Current algorithms in FractionalDiffEq.jl

Fractional Ordinary Differential Equations

Single Term FODE

Missing docstring.

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

Missing docstring.

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

Missing docstring.

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

Missing docstring.

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

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

source
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.PIRectType
solve(prob::MultiTermsFODEProblem, h, PIRect())

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

source
FractionalDiffEq.PITrapType
solve(prob::MultiTermsFODEProblem, h, PITrap())

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

source

System of FODE

FractionalDiffEq.NonLinearAlgType

Usage

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

Nonlinear algorithm for nonlinear fractional differential equations.

References

Dingyu Xue, Northeastern University, China ISBN:9787030543981

source
Missing docstring.

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

Missing docstring.

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

FractionalDiffEq.FLMMBDFType
solve(prob::FODEProblem, 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::FODEProblem, 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::FODEProblem, 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::FODEProblem, 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
source

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}
}
source
Missing docstring.

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

FractionalDiffEq.MatrixFormType

Usage

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

Reference

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

source
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}
}
source

Fractional Differences Equations

Missing docstring.

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

Missing docstring.

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