Skip to content

Commit

Permalink
Format checks and fix the docs
Browse files Browse the repository at this point in the history 
Signed-off-by: ErikQQY <[email protected]>
  • Loading branch information
@ErikQQY
ErikQQY committed Nov 5, 2021
1 parent 4911fb4 commit 50592c9
Show file tree
Hide file tree
Showing 4 changed files with 51 additions and 188 deletions.
6 changes: 3 additions & 3 deletions docs/src/Derivative/derivative.md
View file
Original file line number Diff line number Diff line change
Expand Up @@ -7,7 +7,7 @@ Pages = ["derivative.md"]
To get start with fractional derivative, you need to know that unlike Newtonian derivatives, fractional derivative is defined via integral.

!!! tip "Non-Local Operators"
It is noteworthy that the fractional derivatives are not local operators, which means that we cannot calculate the fractional derivative solely on the basis of function values of $f(x)$ taken from neighborhood of the point $x$. Instead, we have to take the entire history of $f(x)$ (i.e., all function values f(x) for $0<x<a$) into account.
It is noteworthy that the fractional derivatives are not local operators, which means that we cannot calculate the fractional derivative solely on the basis of function values of $f(x)$ taken from neighborhood of the point $x$. Instead, we have to take the entire history of $f(x)$ (i.e., all function values $f(x)$ for $0<x<a$) into account.

## Riemann Liouville sense derivative

Expand All @@ -23,7 +23,7 @@ _tD^\alpha_bf(t)=\frac{d^n}{dt^n}\ _tD^{-(n-\alpha)}_bf(t)=\frac{d^n}{dt^n}\ _tI
We can use FractionalCalculus.jl to compute fractional derivative:

```julia-repl
fracdiff(x->x, 0.5, 1, 0.0001, RLDiff_Approx())
julia> fracdiff(x->x, 0.5, 1, 0.0001, RLDiff_Approx())
```

## Caputo sense derivative
Expand All @@ -32,7 +32,7 @@ In **FractionalCalculus.jl**, let's see, if you want to calculate the $0.4$ orde


```julia-repl
fracdiff(x->x, 0.4, 1, 0.0001, Caputo_Piecewise())
julia>fracdiff(x->x, 0.4, 1, 0.0001, Caputo_Piecewise())
```

There many types of definitions of fractional derivative, Caputo is one of these useful definitions. The Caputo fractional derivative is first be proposed in [Michele Caputo's Paper](https://doi.org/10.1111/j.1365-246X.1967.tb02303.x),
Expand Down
6 changes: 3 additions & 3 deletions docs/src/Integral/integral.md
View file
Original file line number Diff line number Diff line change
Expand Up @@ -11,8 +11,8 @@ _aD_t^{-\alpha}f(t)=\frac{1}{\Gamma(\alpha)}\int_a^t(t-\tau)^{\alpha-1}f(\tau)d\
```
In FractionalCalculus, you can compute the integral of a function with order $\alpha$:

```julia
fracint(x->x, 0.5, 0, 1, 0.0001, RL_Direct())
```julia-repl
julia> fracint(x->x, 0.5, 0, 1, 0.0001, RL_Direct())
```

A tuple contains result and estimating error will return
Expand All @@ -28,7 +28,7 @@ julia> fracint(x->x, 0.5, 0, 1, 0.0001, RL_())

FractionalCalculus.jl support many algorithms to calculate fractional integral, here, I want to highlight the **Triangular Strip Matrix** method proposed by [Prof Igor](http://people.tuke.sk/igor.podlubny/index.html) to discrete fractional derivative.

```julia
```julia-repl
julia> fracdiff(f, α, end_point, h, RLInt_Matrix())
```

Expand Down
151 changes: 29 additions & 122 deletions src/Derivative/Derivative.jl
View file
Original file line number Diff line number Diff line change
Expand Up @@ -136,15 +136,23 @@ struct GL_Finite_Difference <: GL end
"""
Check if the format of nargins is correct.
"""
function checks(α, start_point, end_point)
function checks(f, α, start_point, end_point)
α % 1 != 0 ? nothing : error("α must be a decimal number")
if typeof(end_point) <: Number
if isa(end_point, Number)
start_point < end_point ? nothing : DomainError("Domain error! Start point must smaller than end point")
elseif typeof(end_point) <: AbstractArray
elseif isa(end_point, AbstractArray)
start_point < minimum(end_point) ? nothing : DomainError("Vector domain error! Start point must smaller than end point")
else
ErrorException("Please input correct point you want to compute")
end

end_point == 0 ? 0 : nothing


#The fractional derivative of number is relating with the end_point.
if isa(f, Number)
f == 0 ? 0 : f/sqrt(pi*end_point)
end
end


Expand Down Expand Up @@ -183,20 +191,7 @@ When the input end points is an array, **fracdiff** will compute
Refer to [Caputo derivative](https://en.wikipedia.org/wiki/Fractional_calculus#Caputo_fractional_derivative)
"""
function fracdiff(f::Union{Function, Number}, α, start_point, end_point, h, ::Caputo_Direct)
checks(α, start_point, end_point)

#The fractional derivative of number is relating with the end_point.
if typeof(f) <: Number
if f == 0
return 0
else
return f/sqrt(pi*end_point)
end
end

if end_point == 0
return 0
end
checks(f, α, start_point, end_point)

#Using complex step differentiation to calculate the first order differentiation
g(τ) = imag(f(τ+1*im*h) ./ h) ./ ((end_point-τ) .^α)
Expand Down Expand Up @@ -230,22 +225,7 @@ julia> fracdiff(x->x^5, 0, 0.5, GL_Direct())
Please refer to [Grünwald–Letnikov derivative](https://en.wikipedia.org/wiki/Gr%C3%BCnwald%E2%80%93Letnikov_derivative) for more details.
"""
function fracdiff(f::Union{Function, Number}, α, start_point, end_point, ::GL_Direct)
checks(α, start_point, end_point)

#The fractional derivative of number is relating with the end_point.
if typeof(f) <: Number
if f == 0
return 0
else
return f/sqrt(pi*end_point)
end
end

checks(α, start_point, end_point)

if end_point == 0
return 0
end
checks(f, α, start_point, end_point)

g(τ) = (f(end_point)-f(τ)) ./ (end_point - τ) .^ (1+α)
result = f(end_point)/(gamma(1-α) .* (end_point - start_point) .^ α) .+ quadgk(g, start_point, end_point) .* α ./ gamma(1-α)
Expand Down Expand Up @@ -281,16 +261,6 @@ Return the semi-derivative of ``f(x)=x^5`` at ``x=2.5``.
Compared with **Caputo_Direct** method, this method don't need to specify step size, more precision are guaranteed.
"""
function fracdiff(fd::Function, α, start_point, end_point, ::Caputo_Direct_First_Diff_Known)
checks(α, start_point, end_point)

#The fractional derivative of number is relating with the end_point.
if typeof(f) <: Number
if f == 0
return 0
else
return f/sqrt(pi*end_point)
end
end

g(τ) = (end_point-τ) .^ (-α) * fd(τ)
result = quadgk(g, start_point, end_point) ./gamma(1-α)
Expand Down Expand Up @@ -325,7 +295,6 @@ Return the semi-derivative of ``f(x)=x^5`` at ``x=2.5``.
Compared with **Caputo_Direct** method, this method don't need to specify step size, more precision are guaranteed.
"""
function fracdiff(fd1::Function, fd2, α, start_point, end_point, ::Caputo_Direct_First_Second_Diff_Known)
checks(α, start_point, end_point)

temp1 = fd1(start_point) .* (end_point-start_point) .^ (1-α) ./ gamma(2-α)
g(τ) = fd2(τ) .* ((end_point-τ) .^ (1-α))
Expand Down Expand Up @@ -360,24 +329,12 @@ Return the fractional derivative of ``f(x)=x^5`` at point ``x=2.5``.
"""
function fracdiff(f::Union{Function, Number}, α::Float64, end_point, h::Float64, ::Caputo_Piecewise)::Float64

#The fractional derivative of number is relating with the end_point.
if typeof(f) <: Number
if f == 0
return 0
else
return f/sqrt(pi*end_point)
end
end

if end_point == 0
return 0
end
checks(f, α, 0, end_point)

m = floor(α)+1

summation = 0
n = end_point/h
n = Int64(end_point/h)

for i in range(0, n, step=1)
summation +=W₅(i, n, m, α)*first_order(f, i*h, h)
Expand Down Expand Up @@ -428,22 +385,10 @@ julia> fracdiff(x->x, 0.5, 1, 0.007, GL_Multiplicative_Additive())
The `GL_Multiplicative_Additive` method is not accruate, please use it at your own risk.
"""
function fracdiff(f::Union{Function, Number}, α, end_point, h, ::GL_Multiplicative_Additive)::Float64

#The fractional derivative of number is relating with the end_point.
if typeof(f) <: Number
if f == 0
return 0
else
return f/sqrt(pi*end_point)
end
end

if end_point == 0
return 0
end
checks(f, α, 0, end_point)

summation = 0
n = end_point/h
n = Int64(end_point/h)

for i in range(0, n-1, step=1)
summation+=gamma(i-α)/gamma(i+1)*f(end_point-i*end_point/n)
Expand Down Expand Up @@ -481,19 +426,7 @@ julia> fracdiff(x->x, 0.5, 1, 0.007, GL_Lagrange_Three_Point_Interp())
The `GL_Multiplicative_Additive` method is not accruate, please use it at your own risk.
"""
function fracdiff(f::Union{Function, Number}, α::Float64, end_point, h, ::GL_Lagrange_Three_Point_Interp)::Float64

#The fractional derivative of number is relating with the end_point.
if typeof(f) <: Number
if f == 0
return 0
else
return f/sqrt(pi*end_point)
end
end

if end_point == 0
return 0
end
checks(f, α, 0, end_point)

n = end_point/h
summation=0
Expand Down Expand Up @@ -533,19 +466,7 @@ julia> fracdiff(x->x^5, 0.5, 2.5, 0.0001, RLDiff_Approx())
"""
function fracdiff(f::Union{Number, Function}, α, end_point, h, ::RLDiff_Approx)::Float64

#The fractional derivative of number is relating with the end_point.
if typeof(f) <: Number
if f == 0
return 0
else
return f/sqrt(pi*end_point)
end
end

if end_point == 0
return 0
end
checks(f, α, 0, end_point)

summation = 0
n = end_point/h
Expand All @@ -567,7 +488,7 @@ function fracdiff(f::Union{Number, Function}, α, end_point::AbstractArray, h, :
end

function fracdiff(f::Union{Number, Function}, α, end_point, h, ::RLDiff_Approx_2)::Float64

#The fractional derivative of number is relating with the end_point.
if typeof(f) <: Number
if f == 0
Expand Down Expand Up @@ -663,15 +584,7 @@ julia> fracdiff(x->x, 0.5, 1, 0.01, GL_Finite_Difference())
`GL_Finite_Difference` method is not accruate, please use it at your own risk.
"""
function fracdiff(f::Union{Number, Function}, α::Float64, end_point::Real, h, ::GL_Finite_Difference)::Float64

#The fractional derivative of number is relating with the end_point.
if typeof(f) <: Number
if f == 0
return 0
else
return f/sqrt(pi*end_point)
end
end
checks(f, α, 0, end_point)

n = end_point/h
result=0
Expand All @@ -680,7 +593,7 @@ function fracdiff(f::Union{Number, Function}, α::Float64, end_point::Real, h, :
result+=(-1)^i*gamma(α+1)/(gamma(i+1)*gamma(α-i+1))*f(end_point-i*h)
end

result1=result/h^α
result1=result*h^(-α)
return result1
end

Expand Down Expand Up @@ -711,6 +624,8 @@ julia> fracdiff(x->x, 0.5, 1, 0.007, Caputo_Diethelm())
0 < α < 1
"""
function fracdiff(f::Union{Function, Number}, α::Float64, end_point, h, ::Caputo_Diethelm)
checks(f, α, 0, end_point)

N = end_point/h

result=0
Expand All @@ -722,15 +637,14 @@ function fracdiff(f::Union{Function, Number}, α::Float64, end_point, h, ::Caput

end
function quadweights(n, N, α)
if n==0
if n == 0
return 1
elseif 0<n<N
elseif 0 < n < N
return (n+1)^(1-α)-2*n^(1-α)+(n-1)^(1-α)
elseif n==N
elseif n == N
return (1-α)*N^(-α)-N^(1-α)+(N-1)^(1-α)
end
end

function nderivative(f, n, end_point, h)
temp = 0

Expand Down Expand Up @@ -766,19 +680,13 @@ function fracdiff(f, α, end_point, h, ::RLDiff_Matrix)
return B(N, α, h)*f.(tspan)
end

"""
eliminator(n, row)
Compute the eliminator matrix Sₖ by omiting n-th row
"""
#Compute the eliminator matrix Sₖ by omiting n-th row
function eliminator(n, row)
temp = zeros(n, n)+I
return temp[Not(row), :]
end

"""
Generating elements in Matrix.
"""
#Generate elements in Matrix.
function omega(n, p)
omega = zeros(n+1)

Expand All @@ -790,7 +698,6 @@ function omega(n, p)
return omega

end

function B(N, p, h)
result=zeros(N, N)
temp=omega(N, p)
Expand Down
Loading

2 comments on commit 50592c9

@ErikQQY
Copy link
Member Author

@ErikQQY ErikQQY commented on 50592c9 Nov 5, 2021

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

@JuliaRegistrator register

@JuliaRegistrator
Copy link

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

Error while trying to register: Version 0.1.1 already exists

Please sign in to comment.