Fractional Differential Equations in Chemical Modeling: From Memory and Anomalous Transport to Mathematically Justified Models

Abstract

Fractional differential equations are increasingly used to model memory and anomalous transport in chemistry and related fields. However, the formal replacement of an integer-order derivative by a fractional one does not by itself justify fractional dynamics. The essential issue is whether nonlocality can be derived from a physical, stochastic, geometric, or reduction mechanism and whether it leads to experimentally testable consequences. This review, written primarily for mathematicians and specialists in fractional analysis, discusses representative applications in electrochemistry, adsorption and sorption, reaction--subdiffusion, chemical and enzyme kinetics, polymer relaxation, and atmospheric chemistry. Particular attention is paid to distinguishing mechanism-based fractional models from phenomenological fractionalizations. Mathematical questions of well-posedness, nonnegativity, conservation laws, dimensional consistency, parameter identifiability, and numerical convergence are emphasized. Experimental electrochemical examples illustrate the need to distinguish genuine fractional effects from model mismatch or non-identifiability. Several open problems connecting fractional analysis with experimentally grounded chemical modeling are formulated.

arXiv comment31 pages, 4 figures