Physics of Fractal Operators

This text describes how fractal phenomena, both deterministic and random, change over time, using the fractional calculus. The intent is to identify those characteristics of complex physical phenomena that require fractional derivatives or fractional integrals to describe how the process changes over time. The discussion emphasizes the properties of physical phenomena whose evolution is best described using the fractional calculus, such as systems with long-range spatial interactions or long-time memory. In many cases, classic analytic function theory cannot serve for modeling complex phenomena; "Physics of Fractal Operators" shows how classes of less familiar functions, such as fractals, can serve as useful models in such cases. Because fractal functions, such as the Weierstrass function (long known not to have a derivative), do in fact have fractional derivatives, they can be cast as solutions to fractional differential equations. The traditional techniques for solving differential equations, including Fourier and Laplace transforms as well as Green's functions, can be generalized to fractional derivatives. Physics of Fractal Operators addresses a general strategy for understanding wave propagation through random media, the nonlinear response of complex materials, and the fluctuations of various forms of transport in heterogeneous materials. This strategy builds on traditional approaches and explains why the historical techniques fail as phenomena become more and more complicated.

Review :
From the reviews: "Have you ever wondered about whether one can define differential derivative of non integer order and how useful these fractal derivatives would be? If the answer is yes this is the book to look at. The book is written by physicists with a pragmatic audience in mind. It contains a very thorough and clearly written discussion of the mathematical foundation as well as the applications to important and interesting mathematical and physical problems. All the topics are very main stream and of great general relevance... "I am glad I got to know this book. I don't know yet whether fractal calculus will be of crucial importance to my own research in statistical mechanics and complex systems. But I got the feeling from this book that this might very well be the case. And if this happens, I now know exactly where to go for a highly readable and thorough introduction to the field. I think the book deserves to be present in mathematics and physics libraries. And I believe many interesting undergraduate and graduate projects in mathematics and its applications can start out from this book." - UK Nonlinear News "The book is written by physicists with a pragmatic audience in mind. It contains a very thorough and clearly written discussion of the mathematical foundation as well as the applications to important and interesting mathematical and physical problems. All the topics are very mainstream and of great general relevance. Obviously, the book is also of great relevance to the researcher who may need to become acquainted with Fractal Calculus . I am glad I got to know this book." (Henrik Jensen, UK Nonlinear News, February, 2004) "Physics of Fractal Operators is a timely introduction that discusses the basics of fractional calculus. ... Physics of Fractal Operators, which actively promotes the use of fractional calculus in physics, may help teachers develop an appropriate curriculum. the book s abundance of material makes it very useful to researchers working