Speaker: Mohammed Al-Refai, Dept. of Mathematics, Yarmouk University, Irbid, Jordan

Time : 16:00 - 17:00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

MS Teams : A MS Teams link will appear here, an hour before the talk. For best experince, please download the app.

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: fractional derivatives, general fractional operators, Sonine condition, maximum principles, fractional differential equations

Abstract: We start with a summary of recent results concerning the estimates of fractional derivatives of  functions at extreme points. The results include the Riemann-Liouville derivative, the Caputo derivative and the general fractional derivatives. We then use these estimates to derive weak and strong maximum principles for related fractional differential inequalities. These principles serve as tools for the analytical study of related linear and nonlinear fractional differential equations and their qualitative properties. The analytical results include deriving necessary conditions for the existence of solutions to linear fractional equations, establishing uniqueness and stability results for linear and nonlinear fractional initial–boundary value problems, and obtaining norm estimates for their solutions. We present several examples to illustrate the effectiveness of the obtained results. At the end, we outline some open problems and propose several directions for further study.

Biography: Prof. Mohammed Al-Refai holds a Ph.D. degree in Applied Mathematics from McGill University in 2000. Since then, he worked at different schools in the Middle East: at Jordan University of Science and Technology, at United Arab Emirates University and currently at Yarmouk University. Prof. Al-Refai is the Editor-in-Chief of the Jordan Journal of Mathematics and Statistics (JJMS).

Prof. Al-Refai published more than 80 papers, majority of them are published in peer-reviewed journals with excellent reputation. He received many research grants and participated in organizing several scientific meetings and delivered talks in more than 35 international scientific conferences in the area of applied mathematics. His research interests include: Numerical and analytical methods for ordinary and partial differential equations, Theory of integral and differential equations and fractional differential equa-
tions, Fractional calculus, Maximum principles, and Non-local problems. His research has led to significant contributions in the field of analysis of differential equations and fractional differential equations, where he has explored both theoretical frameworks and practical applications. He has received over 1,900 citations on Google Scholar https://scholar.google.com/citations?user=nRujkOYAAAAJ&hl=ar&oi=ao and has been recognized in the top 2% of researchers worldwide over the past four years, according to Stanford University Scientists’ Classification.

Bibliography

[1] M. Al-Refai. “On the fractional derivatives at extreme points”. In: Electronic Journal of Qualitative Theory of Differential Equations 55 (2012), pp. 1-5.
[2] M. Al-Refai, Yu. Luchko, “Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications”. In: Fractional Calculus and Applied Analysis 17(2) (2014), pp. 483-498.
[3] M. Al-Refai, Y. Luchko, “On fractional derivatives of Djrbashian–Nersessian type with the nth-level Sonin kernels and their basic properties”. In: Fractional Calculus and Applied Analysis 28(2) (2025), pp. 607-621.
[4] M. Al-Jararha, M. Al-Refai, Y. Luchko, “A self-adjoint fractional Sturm-Liouville problem with the general fractional derivatives”. In: Journal of Differential Equations 413 (2024), pp. 110-128.
[5] M Al-Refai, A Fernandez, “Comparison principles for a class of general integro-differential inequalities with applications”. In: Computational and Applied Mathematics 43(2) (2024), pp. 1-12.
[6] M Al-Refai, Y. Luchko, “The general fractional integrals and derivatives on a finite interval”. In: Mathematics 11(4) 2023, Pages 1031.

[7] M Al-Refai, A Fernandez, “Generalizing the fractional calculus with Sonine kernels via conjugations”. In: Journal of Computational and Applied Mathematics 427 (2023), 115159.
[8] A.N. Kochubei, “General fractional calculus, evolution equations, and renewal processes”. In: Integral Equations and Operator Theory 71 (2011), pp. 583–600.
[9] Y. Luchko, “Maximum principle for the generalized time-fractional diffusion equation”. In: Journal of Mathematical Analysis and Applications 351(2009), pp. 218–223.
[10] Y. Luchko, “General fractional integrals and derivatives with the Sonine kernels”. In: Mathematics 9(6) (2021), 594