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任意阶微积分 Any Order Calculus

将导数从自然数推广到一切数域(即不仅是自然数也包含负数、小数、无理数、虚数或复数、函数, ……)， 就得到任意阶微积分, 一统天下把微分与积分统一到一个公式`d^(o(x))/dx^(o(x))`y, 其中o(x)是阶函数（the order function )。当o(x)是正整数时它是导数微分，当o(x)是负整数时它是积分，负2阶是二重积分，当o(x)是分数时它是分数阶微积分， 当o(x)是变量时它是可变阶导数。这些都可用数学手册计算器计算.

分数阶微积分 Fractional Calculus

分数阶导数简介

二次半微分 = 1阶微分,
`d^0.5/dx^0.5 d^0.5/dx^0.5 x = d/dx x = 1`
对于任意的o(x),由于伽玛函数的参数在实数部为负整数时没有定义，需要在分数微分前先进行整数微分。例如
`D^(3/2)`f(x)=`D^(1/2)D^1`f(x)=`d^(1/2)/dx^(1/2) d/dx` f(x)
这苻合Caputo的定义，Caputo的定义是先微分后积分，对于分数阶微分方程的初值要求与普通微分方程相同，因此本文釆用广义Caputo的定义: 积分下限a为f(x)的反函数`f^(-1)(0)` .


负阶导数

负阶导数相当积分, 负一阶等于积分, 负2阶是二重积分，负1/2阶是半积分.
`d^(-0.5)/dx^(-0.5)` f(x) = `int f(x) (dx)^0.5`

复数阶微积分

以上微分算子的扩展不仅仅局限于实数阶。举个例子， (1-i)阶导数作用后， (1+i)阶导数再作用，可以得到二阶导数。

(1+i)阶微分和(1-i)阶微分 = 2阶微分,
`d^(1+i)/dx^(1+i) d^(1-i)/dx^(1-i) sin(x) = d^2/dx^2 sin(x) = -sin(x)`

可变阶微积分 Variable Order calculus

当阶数o(x)是变量时, 它是可变阶导数微分。
`d^cos(x)/dx^cos(x)` sin(x) = d(sin(x),x,cos(x))

这个动画展示了不同分数微分算子D如何操作在 y=x（o(x)=0阶, 蓝色），结果（分数阶, 绿色）在一般的积分（o(x)= -1阶, y=x^2/2 ，紫色）及一般的一次微分（ o(x)=1阶, y=1 ，红色）间连续变化。

分数阶微分方程 Fractional differential equation

`d^(pi)/dx^(pi)` y - y - 2exp(x) = 0

分数维空间

我们生活在的三维立体空间可以扩展到分数维空间，分数维空间对应分数阶导数, 分数阶导数的几何意义是分数维理论, 分数维理论就是分形理论.

计算

数学手册计算器是三合一 = 数学手册 + 计算器 + 计算机代数系统. 具有机器学习的功能，举一反三，解任意阶(0.5i 阶)微分方程的功能,世上独一无二。在数学手册网上输入数学公式，连续点击计算微积分，解方程，给出数值解，分析解和图解, 制图互动放大。随时随地学习，随时随地计算，随时随地开发。学习数学的好助手。 mathHand.com http://chinese.mathhandbook.com

使用经验

  1. 怎么求函数的分数阶导数？兼谈数学手册计算器_百度经验
  2. 分数阶导数 - qq_34040902的博客 - CSDN博客
  3. 特殊的微分方程的解法_百度经验

例题

应用

近年来分数阶微积分被广泛的应用于反常扩散、信号处理与控制、流体力学、图像处理、软物质研究、地震分析、粘弹性阻尼器、电力分形网络、分数阶正弦振荡器、分形理论、分数阶PID控制器设计, 电化学. 半微分在电化学的应用 [1-3], 分形理论应用于电化学 [4-6].

参考文献

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.

书单

  • 分数阶导数简介
  • 分数阶微分方程1
  • 分数阶微分方程-3

    相关条目

  • 任意阶微积分
  • 任意阶微分方程的解析解
  • 一些重要的偏微分方程(组)
  • 分数阶微分方程解析解法

    

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    参阅
    1. 数学 - 数学符号 - 数学索引
    2. 手册 = 初中数学手册 + 高中数学手册 + 数学手册 + 实用数学手册
    3. 初等数学 = 小学数学 + 中学数学 ( 初中数学 + 高中数学 )
    4. 高等数学 = 基础数学 ( 代数 + 几何 + 分析 ) + 应用数学
    5. 公式 - 定理 - 图 - 函数图 - 曲线图 - 平面图 - 立体图 - 动画 - 画画
    6. 书单 = 数学 + 物理 + 化学 + 计算 + 医学 + 英语 + 教材 - QQ群下载书
    7. 数学手册计算器 = 数学 + 手册 + 计算器 + 计算机代数系统
    8. 检测 - 例题 :

    `(d^0.5y)/dx^0.5 = sin(x-1)*sin(y-1) ` == ? `(d^0.5y)/dx^0.5 -cosh(y)-sinh(y)=0 ` == ? `(d^1.6y)/(dx^1.6)-int y(x) (dx)^(0.8)-y-exp(x)=0` == ? `int y(x) (dx)^0.5 -y-exp(x)`=0 == ? `(d^0.5y)/dx^0.5-exp(y)*x=0` == ? `(d^0.5y)/dx^0.5-exp(y)*y=0` == ? `(d^0.5y)/dx^0.5=cos(x)/x*y` == ? `y*(dy^0.5)/dx^0.5-sqrt(x)-1=0` == ? `(d^1.2y)/(dx^1.2)-2(d^0.6y)/dx^0.6+y-exp(x)=0` == ? `(d^0.5y)/dx^0.5=cos(y)*exp(x)*x` == ? `(d^1.6y)/(dx^1.6)-2(d^0.8y)/dx^0.8+y-exp(x)=0` == ? `(d^0.5y)/dx^0.5-exp(y)*sqrt(x)=0` == ? `(d^1.6y)/(dx^1.6)-3 (d^0.8y)/dx^0.8+2y-exp(x)=0` == ? `(d^0.5y)/dx^0.5` +log(y-1)-exp(x)-x=0 == ? `(d^0.5y)/dx^0.5-exp(y)*sin(x)=0` == ? `(d^0.5y)/dx^0.5 = y*sin(x)/x ` == ? `y^((0.5))(x) -4 exp(x)*y-exp(x)=0` == ? `(dy^0.5)/dx^0.5 = 1/(x-y)` == ? `dy/dx-(d^0.5y)/dx^0.5` - y - exp(x)=0 == ? `(dy)/dx -exp(y-1)-x-x^2=0` == ? `(d^1.2y)/(dx^1.2)-3dy^0.6/dx^0.6+2y-exp(x)=0` == ? `dy/dx-(d^0.5y)/dx^0.5-y-1`=0 == ? `(d^0.5y)/dx^0.5-cos(y)*sin(x)=0` == ? `(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(4x)=0` == ? `dy/dx-exp(y-1)-exp(x)=0` == ? `(dy)/dx - 2(d^0.5y)/dx^0.5-y-exp(x)=0` == ? `(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(x)=0` == ? `(d^0.5y)/dx^0.5 -e^(4x)-y`=0 == ? `y^((0.5))(x) - exp(x)*y-exp(x)=0` == ? `y^((0.5))(x) - exp(x)*y-4exp(x)=0` == ? `(dy)/dx -3(d^0.5y)/dx^0.5 +2y-exp(x)=0` == ? `y*(d^0.5y)/dx^0.5-sqrt(x)-1=0` == ? `y^((1))(x)-exp(y-1)-x=0` == ? `(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-2y-exp(x)=0` == ? `(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(4x)=0` == ? `(d^0.5y)/dx^0.5 - log(y-1) - exp(x) + x=0` == ? `(dy)/dx +asin(y-1) - cos(x)-x=0` == ? `(d^1.6y)/(dx^1.6)-3(d^0.8y)/dx^0.8+2y-exp(x)=0` == ? `(dy)/(dx) -sqrt(y-1)-x-1 =0` == ? ` (dy)/(dx) -exp(y-1)-exp(x) = 0` == ? `(dy)/dx` +asinh(y-1)-cosh(x)-x =0 == ? `((d^(1/2)y)/dx^(1/2))^2 -3y* (dy^0.5)/dx^0.5 + 2y^2 = 0` == ? `(dy^0.5)/dx^0.5 = cos(x)*cos(y-1)` == ? `(d^0.5y)/dx^0.5 +log(y-1)-exp(x)-x=0` == ? `(dy^0.5)/dx^0.5 = sin(x-1)*exp(y-1)` == ? `y*(d^2y)/dx^2-(dy/dx)^2+1=0` == ? `y^((1))(x)-exp(y-1)-log(x)=0` == ? `(d^2y)/dx^2 *exp(x)- exp(y-1)=0` == ? `(d^1.6y)/(dx^1.6)-2 (d^0.8y)/dx^0.8-y-exp(x)=0` == ? `(d^1.6y)/(dx^1.6)-2 (d^0.8y)/dx^0.8+y-exp(x)=0` == ? `(dy)/dx -3 (d^0.5y)/dx^0.5+2y-exp(x)=0` == ? `y^((0.5))(x) - x*y-x=0` == ? `y*(dy^3)/dx^3-x^3-3x^2-3x-1=0` == ? `y^((1.8))(x)-2y^((0.9))(x) +y-1=0` == ? `y^((0.5))(x)=1/(x*y-1)` == ? `y^((2))(x)*y^2-x^2-2x-1=0` == ? `((d^0.5y)/dx^0.5)^2 -5(d^0.5y)/dx^0.5 +6=0` == ? `y^((0.5))(x) -2 exp(x)*y-4exp(x)=0` == ? `(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(x)=0` == ? `y^(0.5)(x)=2y*exp(x)` == ? `y^((0.5))(x)-exp(x)*y^2=0` == ? `(d^1.6y)/(dx^1.6)-2(d^0.8y)/dx^0.8+y-exp(x)=0` == ? `y^((1))(x)-y^2-x*y=0` == ? `y^((1))(x)-y^((0.5))(x) -y-1=0` == ? `y^((2))(x) -y^2-x^2=0` == ? `y^((2))(x) -y^2-x^2-2x*y=0` == ? `y^((0.5))(x) -int y(x) (dx)^0.5-y-exp(x)=0` == ? `d^0.5/dx^0.5 y -2cos(y)*exp(x)=0` == ? `d^0.5/dx^0.5 y -4sin(y)*exp(x)=0` == ? `(d^0.5y)/dx^0.5=sin(x^2)*y` == ? `(d^0.5y)/dx^0.5-sin(x)*sin(y)=0` == ? `(d^0.5y)/dx^0.5-sinh(x)*sinh(y)=0` == ? `y^((1))(x)=exp(x-y)-x` == ? `x*(d^0.5y)/dx^0.5-y-2x=0` == ? `(d^0.5y)/dx^0.5=sinh(x-1)*sinh(y-1)` == ? `y^((0.5))(x)-exp(-x)*y^2=0` == ? `(d^0.5y)/dx^0.5=y/x*sin(x)` == ? `(dy)/dx-sin(x-y)-1=0` == ? `(d^2.5y)/dx^2.5=y*(d^0.5y)/dx^0.5` == ? `(d^0.5y)/dx^0.5=y*(dy)/dx` == ? `(d^(2-i)y)/dx^(2-i)- y+x=0` == ? `(d^2y)/dx^2=y^3*x^2` == ? `y*(d^2y)/dx^2-x^2-3x-1=0` == ? `y*(d^2y)/dx^2-2x^2-3x-1=0` == ? `(y-x-1)*(d^2y)/dx^2-3x-1=0` == ? `y^2*(d^2y)/dx^2-x^2-4x-4=0` == ? `(y-x-1)*(d^2y)/dx^2-x^2-4x-4=0` == ? `y*(d^2y)/dx^2-2x^2-2x-1=0` == ? `y*(d^3y)/dx^3-6x^3-3x^2-3x-1=0` == ? `y^((0))(x)*y^((1))(x)*y^((2))(x)=x^2` == ? `y^((3))(x)*y^((2))(x)=y^((1/2))(x)` == ? `y^((3))(x)=exp(x)*y^((1))(x)*y^((1/2))(x)` == ? `y^((1/2))(x)*y^((3))(x)=exp(x)` == ? `y^((1/2))(x)*y^((2))(x)=exp(x)` == ? `(d^0.5y)/dx^0.5-2x*y-1=0` == ? `y^2*(d^0.5y)/dx^0.5-x^2-4x-4=0` == ? `exp(y-1)*(d^0.5y)/dx^0.5-x=0` == ? `y*(d^2y)/dx^2-(x-2)*(2x-4)=0` == ? `y*(d^3y)/dx^3-6x^3-4x^2-4x-1=0` == ? `exp(y-1)*(d^2y)/dx^2-exp(x)=0` == ? `y^2*(d^2y)/dx^2-x^2-1=0` == ? `1/y^2*(d^2y)/dx^2-x^2-1=0` == ? `(y-x-1)*(d^3y)/dx^3-(x-2)*(2x-4)*(3x-1)=0` == ? `(d^0.5y)/dx^0.5-2x^2*y^2-8x^2=0` == ? `(d^0.5y)/dx^0.5-2x*y^2-8x=0` == ? `(d^0.5y)/dx^0.5-y^2-2y-2=0` == ? `(d^0.5y)/dx^0.5-log(y-1)*exp(x)=0` == ? `y*(d^2y)/dx^2-(dy/dx)^2-1=0` == ? `(d^2y)/dx^2-asin(y-1)-sin(x)-x=0` == ? `dy/dx*(x--y)-x--y-1 = 0` == ?

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