Preface

In ancient times, instead of fractions, the natural numbers were used. However, integers cannot always result when measuring or equally dividing things. As time went by, fractions and then non-integers were gradually understood and applied. With the introduction of fractions and more generally nonintegers, people were able to have a closer look at the beauty of nature around them. For example, people long ago realized that a rectangle of the ‘golden ratio’ 1.618 : 1 is most pleasing. The natural exponential e = 2.71828 . . ., and the ratio of any circle’s circumference to its diameter, π = 3.14159 . . . are widely used in mathematics and engineering. The ‘beauty’ of the fraction was recognized and people came to use the ‘fractional view’ to observe the world, to use ‘fractional thinking’ to understand the natural phenomena, and to use ‘fractional techniques’ to solve the problems at hand. The term ‘fractal’ was introduced by Mandelbrot in 1975 [192]. Fractal refers to the self-similar geometric shape, that is, a shape in which is almost identical to the entire shape except in size [91, 102]. Many objects manifest themselves in fractal shape, such as clouds, coastlines and snow flakes. In fractal theory, the fractal dimension was used to characterize the state of nature. Different from the conventional integer dimension, the fractal dimension can be fractional or any non-integer number. Based on the fractal theory, the traditional concept of three-dimensional space can be extended to the fractal (fractional) dimension (FD) which can be applied to characterize complex objects. Likewise, (integer-order) calculus can be extended to fractional or noninteger order calculus. It should be remarked at this point that due to historical reasons, the term ‘fractional’ we use here and throughout this monograph should actually be understood as ‘non-integer’ or ‘arbitrary real number’ to be precise. Fractional calculus, i.e., fractional-order differentiation and integration, is a part of mathematics dealing with derivatives of arbitrary order [139, 203, 209, 218, 237]. Leibniz raised the question about the possibility of generalizing the operation of differentiation to non-integer-orders in 1695 [237]. Fractional calculus, developed from the field of pure mathematics, has been studied increasingly in various fields [64, 142, 311, 315, 323]. Nowadays, fractional calculus is being applied to many fields of science, engineering, and mathematics [49, 74, 78, 135, 290]. Fractional calculus provides a