Fractal and Fractional Order Modeling of Real-World Phenomena

Dear Colleagues,

In this summary, the term ‘Real World’ is the totality of known and unknown systems existing in the Universe. A mathematical model is a mental construction and usually takes the form of a set of aggregated mathematical tools: geometry, topology, and equations describing a number of variables. We distinguish between continuous models, in which the variables vary continuously in space and time and discrete models whose variables varies discontinuously. Applied mathematicians have a procedure, almost a philosophy, that they apply when building models, for a system of interest that they want to describe or, more importantly, explain. Observations of the system lead, sometimes after a great deal of effort, to a hypothetical mechanism that can (verbally) explain the phenomenon. The purpose of a mathematical model is, then, to formulate a description of the mechanism in quantitative terms. To see more closely how a mathematical model is built, it is advisable to read the geometric model that is today called Mandelbrot's Fractals, the viscous fluid flow model built by Navier–Stokes, or simple electrical circuit models. The majority of the 'technical' papers (stability, complex systems, etc.) published today in the scientific literature are not of this kind. Authors take a system of differential equations with integer-order derivatives, describing a real-world phenomena, for instance, Navier–Stokes equations. They have forgotten that the system in discussion was obtained as a result of the aggregation of different principles in physics, and was transformed in a system of differential equations using classical calculus, respecting the principle of objectivity—the described real-world phenomenon is independent onf the observer, well-posedness and coherence. What they achieve is the replacement of integer-order temporary derivatives with fractional-order temporal derivatives, showing that the results concerning existence, stability, etc., are different. However, they do not observe that, in this replacement, the principle of objectivity and coherence, for example, was violated. The purpose of this Special Issue is not the collection of this kind of papers.

In this Special Issue, we want to collect papers which include the following:

1. The authors describe explicitly, in words, the real-world system they want to model.
2. The authors describe, verbally and explicitly, a hypothetical mechanism that can explain the phenomenon.
3. The authors present, explicitly, the principle in physics which will be aggregated in the model.
4. The authors present, from the set of existing mathematical tools, those which will be used in the model. If necessary, the authors can create new mathematical tools.
5. The authors prove that the mathematical model proposed by them is objective (independent of the observer), is coherent and an initial value problem is well posed.

Prof. Dr. Stefan Balint
Guest Editor

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