FractionalDiffEq.jl
FractionalDiffEq.jl provides FDE solvers to DifferentialEquations.jl ecosystem, including FODE(Fractional Ordianry 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 Julia package manager:
pkg> add FractionalDiffEq
Quick start
Fractional ordinary differential equations
Let's see if we have an initial value problem:
So we can use FractionalDiffEq.jl to solve the problem:
using FractionalDiffEq, Plots
fun(u, p, t) = 1-u
u0 = [0, 0]; tspan = (0, 20); h = 0.001;
prob = SingleTermFODEProblem(fun, 1.8, u0, tspan)
sol = solve(prob, h, PECE())
plot(sol)
And if you plot the result, you can see the result of the above IVP:
A sophisticated example
Let's see if the initial value problem like:
using FractionalDiffEq, Plots
h=0.01; 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, h, PIEX())
plot(sol, legend=:bottomright)
Or use the example file 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 system of fractional differential equations, if you are familiar with DifferentialEquations.jl, it would be just like out of the box.
Let's see if we have a Chua chaos system:
By using the NonLinearAlg
algorithms to solve this problem:
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];
h = 0.01; tspan = (0, 100);
p = [10.725, 10.593, 0.268, -1.1726, -0.7872]
prob = FODESystem(chua!, α, x0, tspan, p)
sol = solve(prob, h, NonLinearAlg())
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.
With history function:
using FractionalDiffEq, Plots
ϕ(x) = x == 0 ? (return 19.00001) : (return 19.0)
f(t, y, ϕ) = 3.5*y*(1-ϕ/19)
h = 0.05; α = 0.97; τ = 0.8; T = 56
fddeprob = FDDEProblem(f, ϕ, α, τ, T)
V, y = solve(fddeprob, h, DelayPECE())
plot(y, V, xlabel="y(t)", ylabel="y(t-τ)")
Lyapunov exponents of fractional order system
FractionalDiffEq.jl is capable of generating lyapunov exponents of a fractional order system:
Rabinovich-Fabrikant system:
julia>LE, tspan = FOLyapunov(RF, 0.98, 0, 0.02, 300, [0.1; 0.1; 0.1], 0.005, 1000)
Available Solvers
For more performant solvers, please refer to the FractionalDiffEq.jl Solvers 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!!