Weighted fractional ultrahyperbolic diffusion on geometrically deformed domains

TL;DR

This paper introduces a new spectral method to derive the fundamental solution of a weighted fractional ultrahyperbolic operator, effectively separating medium heterogeneity from geometric deformation in anomalous transport models.

Contribution

It develops a novel spectral approach using the Weighted Fourier Transform to explicitly decouple medium density from geometric deformation in fractional diffusion models.

Findings

Derived the fundamental solution in closed form using Fox H-function.

Discovered a geometry-independent drift mechanism driven by medium inhomogeneity.

Provided a unified framework for anomalous transport in deformed media.

Abstract

Standard fractional models on manifolds often conflate geometric anisotropy with medium heterogeneity. In this Letter, we overcome this rigidity by deriving the fundamental solution for a weighted space-time fractional ultrahyperbolic operator, denoted by $(-\Box_{\phi,\omega})^{\beta}$. Using a novel spectral approach based on the Weighted Fourier Transform, we explicitly \textbf{decouple the medium density from the geometric deformation}. A crucial finding is the emergence of a \textbf{geometry-independent drift mechanism} driven purely by the inhomogeneity of the medium. The Green's function is obtained in closed form via the Fox H-function, providing a unified and computable framework for anomalous transport in complex, structurally deformed media.