Introduction to Fractional Calculus

At the beginning of the development of the theory of classical calculus (called integer-order calculus in this book), the British scientist Isaac Newton and the German mathematician Gottfried Wilhelm Leibniz used different symbols for different orders of derivatives. For example, Newton used the notation $$\dot{y}(x)$$ y ˙ ( x ) , $$\ddot{y}(x)$$ y ¨ ( x ) and $$\dddot{y}(x)$$ y ⃛ ( x ) , while Leibniz used the notation $$\textrm{d}^n y(x)/\textrm{d}x^n$$ d n y ( x ) / d x n , where n is a positive integer. A natural question is how to extend n into fractions or even complex numbers. In a letter written by the French mathematician Marquis de l’Hôpital to Leibniz in 1695, he asked question “what would be the meaning if $$n = 1/2$$ n = 1 / 2 in the $$\textrm{d}^n y(x)/\textrm{d}x^n$$ d n y ( x ) / d x n notation”. In a letter dated 30 September 1695, Leibniz replied, “Thus it follows that $$\textrm{d}^{1/2} x$$ d 1 / 2 x will be equal to $$x\sqrt{\textrm{d}x:x}$$ x d x : x . This is an apparent paradox from which, one day, useful consequences will be drawn” [1]. In this chapter, a brief historic view of fractional calculus is presented. The tools for fractional calculus are summarized.