Exact solutions of fractional differential Stein matrix equations via diagonalization and Mittag–Leffler functions

Lakhlifa Sadek; Ali Algefary; Lakhlifa Sadek; Ali Algefary

doi:10.3934/math.2026688

AIMS Mathematics

2026, Volume 11, Issue 6: 16763-16787. doi: 10.3934/math.2026688

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Research article

Exact solutions of fractional differential Stein matrix equations via diagonalization and Mittag–Leffler functions

Lakhlifa Sadek ^1,2,
Ali Algefary ^{3
,
,}

1.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamilnadu, India
2.
Department of Mathematics, Faculty of Sciences and Technology, BP 34. Ajdir 32003 Al-Hoceima, Abdelmalek Essaadi University, Tetouan, Morocco
3.
Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia

Received: 16 March 2026 Revised: 03 June 2026 Accepted: 08 June 2026 Published: 11 June 2026
MSC : 34A08, 15A24, 26A33, 15A18

In this work, an explicit formula in terms of the Hadamard product was derived for the solutions to fractional differential Stein matrix equations (FDSMEs), assuming that the coefficient matrices are separately diagonalizable. To begin with, we gave a general formula using the Kronecker product notation, which does not need any diagonalizability assumption to show the existence and uniqueness of the solution. In the case of diagonalizable coefficient matrices, the matrix equation became decoupled scalar Caputo fractional differential equations. Solutions to these differential equations were given explicitly in terms of two-parameter Mittag–Leffler functions and put together using the Hadamard product. The results for integer-order differential equations are obtained as special cases. Finally, we give three examples.
- fractional derivative,
- fractional Stein matrix equations,
- Mittag–Leffler function,
- Diagonalization,
- Hadamard product
Citation: Lakhlifa Sadek, Ali Algefary. Exact solutions of fractional differential Stein matrix equations via diagonalization and Mittag–Leffler functions[J]. AIMS Mathematics, 2026, 11(6): 16763-16787. doi: 10.3934/math.2026688

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Abstract

In this work, an explicit formula in terms of the Hadamard product was derived for the solutions to fractional differential Stein matrix equations (FDSMEs), assuming that the coefficient matrices are separately diagonalizable. To begin with, we gave a general formula using the Kronecker product notation, which does not need any diagonalizability assumption to show the existence and uniqueness of the solution. In the case of diagonalizable coefficient matrices, the matrix equation became decoupled scalar Caputo fractional differential equations. Solutions to these differential equations were given explicitly in terms of two-parameter Mittag–Leffler functions and put together using the Hadamard product. The results for integer-order differential equations are obtained as special cases. Finally, we give three examples.

References

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