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닫기 International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 144-154

Published online June 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.2.144

© The Korean Institute of Intelligent Systems

## Solutions of a System of Fuzzy Fractional Differential in Terms of the Matrix Mittag-Leffler Functions

1Vellore Institute of Technology, Chennai Campus, India
2New Prince Shri Bhavani Arts and Sciences College, Chennai, India
3Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai Campus, India

Correspondence to :

Received: August 18, 2021; Revised: August 18, 2021; Accepted: April 28, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

The matrix Mittag-Leffler functions play a crucial role in several applications related to systems with fractional dynamics. These functions represent a generalization for fractional-order systems of the matrix exponential function involved in integer-order systems. Computational techniques for evaluating the matrix Mittag-Leffler functions are therefore of particular importance. In this study, a fuzzy system of differential equations with fractional derivatives was solved in terms of the matrix Mittag-Leffler functions. The matrix Mittag-Leffler function was evaluated based on the Jordan canonical form and the minimal polynomial of the matrix.

Keywords: Mittag-Leffler function, Fuzzy calculus, Fractional calculus, System of fuzzy fractional differential equations, Fractional differential equations

### Biographies V. Padmapriya is a Ph.D. research scholar studying at the Division of Mathematics, Vellore Institute of Technology-Chennai Campus, India. She is currently working as an assistant professor in the Department of Statistics, New prince Shri Bhavani Arts and Science college, Chennai, India. Her research interests include fuzzy differential and fuzzy fractional differential M. Kaliyappan is working as a professor of Mathematics at the Division of Mathematics, School of Advanced Sciences, VIT Chennai, India. He has over 23 years of experience in teaching and research. His research interests include approximation theory, numerical computing, differential equations, fuzzy differential equations, fractional differential equations, fuzzy fractional differential equations, and optimization theory.

E-mail: kaliyappan.m@vit.ac.in

### Article

#### Original Article International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 144-154

Published online June 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.2.144

## Solutions of a System of Fuzzy Fractional Differential in Terms of the Matrix Mittag-Leffler Functions

1Vellore Institute of Technology, Chennai Campus, India
2New Prince Shri Bhavani Arts and Sciences College, Chennai, India
3Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai Campus, India

Received: August 18, 2021; Revised: August 18, 2021; Accepted: April 28, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

The matrix Mittag-Leffler functions play a crucial role in several applications related to systems with fractional dynamics. These functions represent a generalization for fractional-order systems of the matrix exponential function involved in integer-order systems. Computational techniques for evaluating the matrix Mittag-Leffler functions are therefore of particular importance. In this study, a fuzzy system of differential equations with fractional derivatives was solved in terms of the matrix Mittag-Leffler functions. The matrix Mittag-Leffler function was evaluated based on the Jordan canonical form and the minimal polynomial of the matrix.

Keywords: Mittag-Leffler function, Fuzzy calculus, Fractional calculus, System of fuzzy fractional differential equations, Fractional differential equations

### Fig 1. Figure 1.

Solutions 1(t) and 2(t) of Example 1 for different values of (a) α = 0.6, (b) α = 0.8, (c) α = 1.0 at t = 1.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 144-154https://doi.org/10.5391/IJFIS.2022.22.2.144

### Fig 2. Figure 2.

Solutions 1(t), 2(t), and 3(t) of Example 2 for different values of (a) α = 0.6, (b) α = 0.8, (c) α = 1.0 at t = 1.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 144-154https://doi.org/10.5391/IJFIS.2022.22.2.144