About this Research Topic
As an extension of the classical integer-order modeling approach, fractional calculus has long been recognized as an efficient and valuable tool in modeling complex phenomena, such as anomalous diffusion, viscoelastic behaviors, heat conduction, chaos, and magnetic resonance imaging. Nowadays, various numerical methods also have been conducted to reveal the underground physical interpretations for the fractional models. Nevertheless, the fractional operator is well-known as a non-local one, which will bring in remarkable computational costs and memory requirements for the numerical simulation.
Hausdorff fractal derivative, one kind of fractal derivatives, has also been proposed to describe a variety of complex problems. Fractal operator significantly extends the application scope of the classical calculus modeling approach under the framework of continuum mechanics to fractal materials. The Hausdorff derivative is mathematically simple and numerically easy to implement with clear physical significance and real-world applications.
Nowadays, various modeling formalisms, including fractional derivative and Hausdorff fractal derivative operators, have been proposed to characterize anomalous physical and engineering behaviors. The fractional derivative operator is well suitable for non-local phenomena and long-term interactions, while Hausdorff fractal derivative underlines the Non-Euclidean distance and temporal scale effect. The inherent relationships and comparisons between these models lack detailed discussions.
Moreover, the existing models have been found to well describe some specific problems with data fitting or qualitative analysis. The underlying physical interpretations of the modeling formalisms or identifications of parameters still require intensive attention.
This Research Topic aims to collect the up-to-the-minute developments in such two modeling operators, including modeling and numerical simulation.
The related research areas are listed but not limited to the topics below:
• Anomalous diffusion: ultra-slow diffusion, sub- and super diffusion;
• Complex rheological behaviors;
• Power-law acoustic attenuation;
• Non-Newtonian fluid;
• Biomedical engineering;
• Signal Processing;
• Other applications of the abovementioned two kinds of models;
• Advanced numerical methods for fractional derivative models;
• Physical analysis of fractional derivative and Hausdorff fractal derivative models;
• Recent development of variable-order fractional derivative and local derivative modeling formalisms, and the corresponding applications.
Keywords: fractional derivative, Hausdorff fractal derivative, non-local operator, anomalous behaviors
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