Scaling (power-type) laws and self-similarity reveal some featuring properties of physical-chemical objects and can be easily noticed in nature. Moreover, also some mathematical abstract objects, such as nondifferentiable functions and fractals, enjoy scaling and self-similarity. Experimental data often show some characteristic power law and self-similarity. A self-similar (scaling) object repeats itself at different scales in space or time. The property of self-similarity gives us a better opportunity to study phenomena from all analytical and computational aspects.
Scale dependence and multiscale analysis are peculiar properties of some families of special functions and can be observed in nature. A continuous scale transformation from one scale to another implies a generalization and suitable extension of differential operator, as it happens with fractional derivatives.
Dynamical processes and systems of fractional order attract researchers from many areas of sciences and technologies, ranging from mathematics and physics to computer science. From analytical point of view, these kinds of problems often lead us to deal with the concepts of scales, fractals, and fractional operators. For instance, medical images nowadays play an essential role in detection and diagnosis of numerous diseases and a suitable scale-depending interpretation of the images is a fundamental aspect of the clinical investigation. Nonlinear analysis of data, collected by modern devices, offers still unsolved analytical problems related to not only complex physics and abstract mathematical theories but also nonlinear science.
The focus of this special issue is on both the abstract mathematical models on scaling and self-similarity and the applied computations on those dynamical processes and systems of fractional order towards the applications in all aspects of theoretical and practical study in analysis.
Scaling and self-similarity characterize several mathematical topics: self-similar analytical problems: scale-depending theoretical and applied analytical problems; fractals, nondifferentiable functions: theoretical and applied analytical problems of fractal type;
fractional differential/integral equations, fractional operators: systems of fractional order; complex systems, nonlinear processing; wavelets; scaling and self-similarity in applications by focusing on theoretical and analytical aspects arising, for example, in nonlinear analysis of data, image analysis, data science, and system science.
This special issue contains 17 papers.
In the category of scale-depending problems, fractals and, self-similarity there are many papers devoted to interesting problem.
H. Zhai proposes some discussion on certain modular equations about infinite products of Ramanujan. The paper of J. Leng and T. Huang deals with the “
In their paper, S. Hu, and P. Liang propose a smart model to detect and locate singularities by using the theory analysis of left-handed Grünwald-Letnikov formula with
J. Yang et al. investigate the “
Several papers are dealing more specifically with fractional calculus-systems of fractional order.
Transforms within the theory of local fractional calculus are considered, respectively, in the paper of X. Yang et al. by focusing on the continuous wavelet transform and the paper of K. Liu et al. which is dealing with the “
Approximate solution of fractional differential equations is considered in the paper “
A new definition of fractional derivatives based on truncated left-handed Grünwald-Letnikov formula with
Z. Deng and X. Yang propose a “
Mathematical models arising in the fractal forest gap via local fractional calculus are studied in the paper of C. Long et al.
Of the 36 submissions, 17 papers are accepted in this special issue (with the acceptance rate being 47.2%). All papers are dealing with current problems in the topics; however, they are not an exhaustive representation of the area of fractional order systems where the concepts of scale, self-similarity, and fractional order interact. In all papers, the authors have focused on the main aspects of the theory and although they have proposed some solutions and models, most problems remain open, thus giving the opportunity to readers for further research and discussions in this field.
Thanks to the excellent authors’ contributions, all of the key aspects raised have been addressed. The authors would also thank the reviewers in helping improvement of the papers and the publisher for continuous professional assistance. One editor (Ming Li) acknowledges the supports in part by the National Natural Science Foundation of China under the Project Grants nos. 61272402, 61070214, and 60873264.