Fractional derivative API

Caputo sense fractional derivative.

fracdiff(f::Function, α, start_point, end_point, step_size ::Caputo)

Example:

julia> fracdiff(x->x^5, 0.5, 0, 2.5, 0.0001, Caputo_Direct())

Returns a tuple (result, error), which means the value of this derivative is 141.59714979764541, and the error estimate is 1.1532243848672914e-6.

When the input end points is an array, fracdiff will compute

Refer to Caputo derivative

Caputo sense fractional derivative with first derivative known.

fracdiff(fd, α, start_point, end_point, Caputo_Direct_First_Diff_Known())

If the first order derivative of a function is already known, we can use this method to compute the fractional order derivative more precisely.

The inout function should be the first order derivative of a function

Example

julia> fracdiff(x->5*x^4, 0.5, 0, 2.5, Caputo_Direct_First_Diff_Known())

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.

Caputo sense fractional derivative with first and second derivative known.

fracdiff(fd1, fd2, α, start_point, end_point, Caputo_Direct_First_Second_Diff_Known)

If the first and second order derivative of a function is already known, we can use this method to compute the fractional order derivative more precisely.

Example

julia> fracdiff(x->5*x^4, x->20*x^3, 0.5, 0, 2.5, Caputo_Direct_First_Second_Diff_known())

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.

Caputo sense Piecewise algorithm

fracdiff(f, α, end_point, h, Caputo_Piecewise())

Using the piecewise algorithm to obtain the fractional derivative at a specific point.

Example

julia> fracdiff(x->x^5, 0.5, 2.5, 0.001, Caputo_Piecewise())

Return the fractional derivative of $f(x)=x^5$ at point $x=2.5$.

Caputo sense Diethelm computation

fracdiff(f, α, end_point, h, Caputo_Diethelm())

Using quadrature weights(derived from product trapezoidal rule) to approximate the derivative.

Example

julia> fracdiff(x->x, 0.5, 1, 0.007, Caputo_Diethelm())
1.128378318687192

!!!info "Scope" 0 < α < 1

fracdiff()

Use the high precision algorithm to compute the Caputo sense fractional derivative.

Grünwald–Letnikov sense fractional dervivative.

fracdiff(f, α, start_point, end_point, GL_Direct())

Example:

julia> fracdiff(x->x^5, 0, 0.5, GL_Direct())

Please note Grunwald-Letnikov sense fracdiff only support 0 < α < 1.

Please refer to Grünwald–Letnikov derivative for more details.

Grünwald Letnikov sense derivative approximation

fracdiff(f, α, end_point, h, GL_Multiplicative_Additive())

Grünwald–Letnikov multiplication-addition-multiplication-addition··· method to approximate fractional derivative.

Example

julia> fracdiff(x->x, 0.5, 1, 0.007, GL_Multiplicative_Additive())
1.127403405642918

Grünwald Letnikov sense derivative approximation

fracdiff(f, α, end_point, h, GL_Lagrange_Three_Point_Interp())

Using Lagrange three poitns interpolation to approximate the fractional derivative.

Example

julia> fracdiff(x->x, 0.5, 1, 0.007, GL_Lagrange_Three_Point_Interp())
1.1283963376811044

Grünwald Letnikov sense derivative approximation

fracdiff(f::Union{Function, Number}, α::AbstractArray, end_point, h, ::GL_Finite_Difference)::Vector

Use finite difference method to obtain Grünwald Letnikov sense fractional derivative.

Example

julia> fracdiff(x->x, 0.5, 1, 0.01, GL_Finite_Difference())
1.1269695801851276

fracdiff(f, α, point, h, GL_High_Precision())

Use the high precision algorithms to compute the Grunwald letnikov fractional derivative.

fracdiff(f, α, a, x, h, Hadamard_LRect())

Compute Hadamard sense using left rectangular formula.

fracdiff(f, α, a, x, h, Hadamard_Trap())

Compute Hadamard sense using right rectangular formula.

fracdiff(f, α, a, x, h, Hadamard_Trap())

Compute Hadamard sense using Trapezoidal formula.

fracdiff(f, α, end_point, h, Riesz_Symmetric())

Compute fractional derivative of Riesz sense using Triangular Strip Matrix algorithm.

Riemann Liouville sense derivative approximation

fracdiff(f, α, end_point, h, RLDiff_Approx())

Using approximation to obtain fractional derivative value.

Example

julia> fracdiff(x->x^5, 0.5, 2.5, 0.0001, RLDiff_Approx())

Riemann Liouville sense derivative using Triangular Strip Matrix to discrete and compute.

fracdiff(f, α, end_point, h, RLDiff_Matrix())

Using Triangular Strip Matrix to approximate fractional derivative.

Example

julia> fracdiff(x->x^5, 0.5, 2.5, 0.0001, RLInt_Matrix())

Try to set α as an integer, arbitrary integer of course! I promise you would enjoy it😏

Numerical methods for fractional calculus by Li, Changpin Page 57

Linear Spline Interpolation

Base type of fractional differentiation algorithms, in FractionalCalculus.jl, all of the fractional derivative algorithms belong to FracDiffAlg

Caputo sense fractional derivative algorithms, please refer to Caputo derivative for more details.

Grünwald–Letnikov sense fractional derivative algorithms, please refer to Grünwald–Letnikov derivative for more details

Riemann-Liouville sense fractional derivative algorithms, please refer to Riemann-Liouville derivative

Hadamard sense fractional derivative algorithms.

Riesz sense fractional derivative algorithms.

Note Caputo Direct algorithms belong to direct computing, precise are ensured, but maybe cause more memory allocation and take more compilation time.

Using the direct mathematic expression:

\[^CD_t^α=\frac{1}{\Gamma(n-α)}\int_0^t\frac{f^{(n)}(τ)}{(t-τ)^{α+1-n}}\]

As for the derivative inside the integral, we use the Complex Step Differentiation to obtain a more accurate value.

With first derivative known, we can direct use the derivative to compute a more accurate result.

With first and second derivative known, we can direct use derivative and second derivative to compute a more accurate result.

Using piecewise linear interpolation function to approximate input function and combining Caputo derivative then implement summation.

Diethelm's method for computingn Caputo sense fractional derivative using product trapezoidal rule and Richardsin extrapolation

Grunwald Letnikov direct compute method to obtain fractional derivative, precision are guaranteed but cause more memory allocation and compilation time.

Using Grünwald–Letnikov finite difference method to compute Grünwald–Letnikov sense fractional derivative.

Using Grünwald–Letnikov multiplication-addition-multiplication-addition··· method to compute fractional derivative in Grünwald Letnikov sense.

Using Lagrange three points interpolation method to compute fractional derivative in Grünwald Letnikov sense.

Using Linear interpolation to approximate fractional derivative in Riemann Liouville fractional derivative sense.

Using Triangular Strip Matrix to discrete the derivative.

Using finite part integral of left rectangular formula to compute the fractional hadamard derivative.

Using finite part integral of right rectangular formula to compute the fractional hadamard derivative.

Using finite part integral of trapezoidal formula to compute the fractional hadamard derivative.

Using the Triangular Strip Matrices to compute the fractional derivative Riesz derivative.

@fracdiff(f, α, point)

Return the α-order derivative of f at specific point.

julia> @fracdiff(x->x, 0.5, 1)
1.1283791670955188

@semifracdiff(f, point)

Return the semi-derivative of f at spedific point.

julia> @semifracdiff(x->x, 1)
1.1283791670955188