arXiv subjects

William Cvetko

Publications and source records attributed to William Cvetko.

3 recordsLinked to original sources

arXiv2026

Anomalous diffusion memory factorization: Characteristic timescales and application to inverse problem

Memory effects and anomalous diffusion arising in models of transport in complex media, fractals, and viscoelastic materials can be described by fractional-differential equations, such as the time-fractional diffusion equation $D^\alpha_t u - \Delta_x u = 0$. The paper develops a decomposition of solutions of these equations into a product of spatial and temporal components, corresponding to a freeze-out at long times. For fractional diffusion with the Caputo derivative the spatial factor is the inverse-Laplacian $(-\Delta)^{-1}$ of the initial data, while for the Riemann-Liouville derivative it is the inverse bi-Laplacian $ (-\Delta)^{-2}$. In both cases, the temporal factor of the solution is a scaled negative power of time. This memory artifact explicitly encodes the initial data, which gives a simple and robust way to reconstruct the initial conditions in the backward-in-time inverse problem with solution values measured at long times. We derive characteristic timescales for Mittag-Leffler functions, which correspond to such factorization in anomalous diffusion. This also enables an accurate approximation of the number of real zeros of the Mittag-Leffler function. We apply these results to heat transfer on a comb, a model which manifests subdiffusion arising from the comb's fractal structure.

math.AP↗

arXiv2026

Convolution-to-sum identities for Mittag-Leffler type functions

Product-to-sum identities for trigonometric functions play a fundamental role in function theory and numerous applications. In this spirit, we present convolution-to-sum identities for Mittag-Leffler type functions. Using a Laplace domain analysis of fractional operators, we identify a family of Mittag-Leffler type functions that encapsulates the eigenfunctions of Riemann-Liouville and Caputo fractional derivatives. We work with two closely-related parameterizations of this class, $R_{\alpha,v}$ and $P_{\alpha,w}$. The convolution of two such functions can be expressed as a series of them. Moreover, if the functions share the same order $\alpha$, the convolution can be reduced to a sum of two $P/R$ functions through a partial-fraction decomposition in the Laplace domain. Furthermore, $R$ and $P$ functions satisfy a generalization of Euler's identity, which expands the scope of the previous result to convolutions of $P/R$ functions whose orders $\alpha_1,\alpha_2$ are related by a rational factor. For $\frac{\alpha_1}{\alpha_2} = \frac{n}{m}$, the resulting sum has $n+m$ terms. The foundational results and methods developed here are illustrated by their application to forced subdiffusion and to a fractionally attenuated wave equation (the Caputo-Wismer-Kelvin, or the fractional Kelvin-Voigt model).

math.AP↗

arXiv2025

InterACTing dark radiation models after ACT

In this work we assess the implications of the Atacama Cosmology Telescope DR6 measurements for two interacting dark radiation scenarios previously shown to mitigate the Hubble tension. The first model, Wess-Zumino dark radiation (WZDR), features a mass threshold in the dark sector that induces a step-like reduction in the dark radiation abundance as the dark temperature evolves. The second model, new atomic dark matter (nuADaM), introduces dark radiation that remains coupled to a subcomponent of dark matter until shortly before matter-radiation equality. Earlier analyses using Planck data demonstrated that these interactions significantly relax constraints on the dark radiation density and allow values of $H_0$ consistent with local distance-ladder determinations. Incorporating ACT DR6, which extends CMB measurements deep into the high-$\ell$ damping tail, we find that constraints on the additional radiation component tighten substantially in both scenarios, closing most of the parameter space that previously enabled higher values of $H_0$. We further analyze a generalized model including both free-streaming and self-interacting dark radiation, and show that the resulting constraints are consistent with ACT's findings for the limiting cases of purely free-streaming or purely self-interacting radiation. Overall, ACT DR6 significantly restricts interacting dark radiation as a solution to the Hubble tension.

astro-ph.CO↗