, Math. Methods Appl. Sci., 47 (2024), 535–555. https://doi.org/10.1002/mma.9669 doi: 10.1002/mma.9669
[26]	M. M. Arjunan, R. Deepa, Existence results for an impulsive fractional differential equations with generalized Riemann-Liouville fractional derivative, Nonlinear Stud., 30 (2023), 1157–1165.
[27]	G. Trivedi, V. Shah, J. Sharma, R. C. Sanghvi, On solution of non-instantaneous impulsive Hilfer fractional integro-differential evolution system, Math. Appl., 51 (2023), 33–50. https://doi.org/10.14708/ma.v51i1.7167 doi: 10.14708/ma.v51i1.7167
[28]	B. Radhakrishnan, M. Tamilarasi, P. Anukokila, A study on the existence and uniqueness of random impulsive Hilfer pantograph differential equations, Iran. J. Sci., 49 (2025), 1415–1422. https://doi.org/10.1007/s40995-025-01816-y doi: 10.1007/s40995-025-01816-y
[29]	R. Shah, N. Irshad, Existence and uniqueness of solutions of Hilfer fractional neutral impulsive stochastic delayed differential equations with nonlocal conditions, J. Nonlinear Complex Data Sci., 27 (2026), 43–63. https://doi.org/10.1515/jncds-2025-0122 doi: 10.1515/jncds-2025-0122
[30]	Q. Guo, J. Liu, H. Nie, D. Wang, W. Wang, Dynamical behaviors of a complex-variable laser system with a memcapacitor, Chaos Soliton. Fract., 200 (2025), 117091. https://doi.org/10.1016/j.chaos.2025.117091 doi: 10.1016/j.chaos.2025.117091
[31]	S. Du, H. Oqaibi, Visual positioning system for marine industrial robot assembly based on complex variable function, Appl. Math. Nonlinear Sci., 7 (2022), 485–492. https://doi.org/10.2478/amns.2021.1.00091 doi: 10.2478/amns.2021.1.00091
[32]	C. J. Sakitis, D. A. Brown, D. B. Rowe, A Bayesian complex-valued latent variable model applied to functional magnetic resonance imaging, J. R. Stat. Soc. Ser. C Appl. Stat., 74 (2025), 100–125. https://doi.org/10.1093/jrsssc/qlae046 doi: 10.1093/jrsssc/qlae046
[33]	R. Hatamleh, A. Al-Husban, S. A. M. Zubair, M. Elamin, M. M. Saeed, E. Abdolmaleki, et al., AI-assisted wearable devices for promoting human health and strength using complex interval-valued picture fuzzy soft relations, Eur. J. Pure Appl. Math., 18 (2025), 5523. https://doi.org/10.29020/nybg.ejpam.v18i1.5523 doi: 10.29020/nybg.ejpam.v18i1.5523
[34]	T. Fang, J. Sun, Existence and uniqueness of solutions to complex-valued nonlinear impulsive differential systems, Adv. Differ. Equ., 2012 (2012), 115. https://doi.org/10.1186/1687-1847-2012-115 doi: 10.1186/1687-1847-2012-115
[35]	T. Phan, G. Todorova, B. Yordanov, Existence, uniqueness and regularity theory for elliptic equations with complex-valued potentials, Discrete Contin. Dyn. Syst., 41 (2021), 1071–1099. https://doi.org/10.3934/dcds.2020310 doi: 10.3934/dcds.2020310
[36]	A. Higgins, C. Piret, B. Fornberg, Numerical computation of fractional derivatives of complex-valued analytic functions, SIAM Undergrad. Res. Online, 15 (2021), 511–525. https://doi.org/10.1137/22S1520566 doi: 10.1137/22S1520566
[37]	K. Afassinou, A. A. Mebawondu, H. A. Abass, O. K. Narain, Existence of solution of differential and Riemann-Liouville equation via fixed point approach in complex valued b-metric spaces, Aust. J. Math. Anal. Appl., 18 (2021), 1–15.
[38]	H. Gissy, J. Ahmad, Solving Riemann-Liouville fractional integral equations by fixed point results in complex-valued suprametric spaces, Fractal Fract., 9 (2025), 826. https://doi.org/10.3390/fractalfract9120826 doi: 10.3390/fractalfract9120826
[39]	H. Zahed, Existence and uniqueness of solutions to fractional differential equations in complex-valued suprametric spaces, Fractal Fract., 10 (2026), 332. https://doi.org/10.3390/fractalfract10050332 doi: 10.3390/fractalfract10050332
[40]	P. Verma, S. Tiwari, Analysis of multi-term time complex fractional diffusion equation with Hilfer-Hadamard fractional derivative, Math. Sci., 18 (2024), 693–705. https://doi.org/10.1007/s40096-024-00525-8 doi: 10.1007/s40096-024-00525-8
[41]	Y. Zhou, Basic theory of fractional differential equations, Singapore: World Scientific, 2023.
[42]	J. R. Wang, Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266 (2015), 850–859. https://doi.org/10.1016/j.amc.2015.05.144 doi: 10.1016/j.amc.2015.05.144
[43]	J. Du, W. Jiang, A. U. K. Niazi, Approximate controllability of impulsive Hilfer fractional differential inclusions, J. Nonlinear Sci. Appl., 10 (2017), 595–611. https://dx.doi.org/10.22436/jnsa.010.02.23 doi: 10.22436/jnsa.010.02.23
[44]	S. P. Bhairat, Existence and continuation of solutions of Hilfer fractional differential equations, J. Math. Model., 7 (2019), 1–20. https://doi.org/10.22124/jmm.2018.9220.1136 doi: 10.22124/jmm.2018.9220.1136
[45]	K. Qiu, M. Fečkan, J. R. Wang, Existence and approximate controllability of Hilfer fractional impulsive evolution equations, Fract. Calc. Appl. Anal., 28 (2025), 146–180. https://doi.org/10.1007/s13540-025-00372-x doi: 10.1007/s13540-025-00372-x
[46]	J. K. Ajay Kumar, K. S. Prabhu, A. S. Veerendra, Y. B. S. Shastry, G. Srinivas, A comprehensive review of aerodynamic performance evaluation for aircraft wings and engines: insights into experimental, numerical, and theoretical approaches, Aerosp. Syst., 2025, 1–30. https://doi.org/10.1007/s42401-025-00366-w
[47]	F. Beaumont, S. Murer, F. Bogard, G. Polidori, Aerodynamics of flight formations in birds: a quest for energy efficiency, Birds, 6 (2025), 15. https://doi.org/10.3390/birds6020015 doi: 10.3390/birds6020015
[48]	L. Deng, W. Pan, T. Luan, C. Zhang, Y. Leng, Rapid aircraft wake vortex identification model based on optimized image object recognition networks, Aerospace, 11 (2024), 840. https://doi.org/10.3390/aerospace11100840 doi: 10.3390/aerospace11100840
[49]	P. Panagiotou, G. Ioannidis, I. Tzivinikos, K. Yakinthos, Experimental investigation of the wake and the wingtip vortices of a UAV model, Aerospace, 4 (2017), 53. https://doi.org/10.3390/aerospace4040053 doi: 10.3390/aerospace4040053
[50]	K. Mehmood, S. I. Ali Shah, T. Ali Shams, M. N. Mumtaz Qadri, T. A. Khan, D. Kukulka, Flight dynamic characteristics of wide-body aircraft with wind gust and turbulence, Fluids, 8 (2023), 320. https://doi.org/10.3390/fluids8120320 doi: 10.3390/fluids8120320
[51]	W. Yuan, X. Zhang, D. Poirel, A. Wall, Numerical modelling of aerodynamic response to gusts and gust effect mitigation, Aerosp. Sci. Technol., 154 (2024), 109467. https://doi.org/10.1016/j.ast.2024.109467 doi: 10.1016/j.ast.2024.109467
[52]	E. Khatoonabadi, V. Bohlouri, A. R. Kosari, Using a fractional-order controller to improve the performance of a spacecraft attitude control in the presence of uncertainties and disturbances, Proc. Inst. Mech. Eng. Part G, 239 (2025), 1823–1842. https://doi.org/10.1177/09544100251342544 doi: 10.1177/09544100251342544
[53]	X. Shi, J. Peng, Y. Yang, L. Shi, L. Chen, Adaptive finite time fractional-order sliding mode based robust tracking control of quadrotor UAVs in the presence of stochastic disturbances and parametric uncertainties, ISA Trans., 172 (2026), 306–319. https://doi.org/10.1016/j.isatra.2026.02.022 doi: 10.1016/j.isatra.2026.02.022
[54]	S. Sohaili Yekta, C. Sobczak, J. Jackman, J. Clarke, M. Ismail, S. Ahmed Islam, et al., Airfoil design & analysis using conformal transformations, AIAA Region Ⅰ-Ⅶ Student Conferences, 2025. https://doi.org/10.2514/6.2025-97788
[55]	E. R. Love, Fractional derivatives of imaginary order, J. London Math. Soc., s2-3 (1971), 241–259. https://doi.org/10.1112/jlms/s2-3.2.241 doi: 10.1112/jlms/s2-3.2.241
[56]	S. Harikrishnan, E. M. Elsayed, K. Kanagarajan, D. Vivek, A study of Hilfer-Katugampola type pantograph equations with complex order, Ex. Counterexamples, 2 (2022), 100045. https://doi.org/10.1016/j.exco.2021.100045 doi: 10.1016/j.exco.2021.100045

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