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We would use a simple example – Relaxation Oscillation Process to show you how to use FractionalDiffEq.jl to solve fractional differential equations.🙂

The mathematical model of the fractional order relaxation oscillation can be abstracted as an IVP:

\[D^{1.8}y(t)+y(t)=1,\ (t>0)\]


While we can know the analytical solution of this equation is:

\[u(t)=t^{1.8}E_{1.8,\ 2.8}(-t^{1.8})\]

Here $E_{\alpha, \beta}(z)$ is the Mittag Leffler function.

We can solve this problem by the following code using FractionalDiffEq.jl:

using FractionalDiffEq, Plots
fun(u, p, t) = 1-u
α=1.8; tspan = (0, 20); u0 = [0, 0]
prob = FODEProblem(fun, α, u0, tspan)
sol = solve(prob, PIEX(), dt=0.01)

By plotting the numerical result, we can get the approximation result:

Relaxation Oscillation

To provide users with a simple way to solve fractional differential equations, we follow the design pattern of DifferentialEquations.jl

Step 1: Defining a Problem

First, we need to specify the problem we want to solve. Just by passing the parameters —— describing function, orders, initial condition and time span to construct a fractional ordinary differential equation problem:

using FractionalDiffEq
fun(u, p, t) = 1-u
α = 1.8; u0 = [0, 0]; tspan = (0, 20); dt = 0.01;
prob = FODEProblem(fun, α, u0, tspan)

The FODEProblem is a class of fractional differential equations, describing equations with $D^{\alpha}u=f(t, u)$ pattern. For other patterns and classes of fractional differential equations, please refer to Problem types

Step 2: Solving a Problem

After defining a problem, we can solve it by choosing a numerical algorithm and calling the solve function:

sol = solve(prob, Alg(), dt=0.01)

Note that there are different algorithms for different fractional differential equations, such as FODE, FDDE and DODE, we need to choose a suitable algorithm for a specific problem. For all the algorithms, please refer to the algorithms documentation.

Step 3: Analyzing the Solution

Simply call the Plots.jl to visualize the solution:

using Plots

Relaxation Oscillation