Existence and compatibility of positive solutions for boundary value fractional differential equation with modified analytic kernel

Amna Kalsoom; Sehar Afsheen; Akbar Azam; Faryad Ali; Amna Kalsoom; Sehar Afsheen; Akbar Azam; Faryad Ali

doi:10.3934/math.2023390

AIMS Mathematics

2023, Volume 8, Issue 4: 7766-7786. doi: 10.3934/math.2023390

Previous Article Next Article

Research article

Existence and compatibility of positive solutions for boundary value fractional differential equation with modified analytic kernel

1.
Department of Mathematics and Statistics, International Islamic University, H-10, 44000, Islamabad, Pakistan
2.
Department of Mathematics, Grand Asian University, 7KM Pasrur Road, Sialkot, 51310, Punjab, Pakistan
3.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O.Box 90950, Riyadh, 11623, Saudi Arabian

Received: 11 September 2022 Revised: 30 December 2022 Accepted: 16 January 2023 Published: 28 January 2023
MSC : 34B18, 45B05

In this article, a Green's function for a fractional boundary value problem in connection with modified analytic kernel has been constructed to study the existence of multiple solutions of a type of characteristic fractional boundary value problems. It is done here by using a well-known result: Krasnoselskii fixed point theorem. Moreover, a practical example is created to understand the importance of main results regarding the existence of solution of a boundary value fractional differential problem with homogeneous conditions. This example analytically and graphically, explains circumstances under which the Green's functions with different types of differential operator are compatible.
- fractional differential equation,
- boundary value problem,
- positive solution,
- fractional Green's function,
- fixed-point theorem
Citation: Amna Kalsoom, Sehar Afsheen, Akbar Azam, Faryad Ali. Existence and compatibility of positive solutions for boundary value fractional differential equation with modified analytic kernel[J]. AIMS Mathematics, 2023, 8(4): 7766-7786. doi: 10.3934/math.2023390

Related Papers:

Abstract

In this article, a Green's function for a fractional boundary value problem in connection with modified analytic kernel has been constructed to study the existence of multiple solutions of a type of characteristic fractional boundary value problems. It is done here by using a well-known result: Krasnoselskii fixed point theorem. Moreover, a practical example is created to understand the importance of main results regarding the existence of solution of a boundary value fractional differential problem with homogeneous conditions. This example analytically and graphically, explains circumstances under which the Green's functions with different types of differential operator are compatible.

References

[1]	K. S. Miller, B. Ross, An introduction to fractional calculus and fractional diffrential equations, New York: Wiley, 1993.
[2]	K. B. Oldham, J. Spanier, The fractional calculus: Theory and applications of differentiation and integration to arbitrary order, Elsevier, 1974.
[3]	D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204 (1996), 609–625. https://doi.org/10.1006/jmaa.1996.0456 doi: 10.1006/jmaa.1996.0456
[4]	S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 252 (2000), 804–812. https://doi.org/10.1006/jmaa.2000.7123 doi: 10.1006/jmaa.2000.7123
[5]	A. Babakhani, V. Daftardar-Gejji, Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl., 278 (2003), 434–442. https://doi.org/10.1016/S0022-247X(02)00716-3 doi: 10.1016/S0022-247X(02)00716-3
[6]	Z. Bai, H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495–505. https://doi.org/10.1016/j.jmaa.2005.02.052 doi: 10.1016/j.jmaa.2005.02.052
[7]	C. F. Li, X. N. Luo, Y. Zhou, Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Comput. Math. Appl., 59 (2010), 1363–1375. https://doi.org/10.1016/j.camwa.2009.06.029 doi: 10.1016/j.camwa.2009.06.029
[8]	Y. Wang, L. Liu, Y. Wu, Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal. Theor., 74 (2011), 3599–3605. https://doi.org/10.1016/j.na.2011.02.043 doi: 10.1016/j.na.2011.02.043
[9]	B. Ahmad, S. K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Differ. Equ., 2011 (2011), 107384. https://doi.org/10.1155/2011/107384 doi: 10.1155/2011/107384
[10]	J. Wang, Y. Zhou, M. Feckan, On recent developments in the theory of boundary value problems for impulsive fractional differential equations, Comput. Math. Appl., 64 (2012), 3008–3020. https://doi.org/10.1016/j.camwa.2011.12.064 doi: 10.1016/j.camwa.2011.12.064
[11]	M. Feckan, Y. Zhou, J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci., 17 (2012), 3050–3060. https://doi.org/10.1016/j.cnsns.2011.11.017 doi: 10.1016/j.cnsns.2011.11.017
[12]	J. Jiang, L. Liu, Y. Wu, Positive solutions to singular fractional differential system with coupled boundary conditions, Commun. Nonlinear Sci., 18 (2013), 3061–3074. https://doi.org/10.1016/j.cnsns.2013.04.009 doi: 10.1016/j.cnsns.2013.04.009
[13]	J. Henderson, R. Luca, Positive solutions for a system of nonlocal fractional boundary value problems, Fract. Calc. Appl. Anal., 16 (2013), 985–1008. https://doi.org/10.2478/s13540-013-0061-4 doi: 10.2478/s13540-013-0061-4
[14]	K. Shah, R. A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti periodic boundary conditions, Differ. Equ. Appl., 7 (2015), 245–262. https://doi.org/10.7153/dea-07-14 doi: 10.7153/dea-07-14
[15]	Y. Cui, Uniqueness of solution for boundary value problems for fractional differential equations, Appl. Math. Lett., 51 (2016), 48–54. https://doi.org/10.1016/j.aml.2015.07.002 doi: 10.1016/j.aml.2015.07.002
[16]	Y. Zou, G. He, On the uniqueness of solutions for a class of fractional differential equations, Appl. Math. Lett., 74 (2017), 68–73. https://doi.org/10.1016/j.aml.2017.05.011 doi: 10.1016/j.aml.2017.05.011
[17]	B. Ahmad, R. Luca, Existence of solutions for sequential fractional integro-differential equations and inclusions with nonlocal boundary conditions, Appl. Math. Comput., 339 (2018), 516–534. https://doi.org/10.1016/j.amc.2018.07.025 doi: 10.1016/j.amc.2018.07.025
[18]	M. Benchohra, S. Bouriah, J. J. Nieto, Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative, Demonstr. Math., 52 (2019), 437–450. https://doi.org/10.1515/dema-2019-0032 doi: 10.1515/dema-2019-0032
[19]	Z. Yue, Y. Zou, New uniqueness results for fractional differential equation with dependence on the first order derivative, Adv. Differ. Equ., 2019 (2019), 38. https://doi.org/10.1186/s13662-018-1923-1 doi: 10.1186/s13662-018-1923-1
[20]	A. Amara, Existence results for hybrid fractional differential equations with three-point boundary conditions, AIMS Math., 5 (2020), 1074–1088. https://doi.org/10.3934/math.2020075 doi: 10.3934/math.2020075
[21]	H. Afshari, M. S. Abdo, J. Alzabut, Further results on existence of positive solutions of generalized fractional boundary value problems, Adv. Differ. Equ., 2020 (2020), 600. https://doi.org/10.1186/s13662-020-03065-2 doi: 10.1186/s13662-020-03065-2
[22]	H. R. Marasi, H. Aydi, Existence and uniqueness results for two-term nonlinear fractional differential equations via a fixed point technique, J. Math., 2021 (2021), 6670176. https://doi.org/10.1155/2021/6670176 doi: 10.1155/2021/6670176
[23]	A. Tudorache, R. Luca, Positive solutions for a system of fractional boundary value problems with r-Laplacian operators, uncoupled nonlocal conditions and positive parameters, Axioms, 11 (2022), 164. https://doi.org/10.3390/axioms11040164 doi: 10.3390/axioms11040164
[24]	X. Zhang, Y. Tian, Sharp conditions for the existence of positive solutions for a second-order singular impulsive differential equation, Appl. Anal., 101 (2022), 1–13. https://doi.org/10.1080/00036811.2017.1370542 doi: 10.1080/00036811.2017.1370542
[25]	C. E. Wagner, A. C. Barbati, J. Engmann, A. S. Burbidge, G. H. McKinley, Quantifying the consistency and rheology of liquid foods using fractional calculus, Food Hydrocolloid., 69 (2017), 242–254. https://doi.org/10.1016/j.foodhyd.2017.01.036 doi: 10.1016/j.foodhyd.2017.01.036
[26]	L. L. Ferras, N. J. Ford, M. L. Morgado, M. Rebelo, G. H. McKinley, J. M. Nobrega, Theoretical and numerical analysis of unsteady fractional viscoelastic flows in simple geometries, Comput. Fluids, 174 (2018), 14–33. https://doi.org/10.1016/j.compfluid.2018.07.004 doi: 10.1016/j.compfluid.2018.07.004
[27]	A. Stankiewicz, Fractional Maxwell model of viscoelastic biological materials, BIO Web of Conferences, 10 (2018), 02032. https://doi.org/10.1051/bioconf/20181002032 doi: 10.1051/bioconf/20181002032
[28]	Y. A. Rossikhin, M. V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results, Appl. Mech. Rev., 63 (2010), 010801. https://doi.org/10.1115/1.4000563 doi: 10.1115/1.4000563
[29]	H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci., 64 (2018), 213–231. https://doi.org/10.1016/j.cnsns.2018.04.019 doi: 10.1016/j.cnsns.2018.04.019
[30]	A. Carpinteri, F. Mainardi, Fractals and fractional calculus in continuum mechanics, Springer, 2014.
[31]	K. Lazopoulos, Non-local continuum mechanics and fractional calculus, Mech. Res. Commun., 33 (2006), 753–757. https://doi.org/10.1016/j.mechrescom.2006.05.001 doi: 10.1016/j.mechrescom.2006.05.001
[32]	C. S. Drapaca, S. Sivaloganathan, A fractional model of continuum mechanics, J. Elast., 107 (2012), 105–123. https://doi.org/10.1007/s10659-011-9346-1 doi: 10.1007/s10659-011-9346-1
[33]	F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, Vienna: Springer, 1997.
[34]	D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour, A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos Soliton. Fract., 134 (2020), 109705. https://doi.org/10.1016/j.chaos.2020.109705 doi: 10.1016/j.chaos.2020.109705
[35]	V. E. Tarasov, Mathematical economics: Application of fractional calculus, Mathematics, 8 (2020), 660. https://doi.org/10.3390/math8050660 doi: 10.3390/math8050660
[36]	D. Kumar, D. Baleanu, Fractional calculus and its applications in physics, Front. Phys., 7 (2019), 81. https://doi.org/10.3389/fphy.2019.00081 doi: 10.3389/fphy.2019.00081
[37]	Y. Wei, Y. Kang, W. Yin, Y. Wang, Generalization of the gradient method with fractional order gradient direction, J. Frankl. I., 357 (2020), 2514–2532. https://doi.org/10.1016/j.jfranklin.2020.01.008 doi: 10.1016/j.jfranklin.2020.01.008
[38]	K. A. Abro, M. H. Laghari, J. F. Gómez-Aguilar, Application of Atangana-Baleanu fractional derivative to carbon nanotubes based non-Newtonian nanofluid: applications in nanotechnology, J. Appl. Comput. Mech., 6 (2020), 1260–1269. https://doi.org/10.22055/JACM.2020.33461.2229 doi: 10.22055/JACM.2020.33461.2229
[39]	S. Chávez-Vázquez, J. F. Gómez-Aguilar, J. E. Lavín-Delgado, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, Applications of fractional operators in robotics: a review, J. Intell. Robot. Syst., 104 (2022), 63. https://doi.org/10.1007/s10846-022-01597-1 doi: 10.1007/s10846-022-01597-1
[40]	X. Xu, D. Jiang, C. Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Anal., 71 (2009), 4676–4688. https://doi.org/10.1016/j.na.2009.03.030 doi: 10.1016/j.na.2009.03.030
[41]	I. Podlubny, Fractional differential equations, 1998.
[42]	L. Debnath, D. Bhatta, Integral transforms and their applications, Chapman & Hall, 2007.
[43]	A. Fernandez, M. A. Ozarslan, D. Baleanu, On fractional calculus with general analytical kernels, Appl. Math. Comput., 354 (2019), 248–265. https://doi.org/10.1016/j.amc.2019.02.045 doi: 10.1016/j.amc.2019.02.045
[44]	T. R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7–15.
[45]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[46]	F. Jarad, T. Abdeljawad, J. Alzabut, Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Spec. Top., 226 (2017), 3457–3471. https://doi.org/10.1140/epjst/e2018-00021-7 doi: 10.1140/epjst/e2018-00021-7

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)