### AIMS Mathematics

2022, Issue 2: 2281-2317. doi: 10.3934/math.2022130
Research article

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

• 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.

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

### Related Papers:

• 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.

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

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