Current algorithms in FractionalDiffEq.jl

Fractional Ordinary Differential Equations

Single Term FODE

FractionalDiffEq.PECEType

Usage

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}
}
source
FractionalDiffEq.EulerType
solve(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}
}
source
FractionalDiffEq.PIEXType

Usage

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}
}
source

Multi-Term FODE

FractionalDiffEq.FODEMatrixDiscreteType

Usage

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}
}
source
FractionalDiffEq.ClosedFormType

Usage

solve(prob::MultiTermsFODEProblem, ClosedForm())

Use Closed-Form solution to obtain numerical solution at zero initial condition.

Reference:

Dingyu Xue, Northeastern University, China ISBN:9787030543981

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FractionalDiffEq.ClosedFormHankelMType

Usage

solve(prob::MultiTermsFODEProblem, ::ClosedFormHankelM)

Use Closed-Form Hankel matrix algorithm to obtain numerical solution at zero initial condition.

source
FractionalDiffEq.PIPECEType
solve(prob::MultiTermsFODEProblem, h, PIPECE())

Use product integration predictor-corrector method to solve multi-terms FODE.

source
FractionalDiffEq.PIIMRectType
solve(prob::MultiTermsFODEProblem, h, PIIMRect())

Use implicit product integration rectangular type method to solve multi-terms FODE.

source
FractionalDiffEq.PIIMTrapType
solve(prob::MultiTermsFODEProblem, h, PIIMTrap())

Use implicit product integration trapezoidal type method to solve multi-terms FODE.

source

System of FODE

FractionalDiffEq.NonLinearAlgType

Usage

solve(prob::FODESystem, h, NonLinearAlg())

Nonlinear algorithm for nonlinear fractional differential equations.

References

Dingyu Xue, Northeastern University, China ISBN:9787030543981

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Missing docstring.

Missing docstring for FractionalDiffEq.PECE. Check Documenter's build log for details.

FractionalDiffEq.GLType

Usage

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}}
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FractionalDiffEq.FLMMBDFType
solve(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}
}
source
FractionalDiffEq.FLMMNewtonGregoryType
solve(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}
}
source
FractionalDiffEq.FLMMTrapType
solve(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}
}
source
Missing docstring.

Missing docstring for FractionalDiffEq.PIEX. Check Documenter's build log for details.

FractionalDiffEq.NewtonPolynomialType
solve(prob::FODESystem, h, NewtonPolynomial())

Solve FODE system using Newton Polynomials methods.

Tip

Used for the Caputo Fabrizio fractional differential operators.

https://doi.org/10.1016/c2020-0-02711-8
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Fractional Delay Differential Equatinos

FractionalDiffEq.DelayPECEType

Usage

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}
}
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FractionalDiffEq.DelayPIType

Usage

solve(FDDE::FDDEProblem, h, DelayPI())

Use explicit rectangular product integration algorithm to solve an FDDE problem.

References

@article{2020,
   title={On initial conditions for fractional delay differential equations},
   ISSN={1007-5704},
   url={http://dx.doi.org/10.1016/j.cnsns.2020.105359},
   DOI={10.1016/j.cnsns.2020.105359},
   journal={Communications in Nonlinear Science and Numerical Simulation},
   author={Garrappa, Roberto and Kaslik, Eva},
   year={2020},
}
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FractionalDiffEq.MatrixFormType

Usage

solve(prob::FDDEMatrixProblem, h, MatrixForm())

Reference

https://github.com/mandresik/system-of-linear-fractional-differential-delayed-equations

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FractionalDiffEq.DelayABMType
solve(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}
}
source

Distributed Order Differential Equations

FractionalDiffEq.DOMatrixDiscreteType

Usage

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}
}
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Fractional Differences Equations

FractionalDiffEq.PECEDifferenceType

Usage

solve(FDProb::FractionalDifferenceProblem, T, h, PECEDifference())

Use the PECE algorithm to solve fractional difference equations

References

@article{陈福来2011分数阶微分差分方程的,
  title={分数阶微分差分方程的 Matlab 求解},
  author={陈福来 and 王华 and 李势丰},
  journal={湘南学院学报},
  volume={32},
  number={5},
  pages={1--4},
  year={2011}
}
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Missing docstring.

Missing docstring for FractionalDiffEq.GL. Check Documenter's build log for details.