Paper
On the explicit formula linking a function to the order of its fractional derivative
Authors
Vasyl Semenov, Nataliya Vasylyeva
Abstract
In this paper, given a certain regularity of a function $v$, we derive an explicit formula relating the order $ν_0\in(0,1)$ of the leading fractional derivative in a fractional differential operator $\mathbf{D_t}$ with the variable coefficients $r_i=r_i(x,t)$ and the function $v$ on which this operator acts. Moreover, we discuss application of this result in the reconstruction of the memory order of semilinear subdiffusion with memory terms. To achieve this aim, we analyze some inverse problems to multi-term fractional in time ordinary and partial differential equations with smooth local or nonlocal additional measurements for small time. In conclusion, we discuss how this formula may be exploited to numerical computation of $ν_0$ in the case of discrete noisy observation in the corresponding inverse problems. Our theoretical results along with the computational algorithm are supplemented by numerical tests.
Metadata
Related papers
Fractal universe and quantum gravity made simple
Fabio Briscese, Gianluca Calcagni • 2026-03-25
POLY-SIM: Polyglot Speaker Identification with Missing Modality Grand Challenge 2026 Evaluation Plan
Marta Moscati, Muhammad Saad Saeed, Marina Zanoni, Mubashir Noman, Rohan Kuma... • 2026-03-25
LensWalk: Agentic Video Understanding by Planning How You See in Videos
Keliang Li, Yansong Li, Hongze Shen, Mengdi Liu, Hong Chang, Shiguang Shan • 2026-03-25
Orientation Reconstruction of Proteins using Coulomb Explosions
Tomas André, Alfredo Bellisario, Nicusor Timneanu, Carl Caleman • 2026-03-25
The role of spatial context and multitask learning in the detection of organic and conventional farming systems based on Sentinel-2 time series
Jan Hemmerling, Marcel Schwieder, Philippe Rufin, Leon-Friedrich Thomas, Mire... • 2026-03-25
Raw Data (Debug)
{
"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.24149v1</id>\n <title>On the explicit formula linking a function to the order of its fractional derivative</title>\n <updated>2026-03-25T10:17:51Z</updated>\n <link href='https://arxiv.org/abs/2603.24149v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.24149v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>In this paper, given a certain regularity of a function $v$, we derive an explicit formula relating the order $ν_0\\in(0,1)$ of the leading fractional derivative in a fractional differential operator $\\mathbf{D_t}$ with the variable coefficients $r_i=r_i(x,t)$ and the function $v$ on which this operator acts. Moreover, we discuss application of this result in the reconstruction of the memory order of semilinear subdiffusion with memory terms. To achieve this aim, we analyze some inverse problems to multi-term fractional in time ordinary and partial differential equations with smooth local or nonlocal additional measurements for small time. In conclusion, we discuss how this formula may be exploited to numerical computation of $ν_0$ in the case of discrete noisy observation in the corresponding inverse problems. Our theoretical results along with the computational algorithm are supplemented by numerical tests.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.AP'/>\n <category scheme='http://arxiv.org/schemas/atom' term='math.NA'/>\n <published>2026-03-25T10:17:51Z</published>\n <arxiv:primary_category term='math.AP'/>\n <author>\n <name>Vasyl Semenov</name>\n </author>\n <author>\n <name>Nataliya Vasylyeva</name>\n </author>\n </entry>"
}