## Introduction

Extend the derivative from natural numbers to all number domains (that is, not only natural numbers but also negative numbers, decimals, irrational numbers, imaginary or complex numbers, functions, ...), you can get any order calculus, unifying the world and unifying differentiation and integration into one formula `d^(o(x))/dx^(o(x)) y`, where o(x) is the order function. When o(x) is a positive integer, it is a derivative differential. When o(x) is a negative integer, it is an integral. When O(x) is -2, it is a double integral. When o(x) is a fraction, it is a fractional calculus. When When o(x) is a variable, it is a variable order derivative. These can be calculated with a mathHand.com calculator .## Definition

## Property of Fractional Calculus

## Minus order derivative

As we known, `d^(-1)/dx^(-1) sin(x) = int` sin(x) dx## Fractional order derivative

Input sin(x), click the d^0.5/dx^0.5 button for semiderivative, click the d^0.5/dx^0.5 button again for another semiderivative. Two times of semiderivative equal to first derivative, e.g.

`d^0.5/dx^0.5 d^0.5/dx^0.5 sin(x) = d^(0.5+0.5)/dx^(0.5+0.5) sin(x) = d/dx sin(x) = cos(x)`## Complex order derivative

The order can be extented to complex number a+b i, we differentate by the real order first, then by the imag order.

`d^(a+bi)/d^(a+bi) sin(x) = d^(bi)/dx^(bi) d^a/dx^a sin(x)`.

e.g. the (1+i) order derivative and the (1-i) order derivative equal to second derivative,

`d^(1+i)/dx^(1+i) d^(1-i)/dx^(1-i) sin(x) = d^(1+i+1-i)/dx^(1+i+1-i) sin(x)= d^2/dx^2 sin(x) = -sin(x)`## Variable order derivative

The order can be changed as a function, e.g. the cos(x) order changes between -1 to 1.

`d^cos(x)/dx^cos(x)` sin(x) = d(sin(x),x,cos(x))

derivative of x with the cos(x) order changed between -1 to 1.## Difference between Caputo definition and the Riemann-Liouville (R-L) definition

The Caputo (Davison-Essex) and the Riemann-Liouville definitions are different in the following aspect: in the Caputo formula, differentiation is performed first, then integration; but in the R-L formula it is the other way around. The Caputo definition has advantages over the R-L definition:

1. The Caputo definition implemented maps constants to zero, imitating integer order differentiation, while the R-L definition does not.

2. The Caputo definition requires initial value in fractional differential equation is the same as in differential equations, while the R-L definition does not.

These properties of the Caputo definition make it suitable to work with initial value problems for fractional differential equations. So the Caputo definition is used here. If you want to use the Riemann defintion, use the Laplace transform solver lasolve( ).lasolve(y(-0.5,x)=1) give nonzero.

dsolve(y(-0.5,x)=1) give zero.

## Super calculus

## Fractional differential equation

## Fractal Space

Our 3D space can be extented to fractional dimentional space, which is according to fractional differential. It is fractal dimension [5]. The three-dimensional space we live in can be extended to fractal space. Fractal space corresponds to fractional derivatives. The geometric meaning of fractional derivatives is fractal theory.## Example

### Input Fractional Calculus into Calculator

Input sin(0.5,x) for the 0.5th order derivative of sin(x), which is denoted by `sin^((0.5))(x)`. It is differrent from power of sin(x) as `sin(x)^2`. sin(0.5,x) can be computed and plot.

d(sin(x),x,0.5) computes the 0.5th order derivative of sin(x). If you want to hold it (not evaluate), use the derivative holder ds(y,x,0.5).#### semi-differentiate graphically

some functions cannot be differentiated symbolically, but can be semi-differentiated graphically in plot2D. e.g.

d(x=>sin(x),x,0.5) half order derivative## Application

In recent years, fractional calculus has been widely used in abnormal diffusion, signal processing and control, fluid mechanics, image processing, soft matter research, seismic analysis, viscoelastic dampers, power fractal networks, fractional sine oscillators, fractal theory, Fractional PID controller design, electrochemistry. Application of semi-differential in electrochemistry [1-3], application of fractal theory in electrochemistry [4-5].## References

- J. Mo, P. Cai, W. Huang and F. Yun, Study on the multiple semi-differential electroanalysis of electrochemical stripping method with thin mercury film formed in situ, J. Zhongshan Uni. (Zhongshan Daxue Xuebao), 1984, (4), 76-84, CA 103: 115269.
- J. Mo, P. Cai, W. Huang and F. Yun, Theory and application on multiple semidifferential electrochemical stripping analysis with thin mercury film formed in situ, Acta Chimica Sinica, 1984, 42(6), 556-561, CA 101: 162712.
- J. Mo, W. Huang and R.J. Zhang, New Advances in convolution voltammetry (Review), J. Anal. Determ. (Fenxiceshi Tongbao), 1985, 4(3), 1-8, CA 105: 163910.
- W. Huang and B. Hibbert, Computer modelling of electrochemical growth with convection and migration in rectangular cell, Phys. Rev. E, 1996, 53(1), 727-730.
- J. Jiang, W. Huang and B. Hibbert, Determining fractal dimensions of DLA structures using cumulative randic indices, Physica A, 1996, 233(3-4), 884-887.
- W. Huang and B. Hibbert, Fast fractal growth with diffusion, convection, and migration by computer simulation: Effects of voltage on probability, morphology and fractal dimension of electrochemical growth in a rectangular cell, Physica A, 1996, 233(3-4), 888-896.
- Roubíček, T. (2013), Nonlinear Partial Differential Equations with Applications (2nd ed.), Basel, Boston, Berlin: Birkhäuser, doi:10.1007/978-3-0348-0513-1, ISBN 978-3-0348-0512-4, MR 3014456.
- Samko, S. G. (1987), Fractional Integrals And Derivatives - Theory and Applications, 1987.
- Loverro, Adam (2004), Fractional Calculus - History, Definitions, and Applications for the Engineer, 2004.
- Baleanu, D. and Kumar, D. (2019), fractional calculus and its applications in physics. 2019.

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