https://github.com/scifracx/fractionalcalculus.jl

FractionalCalculus.jl: A Julia package for high performance, comprehensive and high precision numerical fractional calculus computing.
https://github.com/scifracx/fractionalcalculus.jl

algorithms caputo differentiation differintegral fractional-calculus grunwald-letnikov integration julia matrix-discrete numerical riemann-liouville

Last synced: 11 days ago
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FractionalCalculus.jl: A Julia package for high performance, comprehensive and high precision numerical fractional calculus computing.

• Host: GitHub
• URL: https://github.com/scifracx/fractionalcalculus.jl
• Owner: SciFracX
• License: mit
• Created: 2021-06-29T16:25:04.000Z (over 3 years ago)
• Default Branch: master
• Last Pushed: 2024-04-07T16:39:59.000Z (8 months ago)
• Last Synced: 2024-08-03T20:04:35.834Z (4 months ago)
• Topics: algorithms, caputo, differentiation, differintegral, fractional-calculus, grunwald-letnikov, integration, julia, matrix-discrete, numerical, riemann-liouville
• Language: Julia
• Homepage: http://scifracx.org/FractionalCalculus.jl/dev/
• Size: 4.69 MB
• Stars: 32
• Watchers: 2
• Forks: 6
• Open Issues: 8
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

        
# 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

```

Current Algorithms

├── FracDiffAlg

│   ├── Caputo

|   |   ├── CaputoDirect

|   |   ├── CaputoTrap

|   |   ├── CaputoDiethelm

|   |   ├── CaputoHighPrecision

|   |   ├── CaputoL1

|   |   ├── CaputoL2

|   |   └── CaputoHighOrder

|   |

│   ├── Grünwald Letnikov

|   |   ├── GLDirect

|   |   ├── GLMultiplicativeAdditive

|   |   ├── GLLagrangeThreePointInterp

|   |   └── GLHighPrecision

|   |

|   ├── Riemann Liouville

|   |    ├── RLDiffL1

|   |    ├── RLDiffL2

|   |    ├── RLDiffL2C

|   |    ├── RLLinearSplineInterp

|   |    ├── RLDiffMatrix

|   |    ├── RLG1

|   |    └── RLD

|   | 

|   ├── Hadamard

|   |    ├── HadamardLRect

|   |    ├── HadamardRRect

|   |    └── HadamardTrap

|   |

|   ├── Riesz

|   |    ├── RieszSymmetric

|   |    └── RieszOrtigueira

|   |

|   ├── Caputo-Fabrizio

|   |    └── CaputoFabrizioAS

|   |

|   └── Atanagana Baleanu

|        └── AtanganaSeda

|

└── FracIntAlg

    ├── Riemann Liouville

    |   ├── RLDirect

    |   ├── RLPiecewise

    |   ├── RLLinearInterp

    |   ├── RLIntApprox

    |   ├── RLIntMatrix

    |   ├── RLIntSimpson

    |   ├── RLIntTrapezoidal

    |   ├── RLIntRectangular

    |   └── RLIntCubicSplineInterp

    |

    └── Hadamard

        └── HadamardMat

```

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](/docs/src/assets/semiderivativeplot.png)

We can see that computing retains high precision⬆️.

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

![Different Order](/docs/src/assets/different_order_x_derivative.png)

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

![Different Order of sin](/docs/src/assets/different_order_sin_derivative.png)

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

![Different Order Of x](/docs/src/assets/different_order_x_integral.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!!