Fractional integral API

fracint(f, α, point, h, FracIntAlg())

The general function used for fractional integral computing.

Example

julia> fracint(x->x^2, 0.5, 1, 0.001, RLInt_Approx())
0.6017877646029814

There are many algorithms can be used to computing fractional integral, please refer to our manual for more details: manual.

Riemann-Liouville sense fractional integral algorithms

As the general fractional order integral type, RLInt is the parent type of all Riemann Liouville sense numerical methods.

Riemann Liouville sense direct compute

fracint(f::Function, α, start_point, end_point, h, RLDirect())

Riemann_Liouville fractional integral using complex step differentiation.

\[(J^αf)(t)=\frac{1}{\Gamma(α)} \int_0^t(t-τ)^{α-1}f(τ)dτ\]

By using QuadGK calculate the integration and obtain the value.

Example:

julia> fracint(x->x^5, 0.5, 0, 2.5, 1e-8, RLDirect())
(64.36234189727955, 7.742994054849976e-7)

Returns a tuple (1.1639316474512205, 1.0183453796725215e-8), which contains the value of this derivative is 1.1639316474512205, and the error estimate is 1.0183453796725215e-8

Riemann Liouville sense fractional integral using piecewise interpolation.

fracint(f, α, end_point, h, RLPiecewise())

Example

julia> fracint(x->x^5, 0.5, 2.5, 0.0001, RLPiecewise())
64.36234206345209

By deploying Piecewise interpolation to approximate the original function, with small step size, this method is fast and take little memory allocation.

Using piecewise linear interpolation:

\[ y_n(t)=\frac{t-t_{i+1}}{t_i-t_{i+1}}y(t_i)+\frac{t-t_i}{t_{i+1}-t_i}y(t_{i+1})\]

Constitute the original function with the interpolation and implement the summation.

Riemann Liouville sense fractional integral approximation.

fracint(f, α, end_point, h, RLIntApprox())

Example

julia> fracint(x->x^5, 0.5, 2.5, 0.0001, RLIntApprox())
64.36229646291393

Using the Staircase approximation to approximate the original function and implement the summation.

Riemann Liouville sense fractional integral using Linear interpolation.

fracint(f, α, end_point, h, RLLinearInterp())

Example

julia> fracint(x->x^5, 0.5, 2.5, 0.0001, RLLinearInterp())
64.36234206434942

RLLinearInterp is more complex compared with RLIntApprox but more precise.

Deploying the Linear Interpolation between $f_{j+1}$ and $f_j$, RLLinearInterp method is more precise than RLIntApprox.

Riemann Liouville sense integral using Triangular Strip Matrix to discrete.

fracint(f, α, end_point, h, RLIntMatrix())

Using Triangular Strip Matrix to approximate fractional integral.

Example

julia> fracint(x->x^5, 0.5, 2.5, 0.0001, RLIntMatrix())

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

Using Triangular Strip Matrix to discrete the integral.

Riemann Liouville sense fractional integral using fractional Simpson's formula to approximate.

Example

julia> fracint(x->x, 0.5, 1, 0.000001, RLIntSimpson())
0.7516516520541335

@inproceedings{Li2015NumericalMF,
  title={Numerical Methods for Fractional Calculus},
  author={Changpin Li and Fanhai Zeng},
  year={2015}
}

Using fractional Simpson's formula to discrete fractional integral.

Riemann Liouville sense fractional integral using fractional trapezoidal formula to approximate.

Example

julia> fracint(x->x, 0.5, 1, 0.001, RLIntTrapezoidal())
0.7522527780636226

《Numerical methods for fractional calculus》. Using Trapezoidal method to approximate fractional integral

Riemann Liouville sense fractional integral using fractional rectangular formula to approximate.

Example

julia> fracint(x->x, 0.5, 1, 0.001, RLIntRectangular())
0.7516812175993778

Riemann Liouville sense fractional integral using cubic spline interpolation to approximate.

Example

julia> fracint(x->x, 0.5, 1, 0.0001, RLIntCubicSplineInterp())
0.7529119186452693

Error estimate of this method is $\mathcal{O(h^4)}$, it is determined by the error of the cubic spline interpolation.

Numerical Methods for Fractional Calculus Page 34

@fracint(f, α, point)

Return the α-order integral of $f$ at specific point.

julia> @fracint(x->x, 0.5, 1)
0.7522525439593486

@semifracint(f, point)

Return the semi-integral of $f$ at specific point.

julia> @semifracint(x->x, 1)
0.7522525439593486

Symbolic fractional integral

semiint(fun)

semiint uses SymbolicUtils.jl and Symbolics.jl to compute symbolic fractional integral.

Example

julia> using SymbolicUtils
julia> @syms x
julia> semiint(x^4)
0.45851597901024005(x^4.5)