Yiming Wang

Water pollution from industrial and domestic sewage demands the accurate modeling of wastewater treatment processes. While the Lawrence–McCarty model is widely used for activated sludge systems, its integer-order formulation cannot fully capture the fractal characteristics of microbial aggregation. This study proposed a fractal Lawrence–McCarty model (FLMM) by incorporating local fractional derivatives (α = ln2/ln3) to describe microbial growth dynamics on Cantor sets. Theoretical analysis reveals that the FLMM exhibits Mittag-Leffler-type solutions, which naturally generate step-wise growth curves—consistent with the phased behavior (lag, rapid growth, and stabilization) observed in real sludge systems. Compared with classical models, the FLMM’s fractional-order structure provides a more flexible framework to represent memory effects and spatial heterogeneity in microbial communities. These advances establish a mathematical foundation for future experimental validation and suggest potential improvements in predicting nonlinear biomass accumulation patterns.

A type of general fractional calculus with respect to another function is introduced in this paper. The kernel function within the fractional derivative and integral is specified as the Exponential function in a typical way. Based on the excellent property of the newly defined general fractional calculus, we proposed two viscoelastic models in the fractional manner compared with the classical calculus.

Building upon the foundations of classical fractional derivatives, the general fractional derivative emerges as a significant advancement in the development of constitutive models, especially for materials with complex properties. This derivative distinguishes itself through a kernel function of variable form, enabling it to encapsulate diverse characteristics of the creep process more effectively than its classical counterpart. This study introduces a general-variable order fractional creep constitutive model, ingeniously linking the order of the fractional derivative to Talbot gradation, which describes the aggregate gradation of cemented backfill materials, alongside dosage and confining pressure parameters. The model's innovative design synergizes the kernel function's diversity from the general fractional derivative with the phase adaptability inherent in the variable-order derivative. This integration permits a comprehensive description of each stage of the creep curve for cementitious filling materials in varying compositions, leveraging the Gamma function's properties within the positive real number domain. The model's rationality and validity are substantiated through a comparative analysis between experimental creep curves and theoretical predictions, affirming its relevance and accuracy in practical applications. This approach represents a notable contribution to the understanding of cemented backfill materials' behavior, offering a robust tool for engineering analysis and design.