Current algorithms in FractionalDiffEq.jl
Fractional Ordinary Differential Equations
Single Term FODE
FractionalDiffEq.PECE
— TypeUsage
solve(prob::SingleTermFODEProblem, h, PECE())
Predict-Evaluate-Correct-Evaluate algorithm.
For more details, please refer to Predictor-Corrector algorithms
This PECE algorithm is taken from Diethelm's paper.
References
@article{
title={A predictor-corrector approach for the numerical solution of fractional differential equations},
author={Diethelm, Kai and Ford, Neville J. and Freed, Alan D.}
doi={https://doi.org/10.1023/A:1016592219341}
}
FractionalDiffEq.Euler
— Typesolve(prob, h, Euler())
The basic forward Euler method for fractional ordinary differential equations.
@inproceedings{Li2015NumericalMF,
title={Numerical Methods for Fractional Calculus},
author={Changpin Li and Fanhai Zeng},
year={2015}
}
FractionalDiffEq.PIEX
— TypeUsage
solve(prob::SingleTermFODEProblem, h, PIEX())
References
@inproceedings{Garrappa2018NumericalSO,
title={Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial},
author={Roberto Garrappa},
year={2018}
}
FractionalDiffEq.AtanganaSeda
— Typesolve(prob::SingleTermFODEProblem, h, AS())
Atangana-Seda method for Caputo single term FODE.
Multi-Term FODE
FractionalDiffEq.FODEMatrixDiscrete
— TypeUsage
solve(prob::MultiTermsFODEProblem, h, FODEMatrixDiscrete())
Using triangular strip matrices to discrete fractional ordinary differential equations to simple algebra system and solve the system.
References
@inproceedings{Podlubny2000MATRIXAT,
title={MATRIX APPROACH TO DISCRETE FRACTIONAL CALCULUS},
author={Igor Podlubny},
year={2000}
}
FractionalDiffEq.ClosedForm
— TypeUsage
solve(prob::MultiTermsFODEProblem, ClosedForm())
Use Closed-Form solution to obtain numerical solution at zero initial condition.
Reference:
Dingyu Xue, Northeastern University, China ISBN:9787030543981
FractionalDiffEq.ClosedFormHankelM
— TypeUsage
solve(prob::MultiTermsFODEProblem, ::ClosedFormHankelM)
Use Closed-Form Hankel matrix algorithm to obtain numerical solution at zero initial condition.
FractionalDiffEq.PIPECE
— Typesolve(prob::MultiTermsFODEProblem, h, PIPECE())
Use product integration predictor-corrector method to solve multi-terms FODE.
FractionalDiffEq.PIRect
— Typesolve(prob::MultiTermsFODEProblem, h, PIRect())
Use implicit product integration rectangular type method to solve multi-terms FODE.
FractionalDiffEq.PITrap
— Typesolve(prob::MultiTermsFODEProblem, h, PITrap())
Use implicit product integration trapezoidal type method to solve multi-terms FODE.
System of FODE
FractionalDiffEq.NonLinearAlg
— TypeUsage
solve(prob::FODESystem, h, NonLinearAlg())
Nonlinear algorithm for nonlinear fractional differential equations.
References
Dingyu Xue, Northeastern University, China ISBN:9787030543981
Missing docstring for FractionalDiffEq.PECE
. Check Documenter's build log for details.
FractionalDiffEq.GL
— TypeUsage
solve(prob::SingleTermFODEProblem, h, GL())
Grunwald Letnikov method for fractional ordinary differential equations
@INPROCEEDINGS{8742063,
author={Clemente-López, D. and Muñoz-Pacheco, J. M. and Félix-Beltrán, O. G. and Volos, C.},
booktitle={2019 8th International Conference on Modern Circuits and Systems Technologies (MOCAST)},
title={Efficient Computation of the Grünwald-Letnikov Method for ARM-Based Implementations of Fractional-Order Chaotic Systems},
year={2019},
doi={10.1109/MOCAST.2019.8742063}}
FractionalDiffEq.FLMMBDF
— Typesolve(prob::FODESystem, h, FLMMBDF())
Use Backward Differentiation Formula generated weights fractional linear multi-steps method to solve system of FODE.
References
@article{Garrappa2015TrapezoidalMF,
title={Trapezoidal methods for fractional differential equations: Theoretical and computational aspects},
author={Roberto Garrappa},
journal={ArXiv},
year={2015},
volume={abs/1912.09878}
}
FractionalDiffEq.FLMMNewtonGregory
— Typesolve(prob::FODESystem, h, FLMMNewtonGregory())
Use Newton Gregory generated weights fractional linear multiple steps method to solve system of FODE.
References
@article{Garrappa2015TrapezoidalMF,
title={Trapezoidal methods for fractional differential equations: Theoretical and computational aspects},
author={Roberto Garrappa},
journal={ArXiv},
year={2015},
volume={abs/1912.09878}
}
FractionalDiffEq.FLMMTrap
— Typesolve(prob::FODEsystem, FLMMTrap())
Use Trapezoidal with generating function $f(x)=\frac{1+x}{2(1-x)^\alpha}$ generated weights fractional linear multiple steps method to solve system of FODE.
References
@article{Garrappa2015TrapezoidalMF,
title={Trapezoidal methods for fractional differential equations: Theoretical and computational aspects},
author={Roberto Garrappa},
journal={ArXiv},
year={2015},
volume={abs/1912.09878}
}
Missing docstring for FractionalDiffEq.PIEX
. Check Documenter's build log for details.
FractionalDiffEq.NewtonPolynomial
— Typesolve(prob::FODESystem, h, NewtonPolynomial())
Solve FODE system using Newton Polynomials methods.
Used for the Caputo Fabrizio fractional differential operators.
https://doi.org/10.1016/c2020-0-02711-8
FractionalDiffEq.AtanganaSedaAB
— Typesolve(prob::FODESystem, h, AtanganaSedaAB())
Solve Atangana-Baleanu fractional order differential equations using Newton Polynomials.
Fractional Delay Differential Equatinos
FractionalDiffEq.DelayPECE
— TypeUsage
solve(FDDE::FDDEProblem, h, DelayPECE())
Using the delayed predictor-corrector method to solve the delayed fractional differential equation problem in the Caputo sense.
Capable of solving both single term FDDE and multiple FDDE, support time varying lags of course😋.
References
@article{Wang2013ANM,
title={A Numerical Method for Delayed Fractional-Order Differential Equations},
author={Zhen Wang},
journal={J. Appl. Math.},
year={2013},
volume={2013},
pages={256071:1-256071:7}
}
@inproceedings{Nagy2014NUMERICALSF,
title={NUMERICAL SIMULATIONS FOR VARIABLE-ORDER FRACTIONAL NONLINEAR DELAY DIFFERENTIAL EQUATIONS},
author={Abdelhameed M. Nagy and Taghreed Abdul Rahman Assiri},
year={2014}
}
@inproceedings{Abdelmalek2019APM,
title={A Predictor-Corrector Method for Fractional Delay-Differential System with Multiple Lags},
author={Salem Abdelmalek and Redouane Douaifia},
year={2019}
}
Missing docstring for FractionalDiffEq.DelayPI
. Check Documenter's build log for details.
FractionalDiffEq.MatrixForm
— TypeUsage
solve(prob::FDDEMatrixProblem, h, MatrixForm())
Reference
https://github.com/mandresik/system-of-linear-fractional-differential-delayed-equations
FractionalDiffEq.DelayABM
— Typesolve(FDDE::FDDEProblem, h, DelayABM())
Use the Adams-Bashforth-Moulton method to solve fractional delayed differential equations.
References
@inproceedings{Bhalekar2011APS,
title={A PREDICTOR-CORRECTOR SCHEME FOR SOLVING NONLINEAR DELAY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER},
author={Sachin Bhalekar and Varsha Daftardar-Gejji},
year={2011}
}
Distributed Order Differential Equations
FractionalDiffEq.DOMatrixDiscrete
— TypeUsage
solve(prob, DOMatrixDiscrete())
Use distributed order strip matrix algorithm to solve distriubted order problem.
References
@inproceedings{Jiao2012DistributedOrderDS,
title={Distributed-Order Dynamic Systems - Stability, Simulation, Applications and Perspectives},
author={Zhuang Jiao and Yang Quan Chen and Igor Podlubny},
booktitle={Springer Briefs in Electrical and Computer Engineering},
year={2012}
}
Fractional Differences Equations
Missing docstring for FractionalDiffEq.PECEDifference
. Check Documenter's build log for details.
Missing docstring for FractionalDiffEq.GL
. Check Documenter's build log for details.