TY - JOUR
T1 - Nonlocal-in-Time Dynamics and Crossover of Diffusive Regimes
AU - Du , Qiang
AU - Zhou , Zhi
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 353
EP - 370
PY - 2023
DA - 2023/03
SN - 20
DO - http://doi.org/10.4208/ijnam2023-1014
UR - https://global-sci.org/intro/article_detail/ijnam/21537.html
KW - Nonlocal model, nonlocal operators, mean square displacement, sub-diffusion, numerical methods.
AB - <p style="text-align: justify;">The aim of this paper is to study a simple nonlocal-in-time dynamic system proposed
for the effective modeling of complex diffusive regimes in heterogeneous media. We present its
solutions and their commonly studied statistics such as the mean square distance. This interesting
model employs a nonlocal operator to replace the conventional first-order time-derivative. It
introduces a finite memory effect of a constant length encoded through a kernel function. The
nonlocal-in-time operator is related to fractional time derivatives that rely on the entire time-history on one hand, while reduces to, on the other hand, the classical time derivative if the
length of the memory window diminishes. This allows us to demonstrate the effectiveness of the
nonlocal-in-time model in capturing the crossover widely observed in nature between the initial
sub-diffusion and the long time normal diffusion.</p>