# The relationship between fractional calculus and fractals

Oct 31, The interest to relationship between fractals and fractional calculus is renewed again. Some original approach (but without proper physical. The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. the possibility of using fractional calculus, a field which deals with derivatives The possible connection between fractals and fractional calculus raises var-.

Besides, the generalizations of the results for a combination of fractal circuits having a discrete set of fractal dimensions were obtained. We suppose that these new results help understand deeper the intimate links between fractals and fractional integrals of different types. These results can be used in different branches of the interdisciplinary physics, where the different equations describing the different physical phenomena and containing the fractional derivatives and integrals are used.

Key Words and Phrases: The interest to relationship between fractals and fractional calculus is renewed again. Some original approach but without proper physical interpretation was outlined in papers [ 1645 ].

One of the basic problems that did not accurately solved yet in the fractional calculus community is the finding of the justified and accurate relationships between the smoothed functions averaged over fractal objects and fractional operators. This problem was solved partly for the time-dependent functions averaged over Cantor sets in monograph [ 9 ] and paper [ 10 ], where the influence of unknown log-periodic function leading finally to the understanding of the meaning of the fractional integral with the complex-conjugated power-law exponents was taken into account.

Possible generalizations helping to understand the role of a spatial fractional integral as a mathematical operator replacing the operation of averaging of the smoothed functions over fractal objects were considered in monograph [ 9 ] as well.

Nevertheless, the basic reason that serves as a specific mathematical obstacle in accurate establishing of the desired relationship between the fractal object and the corresponding fractional integral is the absence of the 2D- and 3D-Laplace transformations.

**Math 312 Fractional Calculus final presentation**

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This article has been cited by other articles in PMC. Abstract This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures Grizzi and Chiriva-Internati,such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well Bassingthwaighte et al.

Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate.

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The observation that human physiology is primarily fractal was first made in the s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series.

This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data.

Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation.

A fractional operator derivative or integral acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a fractal statistical process.

Control of physiologic complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. We emphasize the difference between homeostatic and allometric control mechanisms. Homeostatic control has a negative feedback character, which is both local and rapid.

Allometric control, on the other hand, is a relatively new concept that takes into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the network variable. We hypothesize that allometric control maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks.

Moreover, allometric control can often be described using the fractional calculus to capture the dynamics of complex physiologic networks. Complexity in this context incorporates the recent advances in physiology concerned with the applications of the concepts from fractal geometry, fractal statistics and nonlinear dynamics, to the formation of a new kind of understanding within the life sciences.

### Fractal Physiology and the Fractional Calculus: A Perspective

A parallel development has been the understanding of the dynamics of fractal processes and how those dynamics are manifest in the control of physiologic networks. For a number of years the study of fractals and its application to physiology was restricted to the determination of the fractal dimension of structure, in particular, the static structure of objects and the scaling of time series.

However, now we explore the dynamics of fractal processes using the fractional calculus, and apply this dynamical approach to both regular and stochastic physiologic processes. To understand the need for such an approach a historical perspective is useful.

Homeostasis It is not a coincidence that the modern view of how the human body operates mirrors our understanding of the technological society in which we live, where a thermostat controls the temperature of a home, the sound of a voice can turn the lights on and off, and cruise control determines the speed of a car. It is not clear when this idea of how the body works began to permeate society, but in medicine the concept was introduced by the nineteenth century scientist Claude Bernard — He developed the notion underlying homeostasis in his study of stability of the human body.

The word homeostasis was popularized half a century later by Walter Cannon — in his book The Wisdom of the Body Cannon, Homeostasis is what many consider to be the guiding principle of medicine, whereby every human body has multiple automatic inhibition mechanisms that suppress disquieting influences of the environment. Homeostasis is the evolutionary strategy selected to enable the human body to maintain an internal balance, although it is not always evident how a particular suppressing response is related to a specific antagonism.

Biology teaches that evolution has, over the millennia, reduced homeostatic networks to the bare minimum, so that in the spirit of parsimony, every internal mechanism of a physiological network is necessary to maintain either the structural or functional integrity of the organism.

But why should physiologic networks be homeostatic?

- Fractal Physiology and the Fractional Calculus: A Perspective

In part, nature's choices have to do with the fact that no physiologic network is isolated; these networks are, in fact, made up of a mind-numbing number of subnetworks, the cells. The task of a cell is simple and repetitive, but that of an organ is not. Therefore a complex network like the cardiovascular is made up of a variety of cell types, each type performing a given different function.

If responses to changes in the external environment were at the cellular level, physiology would be much more complicated than it is already, and organs would no doubt be unstable. But nature has found that if the immediate environment of the cells is kept within certain narrowly defined limits, then the cells can continue to perform their specific tasks and no others, even while organs respond to sometimes extravagant external disturbances.

As long as the internal environment stays within a certain operational range the cells continue to function without change.

Thus, homeostasis is the presumed strategy that nature has devised to keep the internal state of the body under control. The level of sophistication of control mechanisms was brought to light with the centrifugal fly-ball governor constructed by J.

## Fractional Calculus and Applied Analysis

Watt for regulating the speed of the rotary steam engine. This artificial control mechanism heralded the onset of the Industrial Revolution. The first mathematical description and consequent understanding of Watt's governor was constructed by the English physicist J. Maxwell inwhen he linearized the differential equations describing the governor's dynamics. The solutions to the linearized differential equations control are stable when the eigenvalues have negative real parts stabilizing feedback and in this way the language for the control of dynamical networks was introduced.