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FractionalCalculus.jl

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FractionalCalculus.jl: A Julia package for high performance, comprehensive and high precision numerical fractional calculus computing.

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# FractionalCalculus.jl

FractionalCalculus.jl provides support for fractional calculus computing.

## 🎇 Installation

If you have already installed Julia, you can install FractionalCalculus.jl in REPL using Julia package manager:

```julia
pkg> add FractionalCalculus
```

## 🦸 Quick start

### Derivative

To compute the fractional derivative in a specific point, for example, compute $\alpha = 0.2$ derivative of $f(x) = x$ in $x = 1$ with step size $h = 0.0001$ using **Riemann Liouville** sense:

```julia
julia> fracdiff(x->x, 0.2, 1, 0.0001, RLDiffL1())
1.0736712740308347
```

This will return the estimated value with high precision.

### Integral

To compute the fractional integral in a specific point, for example, compute the semi integral of $f(x) = x$ in $x = 1$  with step size $h = 0.0001$ using **Riemann-Liouville** sense:

```julia
julia> fracint(x->x, 0.5, 1, 0.0001, RLIntApprox())
0.7522525439593486
```

This will return the estimated value with high precision.

## 💻 All algorithms

For detailed usage, please refer to [our manual](https://scifracx.org/FractionalCalculus.jl/dev/Derivative/derivativeapi/).

## 🖼️ Example

Let's see examples here:

Compute the semi-derivative of $f(x) = x$ in the interval $\left[0, 1\right]$:

![Plot](https://raw.githubusercontent.com/SciFracX/FractionalCalculus.jl/master/docs/src/assets/semiderivativeplot.png)

We can see that computing retains high precision⬆️.

Compute different order derivative of $f(x) = x$:

![Different Order](https://raw.githubusercontent.com/SciFracX/FractionalCalculus.jl/master/docs/src/assets/different_order_x_derivative.png)

Also different order derivative of $f(x) = \sin(x)$:

![Different Order of sin](https://raw.githubusercontent.com/SciFracX/FractionalCalculus.jl/master/docs/src/assets/different_order_sin_derivative.png)

And also different order integral of $f(x) = x$:

![Different Order Of x](https://raw.githubusercontent.com/SciFracX/FractionalCalculus.jl/master/docs/src/assets/different_order_x_integral.png)

<!---

Or arbitrary order derivative? A piece of cake!!😉

![Arbitrary](https://raw.githubusercontent.com/SciFracX/FractionalCalculus.jl/master/docs/src/assets/arbitrary_order_derivative.png)

-->

## 🧙 Symbolic Fractional Differentiation and Integration

Thanks to SymbolicUtils.jl, FractionalCalculus.jl can do symbolic fractional differentiation and integration now!!

```julia
julia> using FractionalCalculus, SymbolicUtils
julia> @syms x
julia> semidiff(log(x))
log(4x) / sqrt(πx)
julia> semiint(x^4)
0.45851597901024005(x^4.5)
```

## 📢 Status

Right now, FractionalCalculus.jl has only supports for little algorithms:

Fractional Derivative:

- [x] Caputo fractional derivative
- [x] Grunwald-Letnikov fractional derivative
- [x] Riemann-Liouville fractional derivative 
- [x] Riesz fractional derivative
- [x] Hadamard  fractional derivative
- [x] Caputo-Fabrizio fractional derivative
- [x] Atangana-Baleanu fractional derivative
- [ ] Marchaud fractional derivative
- [ ] Weyl fractional derivative
- [ ] ......

Fractional Integral:
- [x] Riemann-Liouville fractional integral
- [x] Hadamard fractional integral
- [ ] Atangana-Baleanu fractional integral
- [ ] ......

## 📚 Reference

FractionalCalculus.jl is built upon the hard work of many scientific researchers, I sincerely appreciate what they have done to help the development of science and technology.

## 🥂 Contributing

If you are interested in Fractional Calculus and Julia, welcome to raise an issue or file a Pull Request!!

FractionalCalculus.jl

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