Speaker: Yubin Yan, School of Computer and Engineering Sciences, University of Chester, Chester, CH1 4BJ, UK

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

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

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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 differential equations, Adams method, graded meshes, error estimates

Abstract: We consider a fractional Adams method for solving the nonlinear fractional differential equation ^CD^α_t y(t) = f (t, y(t)), α > 0, equipped with the initial conditions y(k)(0) = y(k) 0 , k = 0, 1, . . . , ⌈α⌉ − 1. Here α may be an arbitrary positive number and
⌈α⌉ denotes the smallest integer no less than α and the differential operator is the Caputo derivative. Under the assumption ^CD^α_t y ∈ C²[0,T], Diethelm et al. [1, Theorem 3.2] introduced a fractional Adams method with the uniform meshes t_n = T (n/N), n = 0, 1, 2, . . . , N
and proved that this method has the optimal convergence order uniformly in t_n, that is O(N −2) if α > 1 and O(N −1−α) if α ≤ 1. They also showed that if ^CD^α_t y(t) /∈ C2[0, T ], the optimal convergence order of this method cannot be obtained with the uniform meshes.
However, it is well known that for y ∈ C^m[0, T] for some m ∈ N and 0 < α < m, the Caputo fractional derivative ^CD^α_t y(t) takes the form “ ^CD^α_t y(t) = ct⌈α⌉−α + smoother terms” [1, Theorem 2.2], which implies that ^CD^α_t y behaves as t^{⌈α⌉−α} which is not in C²[0, T ]. By using the graded meshes t_n = T (n/N )^r, n = 0, 1, 2, . . . , N with some suitable r > 1, we show that the optimal convergence order of this method can be recovered uniformly in tn even if ^CD^α_t y behaves as tσ, 0 < σ < 1. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Biography: Dr. Yubin Yan is a Professor of Mathematics in the School of Computer and Engineering Sciences at the University of Chester, UK. He obtained his PhD in Mathematics from Chalmers University of Technology, Sweden, in 2003. Following his doctoral studies, he worked as a Research Associate at the University of Manchester (2003–2004) and the University of Sheffield (2004–2007) before joining the University of Chester as a full-time academic in 2007. Dr. Yan's research interests lie in numerical analysis for stochastic and deterministic partial differential equations, the finite element method, and numerical methods for fractional differential equations. In 2005, he introduced a novel framework for error analysis of finite element approximations to stochastic parabolic equations, which has since become a widely cited and influential contribution in the field. He has published more than 100 refereed research papers in leading international journals, including SIAM Journal on Numerical Analysis, BIT Numerical Mathematics, and IMA Journal of Numerical Analysis. His work has made significant contributions to numerical analysis, scientific computing, and fractional differential equations.

Dr. Yan has successfully supervised five PhD theses and over thirty MSc dissertations and is currently supervising three PhD students. He has also hosted ten postdoctoral researchers and visiting research scholars from China at the University of Chester, fostering international research collaboration. In addition to his research and teaching activities, Dr. Yan regularly serves as a reviewer for more than thirty international scientific journals. He is a member of the editorial boards of several prestigious mathematics journals, including Applied Numerical Mathematics and Fractional Calculus and Applied Analysis. Furthermore, Dr. Yan has co-organized a number of minisymposia at major international conferences, including the 27th, 29th, 30th, and 31st Biennial Numerical Analysis Conferences held at the University of Strathclyde in 2017, 2021, 2023, and 2025, respectively

Bibliography

[1] K. Diethelm, N.J. Ford and A.D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algor., 36(2004), 31-52.
[2] Y. Liu, J. Roberts and Y. Yan, A Fractional Adams Method for Caputo Fractional Differential Equations with Modified Graded Meshes, Numer. Algorithms (78)2017, 1195-1216
[3] Y. Yang and Y. Yan, A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes, Mathematics (13)2025, 891