Fractional Calculus on Fractal Functions

Abstract

The words fractional calculus were born from a communication between L’Hospital and Leibniz in 1695. By denoting the nth derivative of f with respect to x as $\frac{d^nf}{dx^n}$, Leibniz had written a letter to L’Hospital. In his letter, Leibniz assumed that n takes the value from the positive integers, i.e., $n\in \mathbb {N}$. L’Hospital replied by raising the question of what meaning could be recognized to $\frac{d^nf}{dx^n}$ if n is 1/2, a fraction. This question became a historical statement to pronounce the new name in the theory of mathematic and was also universally accepted as the first existence of the so-called fractional derivative. The name fractional calculus has stayed being used ever since, despite the fact that it is notable at this point that there is no motivation to confine n to the rational numbers. Without a doubt, any real number will do similarly too, even complex numbers might be permitted, yet this is well beyond the aim of this book. Like fractals, the notion of fractional calculus has also been applied to numerous fields of science. Fractional calculus has never experienced the popularity currently enjoyed by fractals since it does not lend itself to produce complex graphical structures. In [13], significant effort has been devoted to relating fractional calculus to fractal geometry. In addition, the relationship between fractional calculus, such as Riemann–Liouville fractional calculus, and the von Koch curve has been revealed. Further, this study found that the value of the fractal dimension of von Koch curve is a linear function of its order of the Riemann–Liouville fractional calculus.