Further studies on ordinary differential equations involving the $ M $-fractional derivative

A. Khoshkenar; M. Ilie; K. Hosseini; D. Baleanu; S. Salahshour; C. Park; J. R. Lee; A. Khoshkenar; M. Ilie; K. Hosseini; D. Baleanu; S. Salahshour; C. Park; J. R. Lee

doi:10.3934/math.2022613

AIMS Mathematics

2022, Volume 7, Issue 6: 10977-10993. doi: 10.3934/math.2022613

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Research article

Further studies on ordinary differential equations involving the $ M $-fractional derivative

1.
Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran
2.
Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
3.
Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara 06530, Turkey
4.
Institute of Space Sciences, Magurele-Bucharest, Romania
5.
Department of Medical Research, China Medical University, Taichung, 40447, Taiwan
6.
Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey
7.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, South Korea
8.
Department of Data Science, Daejin University, Kyunggi 11159, South Korea

Received: 19 November 2021 Revised: 22 February 2022 Accepted: 23 February 2022 Published: 06 April 2022
MSC : 34A08, 35C10

In the current paper, the power series based on the $ M $-fractional derivative is formally introduced. More peciesely, the Taylor and Maclaurin expansions are generalized for fractional-order differentiable functions in accordance with the $ M $-fractional derivative. Some new definitions, theorems, and corollaries regarding the power series in the $ M $ sense are presented and formally proved. Several ordinary differential equations (ODEs) involving the $ M $-fractional derivative are solved to examine the validity of the results presented in the current study.
- $ M $-fractional derivative,
- power series,
- new definitions, theorems and corollaries,
- ordinary differential equations
Citation: A. Khoshkenar, M. Ilie, K. Hosseini, D. Baleanu, S. Salahshour, C. Park, J. R. Lee. Further studies on ordinary differential equations involving the $ M $-fractional derivative[J]. AIMS Mathematics, 2022, 7(6): 10977-10993. doi: 10.3934/math.2022613

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Abstract

In the current paper, the power series based on the $ M $-fractional derivative is formally introduced. More peciesely, the Taylor and Maclaurin expansions are generalized for fractional-order differentiable functions in accordance with the $ M $-fractional derivative. Some new definitions, theorems, and corollaries regarding the power series in the $ M $ sense are presented and formally proved. Several ordinary differential equations (ODEs) involving the $ M $-fractional derivative are solved to examine the validity of the results presented in the current study.

References

[1]	S. Yang, H. Zhou, S. Zhang, L. Wang, Analytical solutions of advective-dispersive transport in porous media involving conformable derivative, Appl. Math. Lett., 92 (2019) 85–92. https://doi.org/10.1016/j.aml.2019.01.004 doi: 10.1016/j.aml.2019.01.004
[2]	K. Hosseini, M. Mirzazadeh, M. Ilie, J. F. Gómez-Aguilar, Biswas-Arshed equation with the beta time derivative: Optical solitons and other solutions, Optik, 217 (2020), 164801. https://doi.org/10.1016/j.ijleo.2020.164801 doi: 10.1016/j.ijleo.2020.164801
[3]	K. Hosseini, M. Ilie, M. Mirzazadeh, A. Yusuf, T. A. Sulaiman, D. Baleanu, S. Salahshour, An effective computational method to deal with a time-fractional nonlinear water wave equation in the Caputo sense, Math. Comput. Simul., 187 (2021), 248–260. https://doi.org/10.1016/j.matcom.2021.02.021 doi: 10.1016/j.matcom.2021.02.021
[4]	T. A. Sulaiman, M. Yavuz, H. Bulut, H. M. Baskonus, Investigation of the fractional coupled viscous Burger's equation involving Mittag-Leffler kernel, Physica A, 527 (2019), 121126. https://doi.org/10.1016/j.physa.2019.121126 doi: 10.1016/j.physa.2019.121126
[5]	R. Khalil, M. Al-Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
[6]	M. Abu Hammad, R. Khalil, Abel's formula and Wronskian for conformable fractional differential equations, Int. J. Differ. Equ. Appl., 13 (2014), 177–183. http://doi.org/10.12732/ijdea.v13i3.1753 doi: 10.12732/ijdea.v13i3.1753
[7]	M. S. Osman, A. Korkmaz, H. Rezazadeh, M. Mirzazadeh, M. Eslami, Q. Zhou, The unified method for conformable time fractional Schrödinger equation with perturbation terms, Chinese J. Phys., 56 (2018), 2500–2506. https://doi.org/10.1016/j.cjph.2018.06.009 doi: 10.1016/j.cjph.2018.06.009
[8]	K. Hosseini, K. Sadri, M. Mirzazadeh, A. Ahmadian, Y. M. Chu, S. Salahshour, Reliable methods to look for analytical and numerical solutions of a nonlinear differential equation arising in heat transfer with the conformable derivative, Math. Method. Appl. Sci., 2021. https://doi.org/10.1002/mma.7582
[9]	A. Atangana, D. Baleanu, A. Alsaedi, Analysis of time-fractional Hunter-Saxton equation: a model of neumatic liquid crystal, Open Phys., 14 (2016), 145–149. https://doi.org/10.1515/phys-2016-0010 doi: 10.1515/phys-2016-0010
[10]	A. Atangana, R. T. Alqahtani, Modelling the spread of river blindness disease via the Caputo fractional derivative and the beta-derivative, Entropy, 18 (2016), 40. https://doi.org/10.3390/e18020040 doi: 10.3390/e18020040
[11]	B. Ghanbari, J. F. Gómez-Aguilar, The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with $\beta$-conformable time derivative, Rev. Mex. Fís., 65 (2019), 503–518.
[12]	K. Hosseini, M. Mirzazadeh, M. Ilie, J. F. Gómez-Aguilar, Soliton solutions of the Sasa-Satsuma equation in the monomode optical fibers including the beta-derivatives, Optik, 224 (2020), 165425. https://doi.org/10.1016/j.ijleo.2020.165425 doi: 10.1016/j.ijleo.2020.165425
[13]	M. Caputo, Linear models 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
[14]	I. Podlubny, Fractional differential equations, Math. Sci. Eng., 198 (1999), 41–119.
[15]	M. Awais, F. S. Alshammari, S. Ullah, M. Altaf Khan, S. Islam, Modeling and simulation of the novel coronavirus in Caputo derivative, Results Phys., 19 (2020), 103588. https://doi.org/10.1016/j.rinp.2020.103588 doi: 10.1016/j.rinp.2020.103588
[16]	A. Yokus, H. Durur, D. Kaya, H. Ahmad, T. A. Nofal, Numerical comparison of Caputo and conformable derivatives of time fractional Burgers-Fisher equation, Results Phys., 25 (2021), 104247. https://doi.org/10.1016/j.rinp.2021.104247 doi: 10.1016/j.rinp.2021.104247
[17]	A. Atangana, D. Baleanu, New fractional derivative with nonlocal and non-singular kernel, theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[18]	D. Avci, A. Yetim, Cauchy and source problems for an advection-diffusion equation with Atangana-Baleanu derivative on the real line, Chaos Soliton. Fract., 118 (2019), 361–365. https://doi.org/10.1016/j.chaos.2018.11.035 doi: 10.1016/j.chaos.2018.11.035
[19]	Z. Korpinar, M. Inc, M. Bayram, Theory and application for the system of fractional Burger equations with Mittag Leffler kernel, Appl. Math. Comput., 367 (2020), 124781. https://doi.org/10.1016/j.amc.2019.124781 doi: 10.1016/j.amc.2019.124781
[20]	K. Hosseini, M. Ilie, M. Mirzazadeh, D. Baleanu, An analytic study on the approximate solution of a nonlinear time-fractional Cauchy reaction-diffusion equation with the Mittag-Leffler law, Math. Method. Appl. Sci., 44 (2021), 6247–6258. https://doi.org/10.1002/mma.7059 doi: 10.1002/mma.7059
[21]	S. G. Samko, B. Ross, Integration and differentiation to a variable fractional order, Integr. Transf. Spec. F., 1 (1993), 277–300. https://doi.org/10.1080/10652469308819027 doi: 10.1080/10652469308819027
[22]	X. Zheng, H. Wang, An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes, SIAM J. Numer. Anal., 58 (2020), 330–352. https://doi.org/10.1137/19M1245621 doi: 10.1137/19M1245621
[23]	X. Zheng, H. Wang, An error estimate of a numerical approximation to a hidden-memory variable-order space-time fractional diffusion equation, SIAM J. Numer. Anal., 58 (2020), 2492–2514. https://doi.org/10.1137/20M132420X doi: 10.1137/20M132420X
[24]	X. Zheng, H. Wang, Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions, IMA J. Numer. Anal., 41 (2021), 1522–1545.
[25]	X. Zheng, H. Wang, A hidden-memory variable-order time-fractional optimal control model: Analysis and approximation, SIAM J. Control Optim., 59 (2021), 1851–1880. https://doi.org/10.1137/20M1344962 doi: 10.1137/20M1344962
[26]	X. Zheng, H. Wang, Analysis and discretization of a variable-order fractional wave equation, Commun. Nonlinear Sci., 104 (2022), 106047. https://doi.org/10.1016/j.cnsns.2021.106047 doi: 10.1016/j.cnsns.2021.106047
[27]	J. Singh, A. Ahmadian, S. Rathore, D. Kumar, D. Baleanu, M. Salimi, et al. An efficient computational approach for local fractional Poisson equation in fractal media, Numer. Meth. Part. D. E., 37 (2021), 1439–1448. ttps://doi.org/10.1002/num.22589 doi: 10.1002/num.22589
[28]	M. Rahaman, S. P. Mondal, A. A. Shaikh, A. Ahmadian, N. Senu, S. Salahshour, Arbitrary-order economic production quantity model with and without deterioration: generalized point of view, Adv. Differ. Equ., 2020 (2020), 16. https://doi.org/10.1186/s13662-019-2465-x doi: 10.1186/s13662-019-2465-x
[29]	S. Ahmad, A. Ullah, K. Shah, S. Salahshour, A. Ahmadian, T. Ciano, Fuzzy fractional-order model of the novel coronavirus, Adv. Differ. Equ., 2020 (2020), 472. https://doi.org/10.1186/s13662-020-02934-0 doi: 10.1186/s13662-020-02934-0
[30]	K. Shah, M. Arfan, I. Mahariq, A. Ahmadian, S. Salahshour, M. Ferrara, Fractal-fractional mathematical model addressing the situation of corona virus in Pakistan, Results Phys., 19 (2020), 103560. https://doi.org/10.1016/j.rinp.2020.103560 doi: 10.1016/j.rinp.2020.103560
[31]	J. Vanterler da C. Sousa, E. Capelas de Oliveira, A new truncated $M$-fractional derivative type unifying some fractional derivative types with classical properties, Int. J. Anal. Appl., 16 (2018), 83–96. https://doi.org/10.28924/2291-8639-16-2018-83 doi: 10.28924/2291-8639-16-2018-83
[32]	A. Yusuf, M. Inc, D. Baleanu, Optical solitons with $M$-truncated and beta derivatives in nonlinear optics, Front. Phys., 7 (2019), 126. https://doi.org/10.3389/fphy.2019.00126 doi: 10.3389/fphy.2019.00126
[33]	Y. S. Özkan, On the exact solutions to Biswas-Arshed equation involving truncated $M$-fractional space-time derivative terms, Optik, 227 (2021), 166109. https://doi.org/10.1016/j.ijleo.2020.166109 doi: 10.1016/j.ijleo.2020.166109
[34]	K. U. Tariq, M. Younis, S. T. R. Rizvi, H. Bulut, $M$-truncated fractional optical solitons and other periodic wave structures with Schrödinger-Hirota equation, Mod. Phys. Lett. B, 34 (2020), 2050427. https://doi.org/10.1142/S0217984920504278 doi: 10.1142/S0217984920504278
[35]	A. Zafar, A. Bekir, M. Raheel, W. Razzaq, Optical soliton solutions to Biswas-Arshed model with truncated $M$-fractional derivative, Optik, 222 (2020), 165355. https://doi.org/10.1016/j.ijleo.2020.165355 doi: 10.1016/j.ijleo.2020.165355
[36]	R. Goreno, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler functions, related topics and applications, Berlin: Springer, 2014.
[37]	T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
[38]	T. M. Apostol, Mathematical analysis, New York: Addison-Wesley Publishing Company, 1981.
[39]	W. Rudin, Principle of mathematical analysis, New York: McGraw-Hill, 1976.
[40]	M. Ilie, J. Biazar, Z. Ayati, Optimal homotopy asymptotic method for first-order conformable fractional differential equations, J. Fract. Calc. Appl., 10 (2019), 33–45.

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