arXiv subjects

C. N. Angstmann

Publications and source records attributed to C. N. Angstmann.

4 recordsLinked to original sources

arXiv2020

Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains

Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a one-dimensional domain that is evolving. The model equations, which have been derived from generalized continuous time random walks, can incorporate complexities such as subdiffusive transport and inhomogeneous domain stretching and shrinking. A method for constructing analytic expressions for short time moments of the position of the particles is developed and moments calculated from this approach are shown to compare favourably with results from random walk simulations and numerical integration of the reaction transport equation. The results show the important role played by the initial condition. In particular, it strongly affects the time dependence of the moments in the short time regime by introducing additional drift and diffusion terms. We also discuss how our reaction transport equation could be applied to study the spreading of a population on an evolving interface.

cond-mat.stat-mech↗

arXiv2017

Subdiffusive discrete time random walks via Monte Carlo and subordination

A class of discrete time random walks has recently been introduced to provide a stochastic process based numerical scheme for solving fractional order partial differential equations, including the fractional subdiffusion equation. Here we develop a Monte Carlo method for simulating discrete time random walks with Sibuya power law waiting times, providing another approximate solution of the fractional subdiffusion equation. The computation time scales as a power law in the number of time steps with a fractional exponent simply related to the order of the fractional derivative. We also provide an explicit form of a subordinator for discrete time random walks with Sibuya power law waiting times. This subordinator transforms from an operational time, in the expected number of random walk steps, to the physical time, in the number of time steps.

math.NA↗

arXiv2017

Generalised fractional diffusion equations for subdiffusion on arbitrarily growing domains

Many physical phenomena occur on domains that grow in time. When the timescales of the phenomena and domain growth are comparable, models must include the dynamics of the domain. A widespread intrinsically slow transport process is subdiffusion. Many models of subdiffusion include a history dependence. This greatly confounds efforts to incorporate domain growth. Here we derive the fractional partial differential equations that govern subdiffusion on a growing domain, based on a Continuous Time Random Walk. This requires the introduction of a new, comoving, fractional derivative.

cond-mat.stat-mech↗

arXiv2015

A fractional order recovery SIR model from a stochastic process

Over the past several decades there has been a proliferation of epidemiological models with ordinary derivatives replaced by fractional derivatives in an an-hoc manner. These models may be mathematically interesting but their relevance is uncertain. Here we develop an SIR model for an epidemic, including vital dynamics, from an underlying stochastic process. We show how fractional differential operators arise naturally in these models whenever the recovery time from the disease is power law distributed. This can provide a model for a chronic disease process where individuals who are infected for a long time are unlikely to recover. The fractional order recovery model is shown to be consistent with the Kermack-McKendrick age-structured SIR model and it reduces to the Hethcote-Tudor integral equation SIR model. The derivation from a stochastic process is extended to discrete time, providing a stable numerical method for solving the model equations. We have carried out simulations of the fractional order recovery model showing convergence to equilibrium states. The number of infecteds in the endemic equilibrium state increases as the fractional order of the derivative tends to zero.

q-bio.PE↗