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Weighted fractional ultrahyperbolic diffusion on geometrically deformed domains

Gustavo Dorrego

TL;DR

This work develops a fundamental solution for the space-time fractional ultrahyperbolic equation on geometrically deformed domains with non-uniform weights, enabling explicit decoupling of medium density from geometric deformation via a weighted Fourier framework. The spatial operator $(-\Box_{\phi,\omega})^{\beta}$ is defined spectrally using a deformed metric, while the time-fractional Hilfer operator ${}^{H}\mathcal{D}_{t,\gamma,\rho}^{\mu,\nu}$ governs temporal dynamics. A key result is that the propagator, expressed in Mellin-Barnes form and as a Fox $H$-function, propagates along a generalized ultrahyperbolic distance $P(\phi(x))$ and includes a density-induced drift $\mathcal{V}_j(x)$ when $\omega \neq |J_\phi|$, yielding geometry-driven, heavy-tailed transport in inhomogeneous media. The framework recovers the classical case when $\phi(x)=x$ and $\omega(x)=1$, and demonstrates how exponential deformations can produce strong anisotropy while isolating metric effects from drift. These results provide a robust mathematical tool for modeling anomalous transport in complex metamaterials and graded media, with a closed-form Green's function furnished by Fox $H$-functions.

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 $(-\Box_{φ,ω})^β$. 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.

Weighted fractional ultrahyperbolic diffusion on geometrically deformed domains

TL;DR

This work develops a fundamental solution for the space-time fractional ultrahyperbolic equation on geometrically deformed domains with non-uniform weights, enabling explicit decoupling of medium density from geometric deformation via a weighted Fourier framework. The spatial operator is defined spectrally using a deformed metric, while the time-fractional Hilfer operator governs temporal dynamics. A key result is that the propagator, expressed in Mellin-Barnes form and as a Fox -function, propagates along a generalized ultrahyperbolic distance and includes a density-induced drift when , yielding geometry-driven, heavy-tailed transport in inhomogeneous media. The framework recovers the classical case when and , and demonstrates how exponential deformations can produce strong anisotropy while isolating metric effects from drift. These results provide a robust mathematical tool for modeling anomalous transport in complex metamaterials and graded media, with a closed-form Green's function furnished by Fox -functions.

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 . 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.
Paper Structure (11 sections, 2 theorems, 31 equations, 1 figure)

This paper contains 11 sections, 2 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Weighted Ultrahyperbolic Operator
  4. Induced Geometry and Causal Cones
  5. Spectral Definition of the Operator
  6. Operational Decomposition and the Induced Drift
  7. Fundamental Solution of the Cauchy Problem
  8. Relevant Particular Cases
  9. Recovery of the Classical Case
  10. Case Study: Exponentially Graded Media
  11. Discussion and Conclusions

Key Result

Theorem 3.3

Given the definition of the Weighted Fourier Transform $\mathcal{F}_{\phi,\omega}$ which incorporates the Jacobian determinant in its measure, the transform of the generalized distance scaled by the inverse density is given by:

Figures (1)

  • Figure 1: Impact of geometric deformation on the characteristic manifolds. (a) Classical ultrahyperbolic geometry ($\lambda \to 0$), showing symmetric hyperbolic characteristics and linear light cones. (b) Deformed geometry under the exponential map with $\lambda=1$. The inhomogeneity induces a curvature in the characteristic curves, breaking the spatial symmetry. This visualizes the structural origin of the bias: although the drift field $\mathcal{V}_j$ is nullified by the choice of $\omega$, the diffusion is 'guided' by the exponential stretching of the underlying space.

Theorems & Definitions (11)

  • Definition 2.1: Weighted Fourier Transform and Inversion DorregoFourier2025
  • Definition 3.1: Deformed Ultrahyperbolic Distance
  • Remark 3.2: The Generalised Characteristic Cone
  • Theorem 3.3: Spectral Identity for the Density-Compensated Distance
  • proof
  • Definition 3.4: Weighted Fractional Ultrahyperbolic Operator
  • Remark 3.5: Independence of Density and Geometry
  • Theorem 4.1: Fundamental Solution
  • proof
  • Remark 4.2: Convergence and Existence Conditions
  • ...and 1 more