Analytical solutions of incommensurate fractional differential equation systems with fractional order $ 1 &lt; \alpha, \beta &lt; 2 $ via bivariate Mittag-Leffler functions

Yong Xian Ng; Chang Phang; Jian Rong Loh; Abdulnasir Isah; Yong Xian Ng; Chang Phang; Jian Rong Loh; Abdulnasir Isah

doi:10.3934/math.2022130

AIMS Mathematics

2022, Volume 7, Issue 2: 2281-2317. doi: 10.3934/math.2022130

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Research article

Analytical solutions of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $ via bivariate Mittag-Leffler functions

1.
Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Malaysia
2.
Foundation in Engineering, Faculty of Science and Engineering, University of Nottingham Malaysia, Semenyih, Selangor, Malaysia
3.
Department of Mathematics Education, Tishk International University, Erbil, Iraq
4.
Center for Computational Applied Mathematics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Malaysia

Received: 03 September 2021 Accepted: 01 November 2021 Published: 10 November 2021
MSC : 34A08, 34A25, 26A33

In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $. The derivation is extended from a recently published paper by Huseynov et al. in ^[1], which is limited for incommensurate fractional order $ 0 < \alpha, \beta < 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 < \alpha, \beta < 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.
- incommensurate fractional order system,
- bivariate Mittag-Leffler function,
- Picard's successive approximations,
- analytical solutions
Citation: Yong Xian Ng, Chang Phang, Jian Rong Loh, Abdulnasir Isah. Analytical solutions of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $ via bivariate Mittag-Leffler functions[J]. AIMS Mathematics, 2022, 7(2): 2281-2317. doi: 10.3934/math.2022130

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Abstract

In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $. The derivation is extended from a recently published paper by Huseynov et al. in ^[1], which is limited for incommensurate fractional order $ 0 < \alpha, \beta < 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 < \alpha, \beta < 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.

References

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