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https://github.com/SciFracX/FractionalDiffEq.jl
Solve Fractional Differential Equations using high performance numerical methods
https://github.com/SciFracX/FractionalDiffEq.jl
caputo differential-equations distributed-order fdde fde fractional-calculus fractional-differencing fractional-differential-equations julia matrix-discrete numerical numerical-methods ode pde riemann-liouville scientific-machine-learning
Last synced: 3 months ago
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Solve Fractional Differential Equations using high performance numerical methods
- Host: GitHub
- URL: https://github.com/SciFracX/FractionalDiffEq.jl
- Owner: SciFracX
- License: mit
- Created: 2021-10-25T11:18:07.000Z (about 3 years ago)
- Default Branch: master
- Last Pushed: 2024-06-26T03:13:56.000Z (5 months ago)
- Last Synced: 2024-07-26T17:13:51.245Z (3 months ago)
- Topics: caputo, differential-equations, distributed-order, fdde, fde, fractional-calculus, fractional-differencing, fractional-differential-equations, julia, matrix-discrete, numerical, numerical-methods, ode, pde, riemann-liouville, scientific-machine-learning
- Language: Julia
- Homepage: https://scifracx.github.io/FractionalDiffEq.jl/dev/
- Size: 11.3 MB
- Stars: 76
- Watchers: 2
- Forks: 11
- Open Issues: 28
-
Metadata Files:
- Readme: README.md
- License: LICENSE
Awesome Lists containing this project
- awesome-sciml - SciFracX/FractionalDiffEq.jl: FractionalDiffEq.jl: A Julia package aiming at solving Fractional Differential Equations using high performance numerical methods
README
# FractionalDiffEq.jl
[](https://github.com/SciML/SciMLStyle)
FractionalDiffEq.jl provides FDE solvers to [DifferentialEquations.jl](https://diffeq.sciml.ai/dev/) ecosystem, including FODE(Fractional Ordinary Differential Equations), FDDE(Fractional Delay Differential Equations) and many more. There are many performant solvers available, capable of solving many kinds of fractional differential equations.
# Installation
If you have already installed Julia, you can install FractionalDiffEq.jl in REPL using the Julia package manager:
```julia
pkg> add FractionalDiffEq
```
# Quick start
### Fractional ordinary differential equations
Let's see if we have an initial value problem:
$$ D^{1.8}y(x)=1-y $$
$$ y(0)=0 $$
So we can use FractionalDiffEq.jl to solve the problem:
```julia
using FractionalDiffEq, Plots
fun(u, p, t) = 1-u
u0 = [0, 0]; tspan = (0, 20);
prob = FODEProblem(fun, 1.8, u0, tspan)
sol = solve(prob, PECE(), dt=0.001)
plot(sol)
```
And if you plot the result, you can see the result of the above IVP:
### A sophisticated example
Let's see if we have an initial value problem:
$$ y'''(t)+\frac{1}{16}{^C_0D^{2.5}_t}y(t)+\frac{4}{5}y''(t)+\frac{3}{2}y'(t)+\frac{1}{25}{^C_0D^{0.5}_t}y(t)+\frac{6}{5}y(t)=\frac{172}{125}\cos(\frac{4t}{5}) $$
$$ y(0)=0,\ y'(0)=0,\ y''(0)=0 $$
```julia
using FractionalDiffEq, Plots
tspan = (0, 30)
rightfun(x, y) = 172/125*cos(4/5*x)
prob = MultiTermsFODEProblem([1, 1/16, 4/5, 3/2, 1/25, 6/5], [3, 2.5, 2, 1, 0.5, 0], rightfun, [0, 0, 0, 0, 0, 0], tspan)
sol = solve(prob, PIEX(), dt=0.01)
plot(sol, legend=:bottomright)
```
Or use the [example file](https://github.com/SciFracX/FractionalDiffEq.jl/blob/master/examples/complicated_example.jl) to plot the numerical approximation, we can see the FDE solver in FractionalDiffEq.jl is amazingly powerful:
### System of Fractional Differential Equations:
FractionalDiffEq.jl is a powerful tool to solve a system of fractional differential equations, if you are familiar with [DifferentialEquations.jl](https://github.com/SciML/DifferentialEquations.jl), it would be just like out of the box.
Let's see if we have a Chua chaos system:
$$ \begin{cases}D^{\alpha_1}x=10.725[y-1.7802x-[0.1927(|x+1|-|x-1|)]\\
D^{\alpha_2}y=x-y+z\\
D^{\alpha_3}z=-10.593y-0.268z\end{cases} $$
By using the ```NonLinearAlg``` algorithms to solve this problem:
```julia
using FractionalDiffEq, Plots
function chua!(du, x, p, t)
a, b, c, m0, m1 = p
du[1] = a*(x[2]-x[1]-(m1*x[1]+0.5*(m0-m1)*(abs(x[1]+1)-abs(x[1]-1))))
du[2] = x[1]-x[2]+x[3]
du[3] = -b*x[2]-c*x[3]
end
α = [0.93, 0.99, 0.92];
x0 = [0.2; -0.1; 0.1];
tspan = (0, 100);
p = [10.725, 10.593, 0.268, -1.1726, -0.7872]
prob = FODEProblem(chua!, α, x0, tspan, p)
sol = solve(prob, BDF(), dt=0.01)
plot(sol, vars=(1, 2), title="Chua System", legend=:bottomright)
```
And plot the result:
## Fractional Delay Differential Equations
There are also many powerful solvers for solving fractional delay differential equations.
$$ D^\alpha_ty(t)=3.5y(t)(1-\frac{y(t-0.74)}{19}) $$
$$ y(0)=19.00001 $$
With history function:
$$ y(t)=19,\ t<0 $$
```julia
using FractionalDiffEq, Plots
ϕ(x) = x == 0 ? (return 19.00001) : (return 19.0)
f(t, y, ϕ) = 3.5*y*(1-ϕ/19)
α = 0.97; τ = 0.8; tspan = (0,56)
fddeprob = FDDEProblem(f, ϕ, α, constant_lags=[τ], tspan)
sol = solve(fddeprob, DelayPECE(), dt=0.05)
plot(sol, xlabel="y(t)", ylabel="y(t-τ)")
```
# Available Solvers
For more performant solvers, please refer to the [FractionalDiffEq.jl Solvers](https://scifracx.org/FractionalDiffEq.jl/dev/algorithms/) page.
# Road map
* More performant algorithms
* Better docs
* More interesting ideas~
# Contributing
If you are interested in Fractional Differential Equations and Julia, welcome to raise an issue or file a Pull Request!!