### Journals Information

**
Mathematics and Statistics Vol. 11(4), pp. 726 - 732 DOI: 10.13189/ms.2023.110414 Reprint (PDF) (568Kb) **

## Fractional Differential Equations and Matrix Bicomplex Two-parameter Mittag-Leffler Functions

**A. Thirumalai , K. Muthunagai ^{*}, M. Kaliyappan **

School of Advanced Sciences, Vellore Institute of Technology, India

**ABSTRACT**

The skew field of Quaternions is the best known extension of the field of Complex numbers. The beauty of the Quaternions is that they form a field but the handicap is loss of commutativity. Thus the four- dimensional algebra called Bicomplex numbers with the set of all Complex numbers as a subalgebra preserving commutativity came into existence, by considering two imaginary units. The conventional calculus is generalized using Fractional calculus which is useful to extend derivatives of integer order to fractional order. Due to their vast applications to various disciplines of Science and Engineering, Mittag- Leffler functions have become prominent. Our contribution here is a combination of all the three streams mentioned above. In our research findings, bicomplex two-parameter Mittag- Leffler functions have been obtained as the solutions for the set of fractional differential equations that are linear in bicomplex space. A block diagonal of a square matrix is a diagonal matrix whose Principal diagonal elements are square matrices and the diagonal elements of lie along the diagonal of . A Jordan block is a matrix that is upper triangular with in the Principal diagonal, 1s just above the Principal diagonal and all other entries as 0. A Jordan Canonical form is a block diagonal matrix where each block is Jordan. A minimal polynomial of a matrix is a polynomial which is monic in with least degree. By using the methods of the minimal polynomial (eigenpolynomial) and Jordan canonical matrix, we have computed matrix Mittag–Leffler functions. The solutions obtained for the numerical examples have been visualized and interpreted using MATLAB.

**KEYWORDS**

Bicomplex Mittag – Leffler Function, Bicomplex Laplace Transform, Fractional Calculus, Fractional Derivative, Fractional Differential Equation

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] A. Thirumalai , K. Muthunagai , M. Kaliyappan , "Fractional Differential Equations and Matrix Bicomplex Two-parameter Mittag-Leffler Functions," Mathematics and Statistics, Vol. 11, No. 4, pp. 726 - 732, 2023. DOI: 10.13189/ms.2023.110414.

(b). APA Format:

A. Thirumalai , K. Muthunagai , M. Kaliyappan (2023). Fractional Differential Equations and Matrix Bicomplex Two-parameter Mittag-Leffler Functions. Mathematics and Statistics, 11(4), 726 - 732. DOI: 10.13189/ms.2023.110414.