Conformable Derivative and Fractal Derivative of Functions on the Interval [0,1]
Keywords:Conformable derivative, Fractal calculus, Fractal dimension, Fα-derivative
AbstractRecently, a calculus-based fractal, called Fα-calculus, has been developed which involve Fα-integral, conjugate to the Riemann integral, and Fα-derivative, conjugate to ordinary derivative, of orders α, 0< α <1, where α is the dimension of F. In F α-calculus the staircase function has a special role. In this paper we obtain fractal Taylor series for fractal elementary functions, sine, cosine, exponential function, etc. and then we compare the graph of these fractal functions with their counterparts in standard calculus on the interval [0,1]. Then, the main part of the paper is discussed about the transition from continuous state to a discrete state when we do fractal differentiation in which characteristic function χC(x) appears. Since Fα-derivative is local, we compare it with a conformable derivative which is also local. Moreover, fractal differential equations for fractal sine, cosine, sine hyperbolic, cosine hyperbolic and exponential functions are deduced. We also represent Pythagorean trigonometric identity for sin and cosine, and hyperbolic sine and hyperbolic cosine in F α-calculus, respectively
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