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Examples of Fractional Calculus Computer Algebra System 例题

Content

• Arithmetic 算术
• Algebra 代数
• Function 函数
• Calculus 微积分
• Equation 方程
• Discrete Math 离散数学
• Definition 定义式
• Numeric math 数值数学
• Number Theory 数论
• Probability 概率
• Statistics 统计
• Multi elements
1. list()
2. vector
3. and
• Animation 动画
• Plot 制图
1. Interactive plot 互动制图
2. parametric plot, polar plot
3. solve equation graphically
4. area plot with integral
5. complex plot
6. Geometry 几何
• plane graph 平面图
1. plane graph 平面图 with plot2D
2. function plot with funplot
3. differentiate graphically with diff2D
4. integrate graphically with integrate2D
5. solve ODE graphically with odeplot
• 3D graph 立体图
1. surface in 3D with plot3D
2. contour in 3D with contour3D
3. wireframe in 3D with wirefram3D
4. complex function in 3D with complex3D
5. a line in 3D with parametric3D
6. a column in 3D with parametric3D
7. the 4-dimensional object (x,y,z,t) in 3D with implicit3D
• programming 编程
• bugs

Arithmetic 算术 >>

Exact computation

1. Fraction 1E2-1/2
2. Big number: add prefix "big" to number
big1234567890123456789

3. mod operation
input mod(3,2) for 3 mod 2

Complex 复数

Complex( 1,2) number is special vector, i.e. the 2-dimentional vector, so it can be operated and plotted as vector.
4. complex numbers in the complex plane
complex(1,2) = 1+2i
5. input complex number in polar(r,theta*degree) coordinates
polar(1,45degree)

6. input complex number in polar(r,theta) coordinates for degree by polard(r,degree)
polard(1,45)

7. input complex number in r*cis(theta*degree) format
2cis(45degree)
8. Convert to complex( )

9. in order to auto plot complex number as vector, input complex(1,-2) for 1-2i, or convert 1-2i to complex(1,-2) by
convert(1-2i to complex) = tocomplex(1-2i)
10. input complex number in polar
tocomplex(polar(1,45degree))

11. Convert complex a+b*i to polar(r,theta) coordinates
convert 1-i to polar = topolar(1-i)

12. Convert complex a+b*i to polar(r,theta*degree) coordinates
topolard(1-i)

13. complex 2D plot
complex2D(x^x)
14. complex 3D plot
complex3D(pow(x,x))

Numerical approximations

15. Convert back by numeric computation n( )
n(polar(2,45degree))
n( sin(pi/4) )
n( sin(30 degree) )
16. sin^((0.5))(1) is the 0.5 order derivative of sin(x) at x=1
n( sin(0.5,1) )
17. sin(1)^(0.5) is the 0.5 power of sin(x) at x=1
n( sin(1)^0.5 )
18. Algebra 代数 >>

19. simplify
taylor( (x^2 - 1)/(x-1) )
20. expand
expand( (x-1)^3 )

21. factorization
factor( x^4-1 )
22. factorizing
factor( x^2+3*x+2 )
23. tangent

24. tangent equation at x=1
tangent( sin(x),x=1 )
25. tangentplot( ) show dynamic tangent line when your mouse over the curve.
tangentplot( sin(x) )

Convert

convert( sin(x) to exp) is the same as toexp(sin(x))
26. convert to exp
toexp( cos(x) )

27. convert to trig
convert exp(x) to trig

28. convert sin(x) to exp(x),
convert sin(x) to exp = toexp( sin(x) )

29. Convert to exp(x)
toexp(Gamma(2,x))
30. inverse function

31. input sin(x), click the inverse button
inverse( sin(x) )
check its result by clicking the inverse button again.
In order to show multi-value, use the inverse equation instead function.

inverse equation

32. inverse equation to show multivalue if it has.
inverse( sin(x)=y )
check its result by clicking the inverse button again.

polynomial

33. the unit polynomial
poly(3,x) gives the unit polynomial x^3+x^2+x+1.

34. Hermite polynomial
hermite(3,x) gives the Hermite polynomial while hermite(3) gives Hermite number.

35. harmonic polynomial
harmonic(-3,1,x) = harmonic(-3,x)
harmonic(-3,x)

36. the zeta polynomial
zeta(-3,x) is the zeta polynomial.

37. expand polynomial
expand(hermite(3,x))

38. topoly( ) convert polynomial to polys( ) as holder of polynomial coefficients,
convert x^2-5*x+6 to poly = topoly( x^2-5*x+6 )
39. activate polys( ) to polynomial
simplify( polys(1,-5,6,x) )
40. polyroots( ) is holder of polynomial roots, topolyroot( ) convert a polynomial to polyroots.
convert (x^2-1) to polyroot = topolyroot(x^2-1)
41. polysolve( ) numerically solve polynomial for multi-roots.
polysolve(x^2-1)
42. nsolve( ) numerically solve for a single root.
nsolve(x^2-1)
43. solve( ) for sybmbloic and numeric roots.
solve(x^2-1)
44. construct polynomial from roots. activate polyroots( ) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify( ) button.
simplify( polyroots(2,3) )

Number

When the variable x of polynomial is numnber, it becomes polynomial number, please see Number_Theory section.
45. Function 函数 >>

Function 函数
46. Trigonometry 三角函数
expand Trigonometry by expandtrig( )

47. inverse function
inverse( sin(x) )

48. plot a multivalue function by the inverse equation
inverse( sin(x)=y )

49. expand
expand( sin(x)^2 )
50. factor
factor( sin(x)*cos(x) )

51. Complex Function 复变函数

complex2D( ) shows the real and imag curves in real domain x, and complex3D( ) shows complex function in complex domain z, for 20 graphes in one plot.

Calculus 微积分 >>

Limit

52. click the lim( ) button to Limit at x->0
lim_(x->0) sin(x)/x  = lim sin(x)/x as x->0 = lim(sin(x)/x)
53. click the nlim( ) button to numeric limit at x->0
54. click the limoo( ) button to Limit at x->oo
lim _(x->oo) log(x)/x = lim( log(x)/x as x->inf )

55. one side limit, left or right side:
lim(exp(-x),x,0,right)

Derivatives

56. Differentiate
d/dx sin(x) = d(sin(x))

57. Second order derivative
d^2/dx^2 sin(x) = d(sin(x),x,2) = d(sin(x) as x order 2)

58. sin(0.5,x) is inert holder of the 0.5 order derivative sin^((0.5))(x), it can be activated by simplify( ):
simplify( sin(0.5,x) )
59. Derivative as x=1
d/dx | _(x->1) x^6 = d( x^6 as x->1 )

60. Second order derivative as x=1
d^2/dx^2| _(x->1) x^6 = d(x^6 as x->1 order 2) = d(x^6, x->1, 2)

Fractional calculus

61. Fractional calculus
62. semiderivative
d^(0.5)/dx^(0.5) sin(x) = d(sin(x),x,0.5) = d( sin(x) as x order 0.5) = semid(sin(x))

63. input sin(0.5,x) as the 0.5 order derivative of sin(x) for
sin^((0.5))(x) = sin^((0.5))(x) = sin(0.5,x)
64. simplify sin(0.5,x) as the 0.5 order derivative of sin(x),
sin^((0.5))(x) = simplify(sin(0.5,x))
65. 0.5 order derivative again
d^(0.5)/dx^(0.5) d^(0.5)/dx^(0.5) sin(x) = d(d(sin(x),x,0.5),x,0.5)
66. Minus order derivative
d^(-0.5)/dx^(-0.5) sin(x) = d(sin(x),x,-0.5)

67. inverse the 0.5 order derivative of sin(x) function
f(-1)( sin(0.5)(x) ) = inverse(sin(0.5,x))

68. Derive the product rule
d/dx (f(x)*g(x)*h(x)) = d(f(x)*g(x)*h(x))

69. … as well as the quotient rule
d/dx f(x)/g(x) = d(f(x)/g(x))

70. for derivatives
d/dx ((sin(x)* x^2)/(1 + tan(cot(x)))) = d((sin(x)* x^2)/(1 + tan(cot(x))))

71. Multiple ways to derive functions
d/dy cot(x*y) = d(cot(x*y) ,y)

72. Implicit derivatives, too
d/dx (y(x)^2 - 5*sin(x)) = d(y(x)^2 - 5*sin(x))

73. the nth derivative formula
 d^n/dx^n (sin(x)*exp(x))  = nthd(sin(x)*exp(x))
74. differentiate graphically

some functions cannot be differentiated or integrated symbolically, but can be semi-differentiated and integrated graphically in diff2D. e.g.

Integrals

75. indefinite integrate int sin(x) dx = integrate(sin(x))

76. enter a function sin(x), then click the ∫ button to integrate
int(cos(x)*e^x+sin(x)*e^x)\ dx = int(cos(x)*e^x+sin(x)*e^x)
int tan(x)\ dx = integrate tan(x) = int(tan(x))

integrator

If integrate( ) cannot do, please try integrator(x)
77. integrator(sin(x))
78. enter sin(x), then click the ∫ dx button to integrator

79. Multiple integrate
int int (x + y)\ dx dy = int( int(x+y, x),y)
int int exp(-x)\ dx dx = integrate(exp(-x) as x order 2)

80. Definite integration
int _1^3 (2*x + 1) dx = int(2x+1,x,1,3) = int(2x+1 as x from 1 to 3)

81. Improper integral
int _0^(pi/2) tan(x) dx =int(tan(x),x,0,pi/2)

82. Infinite integral
int _0^oo 1/(x^2 + 1) dx = int(1/x^2+1),x,0,oo)

83. Definite integration
int_0^1 sin(x) dx = integrate( sin(x),x,0,1 ) = integrate sin(x) as x from 0 to 1

fractional integrate

84. semi integrate, semiint( )
int sin(x) \ dx^(1/2) = int(sin(x),x,1/2) = int sin(x) as x order 1/2 = semiint(sin(x)) = d(sin(x),x,-1/2)

85. indefinite semiintegrate
int sin(x)\ dx^0.5 = d^(-0.5)/dx^(-0.5) sin(x) = int(sin(x),x,0.5) = semiint(sin(x))

86. Definite fractional integration
int_0^1 sin(x) (dx)^0.5 = integrate( sin(x),x,0.5,0,1 ) = semiintegrate sin(x) as x from 0 to 1

int (2x+3)^7 dx = int (2x+3)^7

88. numeric computation by click on the "~=" button
n( int _0^1 sin(x) dx ) = nint(sin(x),x,0,1) = nint(sin(x))

integrate graphically

some functions cannot be differentiated or integrated symbolically, but can be semi-differentiated and integrated graphically in integrate2D. e.g.
89. Equation 方程 >>

inverse an equation

90. inverse an equation to show multivalue curve.
inverse( sin(x)=y )
check its result by clicking the inverse button again.

polynomial equation

91. polyroots( ) is holder of polynomial roots, topolyroot( ) convert a polynomial to polyroots.
convert (x^2-1) to polyroot = topolyroot(x^2-1)

92. polysolve( ) numerically solve polynomial for multi-roots.
polysolve(x^2-1)

93. construct polynomial from roots. activate polyroots( ) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify( ) button.
simplify( polyroots(2,3) )

94. solve( ) for sybmbloic and numeric roots.
solve(x^2-1)
solve( x^2-5*x-6 )

95. solve equation and inequalities, by default, equation = 0 for default unknown x if the unknown omit.
solve( x^2+3*x+2 )

96. Symbolic roots
solve( x^2 + 4*x + a )

97. Complex roots
solve( x^2 + 4*x + 181 )

98. solve equation for x.
solve( x^2-5*x-6=0,x )

99. numerically root
nsolve( x^3 + 4*x + 181 )

100. nsolve( ) numerically solve for a single root.
nsolve(x^2-1)

Algebra Equation

solve( ) algebra equation, e.g. exp( ) equation,
101. Solve nonlinear equations:
solve(exp(x)+exp(-x)=4)

system of equations

102. system of 2 equations with 2 unknowns x and y by default if the unknowns omit.
solve( 2x+3y-1=0,3x+2y-1=0 )

Diophantine equation

103. it is that number of equation is less than number of the unknown, e.g. one equation with 2 unknowns x and y.
solve( 2x-3y-3=0, x,y)

congruence equation

By definition of congruence, a x ≡ b (mod m) if a x − b is divisible by m. Hence, a x ≡ b (mod m) if a x − b = m y, for some integer y. Rearranging the equation to the equivalent form of Diophantine equation a x − m y = b.
x^2-3x-2=2*(mod 2)
x^2-3x-2=2mod(2)

Modulus equation

104. solve( ) Modulus equation for the unknown x inside the mod( ) function, e.g.
input mod(x,2)=1 for
x mod 2 = 1
click the solve button

Probability_equation

105. solve( ) Probability equation for the unknown k inside the Probability function P( ),
solve( P(x>k)=0.2, k)

recurrence_equation

106. rsolve( ) recurrence and functional and difference equation for y(x)
y(x+1)=y(x)+x
y(x+1)=y(x)+1/x
107. fsolve( ) recurrence and functional and difference equation for f(x)
f(x+1)=f(x)+x
f(x+1)=f(x)+1/x

functional_equation

108. rsolve( ) recurrence and functional and difference equation for y(x)
y(a+b)=y(a)*y(b)
y(a*b)=y(a)+y(b)
109. fsolve( ) recurrence and functional and difference equation for f(x)
f(a+b)=f(a)*f(b)
f(a*b)=f(a)+f(b)

difference equation

110. rsolve( ) recurrence and functional and difference equation for y(x)
y(x+1)-y(x)=x
y(x+2)-y(x+1)-y(x)=0
111. fsolve( ) recurrence and functional and difference equation for f(x)
f(x+1)-f(x)=x
f(x+2)- f(x+1)-f(x)=0

Inequalities

112. solve( ) Inequalities for x.
solve( 2*x-1>0 )
solve( x^2+3*x+2>0 )

differential equation

ODE( ) and dsolve( ) and lasove( ) solve ordinary differential equation (ODE) to unknown y.
113. Solve linear ordinary differential equations:
y'=x*y+x
y'= 2y
y'-y-1=0

114. Solve nonlinear ordinary differential equations:
(y')^2-2y^2-4y-2=0
115. 2000 examples of Ordinary differential equation (ODE)
116. more examples in bugs

solve graphically

The odeplot( ) can be used to visualize individual functions, First and Second order Ordinary Differential Equation over the indicated domain. Input the right hand side of Ordinary Differential Equations, y"=f(x,y,z), where z for y', then click the checkbox. by default it is first order ODE.
y''=y'-y for second order ODE

integral equation

indefinite integral equation

117. indefinite integral equation
input ints(y) -2y = exp(x) for
int y dx - 2y = exp(x)

definite integral equation

118. definite integral equation
input integrates(y(t)/sqrt(x-t),t,0,x) = 2y for
int_0^x (y(t))/sqrt(x-t) dt = 2y

differential integral equation

input ds(y)-ints(y) -y-exp(x)=0 for
dy/dx-int y dx -y-exp(x)=0

fractional differential equation

dsolve( ) also solves fractional differential equation
119. Solve linear equations:
d^0.5/dx^0.5 y = 2y
d^0.5/dx^0.5 y -y - E_(0.5) (4x^0.5) = 0
d^0.5/dx^0.5 y -y -exp(4x) = 0
(d^0.5y)/dx^0.5=sin(x)

120. Solve nonlinear equations:
d^0.5/dx^0.5 y = y^2

fractional integral equation

121. d^-0.5/dx^-0.5 y = 2y

fractional differential integral equation

ds(y,x,0.5)-ints(y,x,0.5) -y-exp(x)=0
(d^0.5y)/(dx^0.5)-int y (dx)^0.5 -y-exp(x)=0

complex order differential equation

122. (d^(1-i) y)/dx^(1-i)-2y-exp(x)=0

variable order differential equation

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

system of differential equations

123. system of 2 equations with 2 unknowns x of the 0.5 order and y of the 0.8 order with a variable t.
dsolve( x(1,t)=x,y(1,t)=x-y )

partial differental equation

PDE( ) and pdsolve( ) solve partial differental equation with two variables t and x, then click the plot2D button to plot solution, pull the t slider to change the t value. click the plot3D button for 3D graph.
124. Solve a linear equation:
dy/dt = dy/dx-2y

125. Solve a nonlinear equation:
dy/dt = dy/dx*y^2

fractional partial differental equation

PDE( ) and pdsolve( ) solve fractional partial differental equation.
126. Solve linear equations:
(d^0.5y)/dt^0.5 = dy/dx-2y

127. Solve nonlinear equations:
(d^0.5y)/dt^0.5 = 2* (d^0.5y)/dx^0.5*y^2

128. More examples are in Analytical Solution of Fractional Differential Equations

test solution

test solution for algebaic equation to the unknown x by test(solution,eq,x) or click the test( ) button.

129. test(1,x^2-1=0,x)
test( -1, x^2-5*x-6 )

test solution for differential equation to the unknown y by test( ) or click the test( ) button.

130. test( exp(2x), dy/dx=2y )
131. test( exp(4x), (d^0.5y)/dx^0.5=2y )
132. Discrete Math 离散数学 >>

The default index variable in discrete math is k.
133. Input harmonic(2,x), click the defintion( ) button to show its defintion, check its result by clicking the simplify( ) button, then click the limoo( ) button for its limit as x->oo.

Difference

Δ(k^2) = difference(k^2)
134. Check its result by the sum( ) button

Summation ∑

Indefinite sum

∑ k = sum(k)
135. Check its result by the difference( ) button
Δ sum(k) = difference( sum(k) )
136. In order to auto plot, the index variable should be x,
sum_x x = sum(x,x)

definite sum

137. Definite sum = Partial sum x from 1 to x, e.g.
1+2+ .. +x = sum _(k=1) ^x k = sum(k,k,1,x)
138. Definite sum, sum x from 1 to 5, e.g.
1+2+ .. +5 = ∑(x,x,0,5) = sum(x,x,0,5)
139. Infinite sum x from 0 to inf, e.g.
1/0!+1/1!+1/2!+ .. +1/x! = sum 1/(x!) as x->oo
sum(x^k,k,0,5)

140. Definite sum with parameter x as upper limit

sum(k^2, k,0, x)
141. Check its result by the difference( ) button, and then the expand( ) button.
142. convert to sum series definition
tosum( exp(x) )
143. expand above sum series by the expand( ) button
expand( tosum(exp(x)) )

144. Indefinite sum

∑ k
sum( x^k/k!,k )
145. partial sum of 1+2+ .. + k = ∑ k = partialsum(k)
146. Definite sum of 1+2+ .. +5 = ∑ k

partial sum with parameter upper limit x

sum(1/k^2,k,1,x)

infinite sum

147. if the upper limit go to infinite, it becomes infinite sum
Infinite sum of 1/1^2+1/2^+1/3^2 .. +1/k^2+... = lim(sum( 1/k^2,k,1,x) as x->oo) = sum( 1/k^2,k,1,oo )
Infinite sum of 1/0!+1/1!+1/2! .. +1/k!+... = lim(sum( 1/k!,k,0,x) as x->oo) = sum( 1/k!,k,0,oo )
148. Series 级数

149. convert to sum series definition
tosum( exp(x) ) = toseries( exp(x) )
150. check its result by clicking the simplify( ) button
simplify( tosum( exp(x) ))

151. expand above sum series
expand( tosum(exp(x)) )

152. compare to Taylor series
taylor( exp(x), x=0, 8)
153. compare to series
series( exp(x) )

154. Taylor series expansion as x=0,
taylor( exp(x) as x=0 ) = taylor(exp(x))

by default x=0.

155. series expand not only to taylor series,
series( exp(x) )
but aslo to other series expansion,
series( zeta(2,x) )

Product ∏

156. prod(x,x)

157. prod x

Definition 定义式 >>

158. definition of function
definition( exp(x) )
159. check its result by clicking the simplify( ) button
simplify( def(exp(x)) )
160. series definition

161. convert to series definition
toseries( exp(x) )
162. check its result by clicking the simplify( ) button
simplify( tosum(exp(x)) )
163. integral definition

164. convert to integral definition
toint( exp(x) )
165. check its result by clicking the simplify( ) button
simplify( toint(exp(x)) )

Number Theory 数论 >>

When the variable x of polynomial is numnber, it becomes number, e.g.
167. poly number
poly(3,2)
168. Hermite number
hermite(3,2)
169. harmonic number
harmonic(-3,2)
harmonic(-3,2,4)
harmonic(1,1,4) = harmonic(1,4) = harmonic(4)
170. Bell number
n(bell(5))
171. double factorial 6!!
172. Calculate the 4nd prime prime(4)
173. is prime number? isprime(12321)
174. next prime greater than 4 nextprime(4)
175. binomial number ((4),(2))
176. combination number C_2^4
177. harmonic number H_4
178. congruence equation
3x-1=2*(mod 2)
x^2-3x-2=2mod( 2)
179. modular equation
Enter mod(x-1,10)=2 for (x-1) mod 10=2
180. Diophantine equation
number of equation is less than number of the unknown, e.g. one equation with 2 unkowns x and y,
solve( 3x-2y-2=0, x,y )
181. Probability 概率 >>

182. P( ) is probability of standard normal distribution
P(x<0.8)
183. Phi( ) is standard normal distribution function
Phi(x)
184. solve Probability equation for k
solve(P(x>k)=0.2,k)
185. Statistics 统计

We can sort list( ), add numbers together with total(list()), max(list()), min(list()), size(list()).

Multi elements >>

We can put multi elements together with list(), vector(), and. Most operation in them is the same as in one element, one by one. e.g. +,-,*,/, differentiation, integration, sum, etc. We count its elements with size(), as same as to count elements in function. We only talk about special properties as follows.
186. list( )
The list element can be symbol, formula or function. We can sort list( ), e.g.
sort(list(b,x,sin(x)))
2+list(a,b,c)
187. vector( )
It has direction. the position of the element is fix so we cannot sort it. vector is number with direction. two vector( ) in the same dimention can be operated by +, -, *, /, the result can be checked by its reverse operation. the system auto plot the 2-dimentional vector.
vector(2,4)/vector(1,2) =2
vector(2,4)=vector(1,2)*2
188. and
We can plot multi curves with the and. e.g. plot(x and x*x). the position of the element is fix so we cannot sort it.

Plot 制图 >>

189. plane curve 2D
190. surface 2D

3D graph 立体图 plot 3D >>

191. space curve 3D
192. surface 3D
193. surface 4D

Animation 动画 >>

194. Classication by plot function 按制图函数分类
195. Classication by appliaction 按应用分类

programming 编程 >>

There are many coding :
196. math coding 数学编程
197. html + javaScript coding 网页编程
198. online programming 在线编程
199. plot 函数图
200. rose 玫瑰花
201. check code 验证码
202. calculator 计算器
203. sci calculator 科学计算器
204. color 颜色取色器
205. Chinese calendar 农历日历
206. calendar 日历
more examples

bugs >>

There are over 500 bugs in another software but they are no problem in MathHand.com