Existence and uniqueness of nonlinear fractional differential equations with the Caputo and the Atangana-Baleanu derivatives: Maximal, minimal and Chaplygin approaches

Abdon Atangana; Abdon Atangana

doi:10.3934/math.20241282

AIMS Mathematics

2024, Volume 9, Issue 10: 26307-26338. doi: 10.3934/math.20241282

Previous Article Next Article

Research article Special Issues

Existence and uniqueness of nonlinear fractional differential equations with the Caputo and the Atangana-Baleanu derivatives: Maximal, minimal and Chaplygin approaches

Abdon Atangana ^{1,2,3
,
,}

1.
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, 9301, South Africa
2.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
3.
IT4Innovations, VSB – Technical University of Ostrava, Czech Republic

Received: 25 June 2024 Revised: 06 August 2024 Accepted: 07 August 2024 Published: 11 September 2024
MSC : 34A08, 34A12

This work provided a detailed theoretical analysis of fractional ordinary differential equations with Caputo and the Atangana-Baleanu fractional derivative. The work started with an extension of Tychonoff's fixed point and the Perron principle to prove the global existence with extra conditions due to the properties of the fractional derivatives used. Then, a detailed analysis of the existence of maximal and minimal solutions was presented for both cases. Then, using Chaplygin's approach with extra conditions, we also established the existence and uniqueness of the solutions of these equations. The Abel and the Bernoulli equations were considered as illustrative examples and were solved using the fractional middle point method.
Citation: Abdon Atangana. Existence and uniqueness of nonlinear fractional differential equations with the Caputo and the Atangana-Baleanu derivatives: Maximal, minimal and Chaplygin approaches[J]. AIMS Mathematics, 2024, 9(10): 26307-26338. doi: 10.3934/math.20241282

Related Papers:

Abstract

This work provided a detailed theoretical analysis of fractional ordinary differential equations with Caputo and the Atangana-Baleanu fractional derivative. The work started with an extension of Tychonoff's fixed point and the Perron principle to prove the global existence with extra conditions due to the properties of the fractional derivatives used. Then, a detailed analysis of the existence of maximal and minimal solutions was presented for both cases. Then, using Chaplygin's approach with extra conditions, we also established the existence and uniqueness of the solutions of these equations. The Abel and the Bernoulli equations were considered as illustrative examples and were solved using the fractional middle point method.

References

[1]	M. J. Korenberg, I. W. Hunter, The identification of nonlinear biological systems: Volterra kernel approaches, Ann. Biomed. Eng., 24 (1996), 250–268. https://doi.org/10.1007/BF02648117 doi: 10.1007/BF02648117
[2]	D. K. Campbell, Nonlinear physics: Fresh breather, Nature, 432 (2004), 455–456. https://doi.org/10.1038/432455a doi: 10.1038/432455a
[3]	F. Mosconi, T. Julou, N. Desprat, D. K. Sinha, J. F. Allemand, V. Croquette, Some nonlinear challenges in biology, Nonlinearity, 21 (2008), T131. https://doi.org/10.1088/0951-7715/21/8/T03 doi: 10.1088/0951-7715/21/8/T03
[4]	M. Newman, Power laws, Pareto distributions and Zipf's law, Contemp. Phys., 46 (2005), 323–351. https://doi.org/10.1080/00107510500052444 doi: 10.1080/00107510500052444
[5]	D. M. Mackay, Psychophysics of perceived intensity: A theoretical basis for Fechner's and Stevens' laws, Science, 139 (1963), 1213–1216. https://doi.org/10.1126/science.139.3560.1213.b doi: 10.1126/science.139.3560.1213.b
[6]	J. E. Staddon, Theory of behavioral power functions, Psychol. Rev., 85 (1978), 305–320. https://doi.org/10.1037/0033-295X.85.4.305 doi: 10.1037/0033-295X.85.4.305
[7]	A. Corral, A. Osso, J. E. Llebot, Scaling of tropical cyclone dissipation, Nat. Phys., 6 (2010), 693–696. https://doi.org/doi:10.1038/nphys1725 doi: 10.1038/nphys1725
[8]	R. D. Lorenz, Power law of dust devil diameters on mars and earth, Icarus, 203 (2009), 683–684. https://doi.org/10.1016/j.icarus.2009.06.029 doi: 10.1016/j.icarus.2009.06.029
[9]	W. J. Reed, B. D. Hughes, From gene families and genera to incomes and internet file sizes: Why power laws are so common in nature, Phys. Rev. E., 66 (2002), 067103. https://doi.org/10.1103/PhysRevE.66.067103 doi: 10.1103/PhysRevE.66.067103
[10]	Y. Gerchak, Decreasing failure rates and related issues in the social sciences, Oper. Res., 32 (1984), 537–546. https://doi.org/10.1287/opre.32.3.537 doi: 10.1287/opre.32.3.537
[11]	T. Pritz, Five-parameter fractional derivative model for polymeric damping materials, J. Sound Vibration, 265 (2003), 935–952. https://doi.org/10.1016/S0022-460X(02)01530-4 doi: 10.1016/S0022-460X(02)01530-4
[12]	M. Caputo, Linear model of dissipation whose Q is almost frequency independent–Ⅱ, Geophys. J. R. Astron. Soc., 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
[13]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[14]	W. M. Whyburn, On the fundamental existence theorems for differential systems, Ann. Math., 30 (1929), 31–38. https://doi.org/10.2307/1968263 doi: 10.2307/1968263
[15]	V. Lakshmikantham, S. Leela, Differential and integral inequalities - Theory and applications, Academic Press, 55 (1969), 3–319.
[16]	A. F. Filippov, The existence of solutions of generalized differential equations, Math. Notes Acad. Sci. USSR, 10 (1971), 608–611. https://doi.org/10.1007/BF01464722 doi: 10.1007/BF01464722
[17]	E. E. Viktorovsky, On a general theorem on the existence of solutions of differential equations, connected with the consideration of integral inequalities, Mat. Sb., 31 (1952), 27–33.
[18]	S. A. Chaplygin, A new method of approximate integration of differential equations, Moscow-Leningrad, 1950.
[19]	O. Perron, Über den Integralbegriff, 1914. https://doi.org/10.11588/diglit.37437
[20]	D. V. Griffiths, I. M. Smith, Numerical methods for engineers, 2 Eds., New York: Chapman and Hall/CRC, 2006. https://doi.org/10.1201/9781420010244
[21]	A. A. Tateishi, H. V. Ribeiro, E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion, Front. Phys., 5 (2017), 52. https://doi.org/10.3389/fphy.2017.00052 doi: 10.3389/fphy.2017.00052
[22]	F. Mainardi, Why the Mittag-Leffler function can be considered the queen function of the fractional calculus? Entropy, 22 (2020), 1359. https://doi.org/10.3390/e22121359 doi: 10.3390/e22121359
[23]	J. Sabatier, Fractional-order derivatives defined by continuous Kernels: Are they really too restrictive? Fractal Fract., 4 (2020), 40. https://doi.org/10.3390/fractalfract4030040 doi: 10.3390/fractalfract4030040
[24]	W. Al-Sadi, Z. Wei, I. Moroz, A. Alkhazzan, Existence and stability of solution in Banach space for an impulsive system involving Atangana-Baleanu and Caputo-Fabrizio derivatives, Fractals, 31 (2023), 2340085. https://doi.org/10.1142/S0218348X23400856 doi: 10.1142/S0218348X23400856

Reader Comments

Your name:*

Email:*
© 2024 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)