arXiv subjects

Daniel S. Han

Publications and source records attributed to Daniel S. Han.

7 recordsLinked to original sources

arXiv2026

First Passage Times for Variable-Order Time-Fractional Diffusion

We derive the asymptotic first passage time (FPT) distribution for space-dependent variable-order time-fractional diffusion, where the fractional exponent $\alpha(x)$ varies with position. For any sufficiently smooth $\alpha(x)$ on a finite domain with absorbing and reflecting boundaries, we show that the survival probability decays as $\Psi(t)\sim C\,t^{-\alpha_*}/(\ln t)^{\nu}$, where $\alpha_*$ is the minimum value of the fractional exponent and $\nu$ is determined by the location and shape of the minimum. For a constant fractional exponent $\nu=0$ and this provides a theoretical prediction that can identify spatially heterogeneous anomalous transport in experiments. We validate the theory against exact Laplace-space solutions and Monte Carlo simulations for linear and nonlinear profiles of $\alpha(x)$.

cond-mat.stat-mech↗

arXiv2025

First-Passage Times for the Space-Fractional Spectral Fokker-Planck Equation

We extend the random walk framework to include compounded steps, providing first-passage time (FPT) properties for a new class of superdiffusive processes, which are governed by the space-fractional spectral Fokker-Planck equation. This first-passage process leads to novel FPT properties, different from L\'evy flights, that account for space dependent forces and hitting boundaries throughout the path of a jump. The FPT distribution can be derived for different types of barriers and potentials, for which we also provide specific examples. For the one-sided absorbing boundary with no potential on the semi-infinite line, we find that the FPT density scales asymptotically as $t^{-1/(2\alpha)-1}$ for large times, where the parameter $\alpha \in (0,1]$ relates to the power-law behavior for the distribution of the number of compounded steps. This is in agreement with the method of images but different to the Sparre-Andersen scaling $t^{-3/2}$ for corresponding L\'evy flights of order $2\alpha$. In this case, there exists an optimal space-fractional exponent $\alpha$ to minimize the mean FPT.

cond-mat.stat-mech↗

arXiv2025

The Global Diffusion Limit for the Space Dependent Variable-Order Time-Fractional Diffusion Equation

The diffusion equation and its time-fractional counterpart can be obtained via the diffusion limit of continuous time random walks with exponential and heavy-tailed waiting time distributions. The space-dependent variable-order time-fractional diffusion equation is a generalization of the time-fractional diffusion equation with a fractional exponent that varies over space, modelling systems with spatial heterogeneity. However, there has been limited work on defining a global diffusion limit and an underlying random walk for this macroscopic governing equation, which is needed to make meaningful interpretations of the parameters for applications. Here, we introduce continuous time and discrete time random walk models that limit to the variable-order fractional diffusion equation via a global diffusion limit and space- and time-continuum limits. From this, we show how the master equation of the discrete time random walk can be used to provide a numerical method for solving the variable-order fractional diffusion equation. The results in this work provide underlying random walks and an improved understanding of the diffusion limit for the variable-order fractional diffusion equation, which is critical for the development, calibration and validation of models for diffusion in spatially inhomogeneous media with traps and obstacles.

cond-mat.stat-mech↗

arXiv2024

A compounded random walk for space-fractional diffusion on finite domains

We formulate a compounded random walk that is physically well defined on both finite and infinite domains, and samples space-dependent forces throughout jumps. The governing evolution equation for the walk limits to a space-fractional Fokker-Planck equation valid on bounded domains, and recovers the well known superdiffusive space-fractional diffusion equation on infinite domains. We describe methods for numerical approximation and Monte Carlo simulations and demonstrate excellent correspondence with analytical solutions. This compounded random walk, and its associated fractional Fokker-Planck equation, provides a major advance for modeling space-fractional diffusion through potential fields and on finite domains.

cond-mat.stat-mech↗

arXiv2024

Finite Time Blowup of Integer- and Fractional-Order Time-Delayed Diffusion Equations

In this work, exact solutions are derived for an integer- and fractional-order time-delayed diffusion equation with arbitrary initial conditions. The solutions are obtained using Fourier transform methods in conjunction with the known properties of delay functions. It is observed that the solutions do not exhibit infinite speed of propagation for smooth initial conditions that are bounded and positive. Sufficient conditions on the initial condition are also established such that the finite time blowup of the solutions can be explicitly calculated. Examples are provided that highlight the contrasting behaviours of these exact solutions with the known dynamics of solutions to the standard diffusion equation.

math.AP↗

arXiv2024

Exact Solutions of Time-Delay Integer- and Fractional-Order Advection Equations

Transport phenomena play a vital role in various fields of science and engineering. In this work, exact solutions are derived for advection equations with integer- and fractional-order time derivatives and a constant time-delay in the spatial derivative. Solutions are obtained, for arbitrary separable initial conditions, by incorporating recently introduced delay functions in a separation of variables approach. Examples are provided showing oscillatory and translatory behaviours that are fundamentally different to standard propagating wave solutions.

math.AP↗

arXiv2023

A Stochastic Simulation Method for Fractional Order Compartment Models

Our study focuses on fractional order compartment models derived from underlying physical stochastic processes, providing a more physically grounded approach compared to models that use the dynamical system approach by simply replacing integer-order derivatives with fractional order derivatives. In these models, inherent stochasticity becomes important, particularly when dealing with the dynamics of small populations far from the continuum limit of large particle numbers. The necessity for stochastic simulations arises from deviations of the mean states from those obtained from the governing equations in these scenarios. To address this, we introduce an exact stochastic simulation algorithm designed for fractional order compartment models, based on a semi-Markov process. We have considered a fractional order resusceptibility SIS model and a fractional order recovery SIR model as illustrative examples, highlighting significant disparities between deterministic and stochastic dynamics when the total population is small. Beyond its modeling applications, the algorithm presented serves as a versatile tool for solving fractional order differential equations via Monte Carlo simulations.

cond-mat.stat-mech↗