diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 6252144c..929a315c 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2025-01-03T17:42:30","documenter_version":"1.8.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2025-01-13T15:04:25","documenter_version":"1.8.0"}} \ No newline at end of file diff --git a/dev/ChaosGallery/Gallery/Arneodo/index.html b/dev/ChaosGallery/Gallery/Arneodo/index.html index ea7aa102..333845c5 100644 --- a/dev/ChaosGallery/Gallery/Arneodo/index.html +++ b/dev/ChaosGallery/Gallery/Arneodo/index.html @@ -14,4 +14,4 @@ prob = FODESystem(Arneodo!, alpha, x0, tspan) sol = solve(prob, h, GL()) -plot(sol, vars=(1,2,3), title="Fractional Order Arneodo System")

Arneodo

This page was generated using DemoCards.jl.

+plot(sol, vars=(1,2,3), title="Fractional Order Arneodo System")

Arneodo

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/Brusselator/index.html b/dev/ChaosGallery/Gallery/Brusselator/index.html index f0619956..7f46725e 100644 --- a/dev/ChaosGallery/Gallery/Brusselator/index.html +++ b/dev/ChaosGallery/Gallery/Brusselator/index.html @@ -24,4 +24,4 @@ plot(sol, vars=(1,2)) # Time plane -plot(sol, vars=(0,1,2))

BrusselatorPhase

BrusselatorTime

This page was generated using DemoCards.jl.

+plot(sol, vars=(0,1,2))

BrusselatorPhase

BrusselatorTime

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/Chen/index.html b/dev/ChaosGallery/Gallery/Chen/index.html index e01b32f4..c0dd9dc2 100644 --- a/dev/ChaosGallery/Gallery/Chen/index.html +++ b/dev/ChaosGallery/Gallery/Chen/index.html @@ -18,4 +18,4 @@ prob = FODESystem(Chen!, alpha, x0, tspan) sol = solve(prob, h, GL()) -plot(sol, vars=(1,2,3), title="Fractional Order Chen System")

Chen

This page was generated using DemoCards.jl.

+plot(sol, vars=(1,2,3), title="Fractional Order Chen System")

Chen

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/Duffing/index.html b/dev/ChaosGallery/Gallery/Duffing/index.html index edc27360..b717d05b 100644 --- a/dev/ChaosGallery/Gallery/Duffing/index.html +++ b/dev/ChaosGallery/Gallery/Duffing/index.html @@ -13,4 +13,4 @@ prob = FODESystem(Duffing!, alpha, x0, tspan) sol = solve(prob, h, GL()) -plot(sol, vars=(1,2), title="Fractional Order Duffing System")

Duffing

This page was generated using DemoCards.jl.

+plot(sol, vars=(1,2), title="Fractional Order Duffing System")

Duffing

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/FNeuralNetwork/index.html b/dev/ChaosGallery/Gallery/FNeuralNetwork/index.html index 5a0082da..d182568c 100644 --- a/dev/ChaosGallery/Gallery/FNeuralNetwork/index.html +++ b/dev/ChaosGallery/Gallery/FNeuralNetwork/index.html @@ -13,4 +13,4 @@ prob = FractionalDifferenceSystem(sys!, 0.98, [1, -1, 0]) result = solve(prob, 7, GL()) -plot(result[1, :], result[2, :], result[3, :], seriestype=:scatter)

FNetwork

This page was generated using DemoCards.jl.

+plot(result[1, :], result[2, :], result[3, :], seriestype=:scatter)

FNetwork

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/Genesio-Tesi/index.html b/dev/ChaosGallery/Gallery/Genesio-Tesi/index.html index 7c11b2f9..07bc6f39 100644 --- a/dev/ChaosGallery/Gallery/Genesio-Tesi/index.html +++ b/dev/ChaosGallery/Gallery/Genesio-Tesi/index.html @@ -14,4 +14,4 @@ prob = FODESystem(GenesioTesi!, alpha, x0, tspan) sol = solve(prob, h, GL()) -plot(sol, vars=(1,2,3), title="Fractional Order Genesio-Tesi System")

Genesio-Tesi

This page was generated using DemoCards.jl.

+plot(sol, vars=(1,2,3), title="Fractional Order Genesio-Tesi System")

Genesio-Tesi

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/Hadley/index.html b/dev/ChaosGallery/Gallery/Hadley/index.html index 7ce283af..f7fe815a 100644 --- a/dev/ChaosGallery/Gallery/Hadley/index.html +++ b/dev/ChaosGallery/Gallery/Hadley/index.html @@ -17,4 +17,4 @@ prob = FODESystem(fun, α, u0, (t0, tfinal)) sol = solve(prob, h, AtanganaSedaAB()) -plot(sol, vars=(1,2,3) title="Fractional Order Hadley System")

Hadley

This page was generated using DemoCards.jl.

+plot(sol, vars=(1,2,3) title="Fractional Order Hadley System")

Hadley

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/Labyrinth/index.html b/dev/ChaosGallery/Gallery/Labyrinth/index.html index 93a8326d..fb5891a3 100644 --- a/dev/ChaosGallery/Gallery/Labyrinth/index.html +++ b/dev/ChaosGallery/Gallery/Labyrinth/index.html @@ -17,4 +17,4 @@ prob = FODESystem(fun, α, u0, (t0, tfinal)) sol = solve(prob, h, NewtonPolynomial()) -plot(sol, vars=(1,2,3), title="Fractional Order Labyrinth System")

Labyrinth

This page was generated using DemoCards.jl.

+plot(sol, vars=(1,2,3), title="Fractional Order Labyrinth System")

Labyrinth

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/Liu/index.html b/dev/ChaosGallery/Gallery/Liu/index.html index e9864f66..de26922f 100644 --- a/dev/ChaosGallery/Gallery/Liu/index.html +++ b/dev/ChaosGallery/Gallery/Liu/index.html @@ -14,4 +14,4 @@ prob = FODESystem(Liu!, alpha, x0, tspan) sol = solve(prob, h, GL()) -plot3d(sol, vars=(1,2,3), title="Fractional Order Liu System")

Liu

This page was generated using DemoCards.jl.

+plot3d(sol, vars=(1,2,3), title="Fractional Order Liu System")

Liu

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/LotkaVolterra/index.html b/dev/ChaosGallery/Gallery/LotkaVolterra/index.html index e625b4a0..f2642759 100644 --- a/dev/ChaosGallery/Gallery/LotkaVolterra/index.html +++ b/dev/ChaosGallery/Gallery/LotkaVolterra/index.html @@ -14,4 +14,4 @@ prob = FODESystem(LotkaVolterra!, alpha, x0, tspan) sol = solve(prob, h, GL()) -plot(sol, vars=(1,2,3), title="Fractional Order Lotka Volterra System")

LotkaVolterra

This page was generated using DemoCards.jl.

+plot(sol, vars=(1,2,3), title="Fractional Order Lotka Volterra System")

LotkaVolterra

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/Lu/index.html b/dev/ChaosGallery/Gallery/Lu/index.html index c916f955..956c50f3 100644 --- a/dev/ChaosGallery/Gallery/Lu/index.html +++ b/dev/ChaosGallery/Gallery/Lu/index.html @@ -14,4 +14,4 @@ prob = FODESystem(Lu!, alpha, x0, tspan) sol = solve(prob, h, GL()) -plot(sol, vars=(1,2,3), title="Fractional Order Lu System")

Lu

This page was generated using DemoCards.jl.

+plot(sol, vars=(1,2,3), title="Fractional Order Lu System")

Lu

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/Qi/index.html b/dev/ChaosGallery/Gallery/Qi/index.html index bf4e6078..de357b69 100644 --- a/dev/ChaosGallery/Gallery/Qi/index.html +++ b/dev/ChaosGallery/Gallery/Qi/index.html @@ -22,4 +22,4 @@ prob=FODESystem(Qi!, alpha, x0, tspan) sol = solve(prob, h, GL()) -plot(sol, vars=(1,2,3), title="Fractional Order Qi System")

Rossler

This page was generated using DemoCards.jl.

+plot(sol, vars=(1,2,3), title="Fractional Order Qi System")

Rossler

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/Rossler/index.html b/dev/ChaosGallery/Gallery/Rossler/index.html index f963f6b1..4398ac43 100644 --- a/dev/ChaosGallery/Gallery/Rossler/index.html +++ b/dev/ChaosGallery/Gallery/Rossler/index.html @@ -15,4 +15,4 @@ sol = solve(prob, h, GL()) plot(sol, vars=(1,2,3), title="Fractional Order Rossler System") -

Rossler

This page was generated using DemoCards.jl.

+

Rossler

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/Gallery/VanderPol/index.html b/dev/ChaosGallery/Gallery/VanderPol/index.html index 06168c7c..7f474e6d 100644 --- a/dev/ChaosGallery/Gallery/VanderPol/index.html +++ b/dev/ChaosGallery/Gallery/VanderPol/index.html @@ -13,4 +13,4 @@ prob = FODESystem(VanderPol!, alpha, x0, tspan) sol = solve(prob, h, GL()) -plot(sol, vars=(1,2), title="Fractional Order Van der Pol Oscillator")

VanderPol

This page was generated using DemoCards.jl.

+plot(sol, vars=(1,2), title="Fractional Order Van der Pol Oscillator")

VanderPol

This page was generated using DemoCards.jl.

diff --git a/dev/ChaosGallery/index.html b/dev/ChaosGallery/index.html index 4d40f491..d24b8d2a 100644 --- a/dev/ChaosGallery/index.html +++ b/dev/ChaosGallery/index.html @@ -167,4 +167,4 @@

card image

- + diff --git a/dev/algorithms/index.html b/dev/algorithms/index.html index 6f6bc1b5..88f02aff 100644 --- a/dev/algorithms/index.html +++ b/dev/algorithms/index.html @@ -1,2 +1,2 @@ -Algorithms · FractionalDiffEq.jl

Current algorithms in FractionalDiffEq.jl

Fractional Ordinary Differential Equations

Multi-Term FODE

MatrixDiscrete ClosedForm ClosedFormHankelM MTPIEX MTPIPECE MTPIRect MTPITrap

System of FODE

NonLinearAlg Euler PECE GL BDF NewtonGregory Trapezoid PIEX PIRect PITrap NewtonPolynomial AtanganaSedaAB

Fractional Delay Differential Equations

DelayPECE DelayPI MatrixForm DelayABM

Distributed Order Differential Equations

DOMatrixDiscrete

Fractional Differences Equations

PECEDifference GL

+Algorithms · FractionalDiffEq.jl

Current algorithms in FractionalDiffEq.jl

Fractional Ordinary Differential Equations

Multi-Term FODE

MatrixDiscrete ClosedForm ClosedFormHankelM MTPIEX MTPIPECE MTPIRect MTPITrap

System of FODE

NonLinearAlg Euler PECE GL BDF NewtonGregory Trapezoid PIEX PIRect PITrap NewtonPolynomial AtanganaSedaAB

Fractional Delay Differential Equations

DelayPECE DelayPI MatrixForm DelayABM

Distributed Order Differential Equations

DOMatrixDiscrete

Fractional Differences Equations

PECEDifference GL

diff --git a/dev/comparison/index.html b/dev/comparison/index.html index d7b7a61e..382e84be 100644 --- a/dev/comparison/index.html +++ b/dev/comparison/index.html @@ -1,2 +1,2 @@ -Comparison · FractionalDiffEq.jl

Comparison with other toolboxes

There are also some toolboxes that can be used for fractional differential equations, here we make a survey of all these toolboxes and have a more intuitive perspective.

Matlab

FOTF toolbox

FOTF toolbox is a Matlab toolbox developed by Prof Dingyu Xue, with fractional calculus, fractional differential equations and fractional order systems integrated together. To be honest, FOTF has a great impact on FractionalDiffEq.jl, all of the algorithms in FOTF toolbox is all supported in FractionalDiffEq.jl

corresponding APIs:

FOTF toolboxFractionalDiffEq.jl
fode_sol9ClosedForm
fode_solmClosedFormHankelM
nlfode_vecNonLinearAlg

Matrix approach to discretization of ODEs and PDEs of arbitrary real order

The corresponding matrix discretization paper and toolbox is developed by Prof Igor Podlubny. The corresponding matrix discretization method in FractionalDiffEq.jl is FODEMatrixDiscrete and FPDEMatrixDiscrete.

FLMM2 Toolbox

FLMM2 toolbox is a toolbox developed by Prof Roberto Garrappa together with his paper

ExsitingFractionalDiffEq.jl
FLMM2(method=1)Trap
FLMM2(method=2)NewtonGregory
FLMM2(method=3)BDF
fde12PECE
mt_fde_pi1_exMTPIEX
mt_fde_pi1_immTPIRect
mt_fde_pi2_imMTPITrap
mt_fde_pi12_pcMTPECE

Python

fodeint: With explicit one-step Adams-Bashforth (Euler) method.

R

Don't see any packages for fractional differential equations.

+Comparison · FractionalDiffEq.jl

Comparison with other toolboxes

There are also some toolboxes that can be used for fractional differential equations, here we make a survey of all these toolboxes and have a more intuitive perspective.

Matlab

FOTF toolbox

FOTF toolbox is a Matlab toolbox developed by Prof Dingyu Xue, with fractional calculus, fractional differential equations and fractional order systems integrated together. To be honest, FOTF has a great impact on FractionalDiffEq.jl, all of the algorithms in FOTF toolbox is all supported in FractionalDiffEq.jl

corresponding APIs:

FOTF toolboxFractionalDiffEq.jl
fode_sol9ClosedForm
fode_solmClosedFormHankelM
nlfode_vecNonLinearAlg

Matrix approach to discretization of ODEs and PDEs of arbitrary real order

The corresponding matrix discretization paper and toolbox is developed by Prof Igor Podlubny. The corresponding matrix discretization method in FractionalDiffEq.jl is FODEMatrixDiscrete and FPDEMatrixDiscrete.

FLMM2 Toolbox

FLMM2 toolbox is a toolbox developed by Prof Roberto Garrappa together with his paper

ExsitingFractionalDiffEq.jl
FLMM2(method=1)Trap
FLMM2(method=2)NewtonGregory
FLMM2(method=3)BDF
fde12PECE
mt_fde_pi1_exMTPIEX
mt_fde_pi1_immTPIRect
mt_fde_pi2_imMTPITrap
mt_fde_pi12_pcMTPECE

Python

fodeint: With explicit one-step Adams-Bashforth (Euler) method.

R

Don't see any packages for fractional differential equations.

diff --git a/dev/discrete/index.html b/dev/discrete/index.html index 1a0226f1..77dee877 100644 --- a/dev/discrete/index.html +++ b/dev/discrete/index.html @@ -20,4 +20,4 @@ prob = FractionalDsicreteSystem(sys!, 0.98, [1, -1, 0]) result = solve(prob, 7, GL()) -plot(result[1, :], result[2, :], result[3, :], seriestype=:scatter)

FNN

+plot(result[1, :], result[2, :], result[3, :], seriestype=:scatter)

FNN

diff --git a/dev/dode/index.html b/dev/dode/index.html index de9e8557..d0cafb60 100644 --- a/dev/dode/index.html +++ b/dev/dode/index.html @@ -16,4 +16,4 @@ f(t) = 0 ≤ t ≤ 1 ? (return 8) : (return 0) prob=DODEProblem([1, 1, 1], [2, x->6*x*(1-x), 0], (0, 1), f, [0; 0], t) sol=solve(prob, h, DOMatrixDiscrete()) -plot(sol)

dobagleytorvik

Please see Distributed-Order Dynamic Systems for systematic introduction and knowledge.

+plot(sol)

dobagleytorvik

Please see Distributed-Order Dynamic Systems for systematic introduction and knowledge.

diff --git a/dev/fdde/index.html b/dev/fdde/index.html index fbe0c82f..f80d2f09 100644 --- a/dev/fdde/index.html +++ b/dev/fdde/index.html @@ -60,4 +60,4 @@ prob = FDDEMatrixProblem(α, τ, A, B, f, x0, tspan) sol=solve(prob, h, MatrixForm()) -plot(sol[:, 1], sol[:, 3])

Matrix Form

+plot(sol[:, 1], sol[:, 3])

Matrix Form

diff --git a/dev/fdesystem/index.html b/dev/fdesystem/index.html index 75c49d4f..87960a70 100644 --- a/dev/fdesystem/index.html +++ b/dev/fdesystem/index.html @@ -16,4 +16,4 @@ 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)

Chua

Cheers!🎉🎉🎉

Short memory effect in FDE

Why we use short memory effect in simulation?

While the Chua system is a real life chaos system, when we want to simulate the system more to see the system more clearly, we must increase the simulating time $t_n$, however, limited by the fact that the hardware resources and the computing capability can't increase endlessly, we need to use short memory effect to help us improve the simulating efficiency.

To further elaborate, we can look at how the short memory affects the simulation:

By using the same code above, but set $t_n=500$ and memory length as $L_0=10000$ to see the model more comprehensively but reduce the computing cost same time:

result = solve(prob, h, tn, NonLinearAlg(), 10000)

Chua_short_memory

More examples

For more fractional order chaotic systems, please see Chaos Gallery😉

+plot(sol, vars=(1, 2) title="Chua System", legend=:bottomright)

Chua

Cheers!🎉🎉🎉

Short memory effect in FDE

Why we use short memory effect in simulation?

While the Chua system is a real life chaos system, when we want to simulate the system more to see the system more clearly, we must increase the simulating time $t_n$, however, limited by the fact that the hardware resources and the computing capability can't increase endlessly, we need to use short memory effect to help us improve the simulating efficiency.

To further elaborate, we can look at how the short memory affects the simulation:

By using the same code above, but set $t_n=500$ and memory length as $L_0=10000$ to see the model more comprehensively but reduce the computing cost same time:

result = solve(prob, h, tn, NonLinearAlg(), 10000)

Chua_short_memory

More examples

For more fractional order chaotic systems, please see Chaos Gallery😉

diff --git a/dev/ffode/index.html b/dev/ffode/index.html index f069e33a..527d4f57 100644 --- a/dev/ffode/index.html +++ b/dev/ffode/index.html @@ -30,4 +30,4 @@ prob = FFODEProblem(fun, [alpha, β], u0, (1, tfinal)) sol = solve(prob, h, AtanganaSeda()) -plot3d(sol, vars=(1,2,3))

CFVariable

+plot3d(sol, vars=(1,2,3))

CFVariable

diff --git a/dev/flyapunovexp/index.html b/dev/flyapunovexp/index.html index 75f78f7f..5a4e532f 100644 --- a/dev/flyapunovexp/index.html +++ b/dev/flyapunovexp/index.html @@ -47,4 +47,4 @@ du[3] = -2*u[3]*(0.98+u[1]*u[2]) end LE = FOLyapunov(LE_RF_TEST, [0.995, 0.992, 0.996], 0, 0.1, 1000, [1,1,1], 0.01, 1000) -plot(LE) +plot(LE) diff --git a/dev/get_started/index.html b/dev/get_started/index.html index 7c7a8a01..1299f6bb 100644 --- a/dev/get_started/index.html +++ b/dev/get_started/index.html @@ -8,4 +8,4 @@ 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
-plot(sol)

Relaxation Oscillation

+plot(sol)

Relaxation Oscillation

diff --git a/dev/index.html b/dev/index.html index 931a3502..5b2542ca 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -FractionalDiffEq.jl · FractionalDiffEq.jl

FractionalDiffEq.jl

Hello there👋!

FractionalDiffEq.jl is a Julia package aiming at solving Fractional Differential Equations using high performance numerical methods.

Star Us

If you think FractionalDiffEq.jl is useful, please star us on GitHub, it means a lot to us!

Installation

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

pkg> add FractionalDiffEq

Features

While most fractional differential equations solvers are implemented using Matlab, FractionalDiffEq.jl is totally driven by Julia and licensed with MIT License, ensuring its everlasting development and open source. Benefit from the advancing features of JuliaLang, FractionalDiffEq.jl has impressive performance and high speed, helps the computing more efficient and robust. Also fractionalDiffEq.jl has the most solvers for FDE available.

  • Fractional ordinary differential equations(FODE), single-term FODE, Multi-term FODE, nonlinear system of FODE(Caputo sense, Caputo-Fabrizio sense, Atangana-Baleanu sense, etc), variable order FODE.
  • Fractional delay differential equations(FDDE), single lag FDDE, multiple lags FDDE, time-varying lags FDDE, variable order FDDE, nonlinear system of FDDE.
  • Fractional difference equations, single-term fractional difference equations and nonlinear systems of fractional difference equations.
  • Distributed order differential equations(DODE), single-term DODE and multi-term DODE.

Roadmap

  • More algorithms are planned to support.

  • Improve benchmark.

  • Connect with SciML ecosystem.

  • More interesting ideas~

Contributing

Just by using FractionalDiffEq.jl you're already contributing!

The development of FractionalDiffEq.jl is on GitHub, please report bugs or send pull requests to help FractionalDiffEq.jl grow.

+FractionalDiffEq.jl · FractionalDiffEq.jl

FractionalDiffEq.jl

Hello there👋!

FractionalDiffEq.jl is a Julia package aiming at solving Fractional Differential Equations using high performance numerical methods.

Star Us

If you think FractionalDiffEq.jl is useful, please star us on GitHub, it means a lot to us!

Installation

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

pkg> add FractionalDiffEq

Features

While most fractional differential equations solvers are implemented using Matlab, FractionalDiffEq.jl is totally driven by Julia and licensed with MIT License, ensuring its everlasting development and open source. Benefit from the advancing features of JuliaLang, FractionalDiffEq.jl has impressive performance and high speed, helps the computing more efficient and robust. Also fractionalDiffEq.jl has the most solvers for FDE available.

  • Fractional ordinary differential equations(FODE), single-term FODE, Multi-term FODE, nonlinear system of FODE(Caputo sense, Caputo-Fabrizio sense, Atangana-Baleanu sense, etc), variable order FODE.
  • Fractional delay differential equations(FDDE), single lag FDDE, multiple lags FDDE, time-varying lags FDDE, variable order FDDE, nonlinear system of FDDE.
  • Fractional difference equations, single-term fractional difference equations and nonlinear systems of fractional difference equations.
  • Distributed order differential equations(DODE), single-term DODE and multi-term DODE.

Roadmap

  • More algorithms are planned to support.

  • Improve benchmark.

  • Connect with SciML ecosystem.

  • More interesting ideas~

Contributing

Just by using FractionalDiffEq.jl you're already contributing!

The development of FractionalDiffEq.jl is on GitHub, please report bugs or send pull requests to help FractionalDiffEq.jl grow.

diff --git a/dev/mittagleffler/index.html b/dev/mittagleffler/index.html index 9c1f0db5..84248020 100644 --- a/dev/mittagleffler/index.html +++ b/dev/mittagleffler/index.html @@ -2,4 +2,4 @@ Mittag Leffler Function · FractionalDiffEq.jl

Mittag Leffler function

For more intreresting topics and applications about Mittag Leffler function, we recommend you to read Mittag-Leffler Functions, Related Topics and Applications

Being called "The Queen Function of the Fractional Calculus", Mittag Leffler function plays an important role in fractional order computing and modeling. Here, we classify the Mittag Leffler function into three types:

Classical Mittag Leffler function(Single-Parametric version)

The Mittag Leffler function is defined by Gösta Magnus Mittag-Leffler by a power series as:

\[E_\alpha(z)=\sum_{k=0}^{\infty}\frac{z^k}{\Gamma(\alpha k+1)},\ (\alpha\in\mathbb{C})\]

Two-Parametric Mittag Leffler function

The two-parametric Mittag Leffler function as the generalization of the classical Mittag Leffler function :

\[E_{\alpha,\ \beta}(z)=\sum_{k=0}^{\infty}\frac{z^k}{\Gamma(\alpha k+\beta)},\ (\mathfrak{Re}(\alpha)>0,\ \beta\in\mathbb{C})\]

Three-Parametric Mittag Leffler function

FractionalDiffEq.jl also supports the three-parametric Mittag Leffler function(The Prabhakar function):

\[E_{\alpha,\ \beta}^\gamma(z)=\sum_{k=0}^{\infty}\frac{(\gamma)_k z^k}{k!\Gamma(\alpha k+\beta)},\ (\mathfrak{Re}(\alpha)>0,\ \mathfrak{Re}(\beta)>0,\ \gamma\in\mathbb{C})\]

Here $(\gamma)_k=\frac{\Gamma(\gamma+k)}{\Gamma(\gamma)}$ is the Pochhammer symbol.

In FractionalDiffEq.jl, you can compute the three types of Mittag Leffler functions by calling:

julia> mittleff(α, z)
 julia> mittleff(α, β, z)
 julia> mittleff(α, β, γ, z)

Also, you can compute the derivative of the Mittag Leffler function by calling:

julia> mittleffderiv(α, β, z)

To compute Mittag Leffler function error:

julia> mittlefferr(α, z, ρ)
-julia> mittlefferr(α, β, z, ρ)

Here, ρ is the targeting accuracy.

Different order classical Mittag Leffler function plot($0<\alpha<1$):

MittLeff

And also $1<\alpha<2$:

MittagLeffler

If we input complex number to Mittag Leffler function, for example:

\[E_{0.8, 0.9}(z)\]

MittLeff3D

+julia> mittlefferr(α, β, z, ρ)

Here, ρ is the targeting accuracy.

Different order classical Mittag Leffler function plot($0<\alpha<1$):

MittLeff

And also $1<\alpha<2$:

MittagLeffler

If we input complex number to Mittag Leffler function, for example:

\[E_{0.8, 0.9}(z)\]

MittLeff3D

diff --git a/dev/multiterm/index.html b/dev/multiterm/index.html index dcee572e..ef301064 100644 --- a/dev/multiterm/index.html +++ b/dev/multiterm/index.html @@ -6,4 +6,4 @@ 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)

By solving the equation and plotting the result, we can see its solution here:

Solution

Example in GitHub

This example is an official example in the source code, please see the example folder

+plot(sol, legend=:bottomright)

By solving the equation and plotting the result, we can see its solution here:

Solution

Example in GitHub

This example is an official example in the source code, please see the example folder

diff --git a/dev/ode/index.html b/dev/ode/index.html index b4ca1115..31a11010 100644 --- a/dev/ode/index.html +++ b/dev/ode/index.html @@ -17,4 +17,4 @@ multisol = FractionalDiffEq.solve(multiprob, 0.01, PIPECE()) analytical(x)=exp(4*x)*(3-2*exp(x)) plot(multisol) -plot!(collect(0:0.01:1), analytical.(collect(0:0.01:1)), ls=:dash)

multitermsode

+plot!(collect(0:0.01:1), analytical.(collect(0:0.01:1)), ls=:dash)

multitermsode

diff --git a/dev/problems/index.html b/dev/problems/index.html index 2f8806e0..daab5ffc 100644 --- a/dev/problems/index.html +++ b/dev/problems/index.html @@ -14,7 +14,7 @@ using FractionalDiffEq sol = solve(prob, PIEX()) using Plots; -plot(sol, vars = (1, 2, 3));source

Multi-terms FODE problems

FractionalDiffEq.MultiTermsFODEProblem — Type

Defines an multiple terms linear fractional ordinary differential equation (FODE) problem.

Mathematical Specification of an multi-terms FODE problem

To define an multi-terms FODE Problem, you simply need to given the parameters, their correspoding orders, right hand side function and the initial condition $u_0$ which define an FODE:

\[\frac{du^\alpha}{d^{\alpha}t} = f(u,p,t)\]

Multiple terms fractional order differential equations.

source

FDDE Problem

FractionalDiffEq.FDDEProblem — Type

Defines a fractional delay differential equation (FDDE) problem.

Mathematical Specification of an FDDE problem

To define an FDDE problem, you simply need to given the function $f$, the history function $ϕ$, the fractional order $α$, the delay $τ$ and the initial condition $u₀$ which define an FDDE:

\[D^\alpha_t y(t)=f(t,y(t),y(t-τ)),t ≥ 0\]

\[y(t)=ϕ(t), t ≤ 0\]

There are two different ways of specifying f:

  • f(du,u,h,p,t): The function describing fractional delay differential equations.
  • f(u,h,p,t): returning du. Less memory-efficient way, particularly suitable when mutation is not allowed (e.g. with certain automatic differentiation packages such as Zygote).
  • ϕ: History function
  • α: The fractional order of the differential equations, commensurate and non-commensurate is both supported.
  • τ: The time delay of the differential equations.

Problem Type

Constructors

FODEProblem can be constructed by first building an ODEFunction or by simply passing the FODE right-hand side to the constructor. The constructors are:

  • FDDEProblem(f::ODEFunction,order,u0,ϕ,tspan,p=NullParameters();kwargs...)
  • FDDEProblem{isinplace,specialize}(f,order,u0,ϕ,tspan,p=NullParameters();kwargs...) : Defines the FODE with the specified functions. isinplace optionally sets whether the function is inplace or not. This is determined automatically, but not inferred. specialize optionally controls the specialization level. See the specialization levels section of the SciMLBase documentation for more details. The default is AutoSpecialize.

For more details on the in-place and specialization controls, see the ODEFunction documentation.

Parameters are optional, and if not given, then a NullParameters() singleton will be used which will throw nice errors if you try to index non-existent parameters. Any extra keyword arguments are passed on to the solvers. For example, if you set a callback in the problem, then that callback will be added in every solve call.

For specifying Jacobians and mass matrices, see the ODEFunction documentation.

Fields

  • f: The function in the FDDE.
  • ϕ: The history function in the FDDE.
  • order: The order of the FDDE.
  • τ: The time delay of the FDDE.
  • tspan: The timespan for the problem.
  • p: The parameters.
  • kwargs: The keyword arguments passed onto the solves.

Example Problem

function ϕ(p, t)
+plot(sol, vars = (1, 2, 3));
source

Multi-terms FODE problems

FractionalDiffEq.MultiTermsFODEProblem — Type

Defines an multiple terms linear fractional ordinary differential equation (FODE) problem.

Mathematical Specification of an multi-terms FODE problem

To define an multi-terms FODE Problem, you simply need to given the parameters, their correspoding orders, right hand side function and the initial condition $u_0$ which define an FODE:

\[\frac{du^\alpha}{d^{\alpha}t} = f(u,p,t)\]

Multiple terms fractional order differential equations.

source

FDDE Problem

FractionalDiffEq.FDDEProblem — Type

Defines a fractional delay differential equation (FDDE) problem.

Mathematical Specification of an FDDE problem

To define an FDDE problem, you simply need to given the function $f$, the history function $ϕ$, the fractional order $α$, the delay $τ$ and the initial condition $u₀$ which define an FDDE:

\[D^\alpha_t y(t)=f(t,y(t),y(t-τ)),t ≥ 0\]

\[y(t)=ϕ(t), t ≤ 0\]

There are two different ways of specifying f:

  • f(du,u,h,p,t): The function describing fractional delay differential equations.
  • f(u,h,p,t): returning du. Less memory-efficient way, particularly suitable when mutation is not allowed (e.g. with certain automatic differentiation packages such as Zygote).
  • ϕ: History function
  • α: The fractional order of the differential equations, commensurate and non-commensurate is both supported.
  • τ: The time delay of the differential equations.

Problem Type

Constructors

FODEProblem can be constructed by first building an ODEFunction or by simply passing the FODE right-hand side to the constructor. The constructors are:

  • FDDEProblem(f::ODEFunction,order,u0,ϕ,tspan,p=NullParameters();kwargs...)
  • FDDEProblem{isinplace,specialize}(f,order,u0,ϕ,tspan,p=NullParameters();kwargs...) : Defines the FODE with the specified functions. isinplace optionally sets whether the function is inplace or not. This is determined automatically, but not inferred. specialize optionally controls the specialization level. See the specialization levels section of the SciMLBase documentation for more details. The default is AutoSpecialize.

For more details on the in-place and specialization controls, see the ODEFunction documentation.

Parameters are optional, and if not given, then a NullParameters() singleton will be used which will throw nice errors if you try to index non-existent parameters. Any extra keyword arguments are passed on to the solvers. For example, if you set a callback in the problem, then that callback will be added in every solve call.

For specifying Jacobians and mass matrices, see the ODEFunction documentation.

Fields

  • f: The function in the FDDE.
  • ϕ: The history function in the FDDE.
  • order: The order of the FDDE.
  • τ: The time delay of the FDDE.
  • tspan: The timespan for the problem.
  • p: The parameters.
  • kwargs: The keyword arguments passed onto the solves.

Example Problem

function ϕ(p, t)
     if t == 0
         return 19.00001
     else
@@ -30,4 +30,4 @@
 tspan = (0.0, 2.0)
 dt = 0.5
 prob = FDDEProblem(f, order, u0, ϕ, constant_lags = τ, tspan)
-sol = solve(prob, DelayPIEX(), dt = dt)
source

FFODE Problem

FFPODEProblem FFEODEProblem FFMODEProblem

DODE Problem

FractionalDiffEq.DODEProblem — Type

Defines an distributed order fractional ordinary differential equation (DODE) problem.

Mathematical Specification of an distributed order FODE problem

To define an multi-terms FODE Problem, you simply need to given the parameters, their correspoding orders, right hand side function and the initial condition $u_0$ which define an FODE:

\[\frac{du^\alpha}{d^{\alpha}t} = f(u,p,t)\]

Distributed order differential equations.

source

Fractional Difference Problem

FractionalDifferenceProblem FractionalDifferenceSystem

+sol = solve(prob, DelayPIEX(), dt = dt)source

FFODE Problem

FFPODEProblem FFEODEProblem FFMODEProblem

DODE Problem

FractionalDiffEq.DODEProblem — Type

Defines an distributed order fractional ordinary differential equation (DODE) problem.

Mathematical Specification of an distributed order FODE problem

To define an multi-terms FODE Problem, you simply need to given the parameters, their correspoding orders, right hand side function and the initial condition $u_0$ which define an FODE:

\[\frac{du^\alpha}{d^{\alpha}t} = f(u,p,t)\]

Distributed order differential equations.

source

Fractional Difference Problem

FractionalDifferenceProblem FractionalDifferenceSystem