Preface

Chapter 8 **Factors Affecting Accuracy and Precision in Measuring Material**

**Section 3 Applications, Scattering, Porosity, Turbulence 185**

**Fractal Anisotropic Surface 187**

Alexander A. Potapov

Chapter 10 **Fractal Geometry and Porosity 249**

Opeyemi Alice Oyewo

**Fractal Geometry 267**

Chapter 9 **On the Indicatrixes of Waves Scattering from the Random**

Chapter 11 **Analysis and Application of Decaying Turbulence with Initial**

Oluranti Agboola, Maurice Steven Onyango, Patricia Popoola and

Hiroki Suzuki, Shinsuke Mochizuki, Yasuhiko Sakai and Koji Nagata

**Surfaces 173** Jason A. Griggs

**VI** Contents

Fractal analysis has entered a new era. The applications to different areas of knowledge have been surprising. Let us begin with the fractional calculus-fractal geometry relationship, which allows for modeling with extreme precision of phenomena such as diffusion in po‐ rous media with fractional partial differential equations in fractal objects. Where the order of the equation is the same as the fractal dimension, this allows us to make calculations with enormous precision in diffusion phenomena—particularly in the petroleum industry, for new spillage prevention.

On the subject of fractional calculus, it is a problem proposed 300 years ago. In the I'Hopital letters, Leibniz and Bernoulli asked themselves how to define the fractional derivative and give it a physical and geometrical interpretation. Answers (but only about the definition) have been around since 1720, when Euler generalized the concept of *n!* for real and complex numbers with the Gamma function in order to give us the first answer to it. Then, in 1850, Riemann and Liouville gave us an answer about fractional integrals, so, too, others like Grunwald and Letnikov. But the physical and geometrical interpretation has only appeared in our days now that we understand that there is a relationship between fractional calculus and fractal geometry.

We have several partial answers to this 300-year-old problem:

the order of the fractional partial differential equations turned out to be related to the fractal dimension of the geometry where the phenomena take place. And the success of fractional calculus modeling physical phenomena reveals an underlying fractal nature of reality (S. Butera and M. Di Paola).

We have also made progress with the Nigmatullin-Rutman controversy—and the clarifica‐ tions of Nigmatullin-Le Mehaute.

It's important to keep in mind Metzler, Glockle, and Nonnenmacher, who clarify that the parameters of the fractional partial differential equation that came from the anomalous dif‐ fusion are uniquely determined by the fractal Hausdorff dimension of the underlying object and the anomalous diffusion experiment.

An important note is that the fractional derivatives are not the weak Fourier-Laplace deriva‐ tives nor are the fractional operators pseudo-differential operators.

Fractal analysis is no longer just creating nice images, nor is it a branch of mathematics with little interaction with the other areas.

In the history of fractal geometry, important contributions were made toward seeing nature as it is, rather than as an approximation to classical geometry.

With the mathematics deduced from classical geometry, we could only model approxima‐ tions of nature. Part of the history of fractal geometry was using them to create images of nature on a computer. For example, in 1978, Loren Carpenter achieved unprecedented im‐ ages of mountains by using fractals, for a Boeing Commercial Aircraft. Carpenter based his work on the book *Fractals form, change and dimension*, by Benoit Mandelbrot. And so began the era of fractal images. Consider, for example, the lava scene from the movie Star Trek III —a masterpiece of fractal use.

This story takes us to 1999—the use of fractal antennas for cell phones. N. Cohen construct‐ ed the first fractal antenna for personal use; then, R. Hohlfeld and N. Cohen published an article in which they proved mathematically that in order to receive many frequencies, it is necessary to have fractal antennas.

More recently, while analyzing heart rate charts at Harvard, H. Goldberger realized that fractal analysis of these graphs allowed him to distinguish between a heart in good health and one that is not in good health.

It has also helped us detect changes and abnormalities in blood flow, which allows us to determine whether an organ, such as a kidney, has or will have cancerous tumors. Early cancer detection by use of fractal analysis, basically.

Power laws in nature are deduced from fractal dimension.

Fractals also allow us to know about the health of a forest. When we see a tree, we realize that a similar pattern is repeated all throughout the forest itself.

Now, we enter the modern part of fractal geometry. In this book, we have applications that Mandelbrot would have surely loved to see. Applications for the petroleum industry, nu‐ merical analysis, fractal antennas for cell phones, spacecraft, radars, image processing, meas‐ ure, porosity, turbulence, scattering theory…

This book is divided into three sections, where the research chapters are presented in the following way.

The first part, called "Petroleum Industry, Numerical Analysis, and Fractal History," is about the use of fractal geometry and fractional calculus to model the difference in pressure with which the oil will come out. A history of fractional calculus and its relation accompany this section to the fractal dimension of the medium. That is because porosity can be meas‐ ured using the fractal dimension, as shown by K. Oleschko. We also have in this section the numerical analysis for fractional partial differential equations, a new area of research.

Our second part is called "Industry, Antennas, Spacecraft, Radar, Images, and Measure." It presents applications of fractal analysis that are quite important today, such as fractal anten‐ nas and metamaterials that are used in all cell phones. Such topics include miniaturization and its use for efficiency in industry, spacecraft, microaccelearation, radars, image process‐ ing, industrial applications, time series, and precision measuring.

The third part is "Applications, Scattering, Porosity, and Turbulence." These classical sub‐ jects where mathematical modeling has used quantum mechanics, partial equations, and ge‐ ometry nowadays have benefited from the use of fractal analysis. The applications of scattering theory of porosity and turbulence are of greater precision, for example.

It's impossible to picture today's research without fractal geometry.

**Prof. Fernando Brambila** Mathematics Department, School of Sciences National Autonomous University of Mexico, Mexico **Petroleum Industry, Numerical Analysis and Fractal History**

the era of fractal images. Consider, for example, the lava scene from the movie Star Trek III

This story takes us to 1999—the use of fractal antennas for cell phones. N. Cohen construct‐ ed the first fractal antenna for personal use; then, R. Hohlfeld and N. Cohen published an article in which they proved mathematically that in order to receive many frequencies, it is

More recently, while analyzing heart rate charts at Harvard, H. Goldberger realized that fractal analysis of these graphs allowed him to distinguish between a heart in good health

It has also helped us detect changes and abnormalities in blood flow, which allows us to determine whether an organ, such as a kidney, has or will have cancerous tumors. Early

Fractals also allow us to know about the health of a forest. When we see a tree, we realize

Now, we enter the modern part of fractal geometry. In this book, we have applications that Mandelbrot would have surely loved to see. Applications for the petroleum industry, nu‐ merical analysis, fractal antennas for cell phones, spacecraft, radars, image processing, meas‐

This book is divided into three sections, where the research chapters are presented in the

The first part, called "Petroleum Industry, Numerical Analysis, and Fractal History," is about the use of fractal geometry and fractional calculus to model the difference in pressure with which the oil will come out. A history of fractional calculus and its relation accompany this section to the fractal dimension of the medium. That is because porosity can be meas‐ ured using the fractal dimension, as shown by K. Oleschko. We also have in this section the numerical analysis for fractional partial differential equations, a new area of research.

Our second part is called "Industry, Antennas, Spacecraft, Radar, Images, and Measure." It presents applications of fractal analysis that are quite important today, such as fractal anten‐ nas and metamaterials that are used in all cell phones. Such topics include miniaturization and its use for efficiency in industry, spacecraft, microaccelearation, radars, image process‐

The third part is "Applications, Scattering, Porosity, and Turbulence." These classical sub‐ jects where mathematical modeling has used quantum mechanics, partial equations, and ge‐ ometry nowadays have benefited from the use of fractal analysis. The applications of

**Prof. Fernando Brambila**

Mathematics Department, School of Sciences

National Autonomous University of Mexico, Mexico

scattering theory of porosity and turbulence are of greater precision, for example.

—a masterpiece of fractal use.

VIII Preface

necessary to have fractal antennas.

and one that is not in good health.

cancer detection by use of fractal analysis, basically.

ure, porosity, turbulence, scattering theory…

following way.

Power laws in nature are deduced from fractal dimension.

that a similar pattern is repeated all throughout the forest itself.

ing, industrial applications, time series, and precision measuring.

It's impossible to picture today's research without fractal geometry.
