Choosing a method

Time memory

BirkSong is the general default because it has fixed storage and accepts variable timesteps. Use FullHistory as a direct variable-step reference. Use BDF2 LubichCQ for classical uniform-grid convolution quadrature, and AlikhanovL21Sigma when the offset second-order PDE formula is appropriate. Use fast-oblivious CQ for long uniform-grid histories where its logarithmic scaling outweighs its contour-state constants.

Diethelm2008 is its direct alternative for narrow rate spans. SumOfExponentials is a supported alternative when the time interval and an absolute kernel tolerance are known in advance. The remaining diffusive representations are for literature comparisons. For single-exponential memory, see Exponential memory and Caputo-Fabrizio.

Spatial operators

The three spatial operators are different realizations, not interchangeable discretizations. The model determines which one applies. See choosing a spatial realization.

Method map

Operator	Method	What it is	Use	Primary source
Caputo and Riemann-Liouville	Cayley	Diffusive modes, Gauss-Jacobi after a Cayley map of selectable exponent	Family of the default BirkSong. Requires power or a declared time range	Generalizes Diethelm (2008) and Birk and Song (2010)
	Jacobi	Diffusive modes with independent low-rate and high-rate map exponents	Expert option for measured asymmetric rate ranges	Diethelm (2023)
	SumOfExponentials	Tolerance-driven positive exponential sum on a declared time interval	Supported alternative	Jiang et al. (2017)
	Diethelm2022	Truncated log-rate quadrature	Comparison only	Diethelm (2022), (2023)
	YuanAgrawal	Original Gauss-Laguerre rule	Comparison only	Yuan and Agrawal (2002)
	SineDiffusive	Generalized Gauss-Laguerre quadrature of undamped sine modes	Comparison only	Khosravian-Arab and Dehghan (2024)
	LubichCQ	Uniform-grid BDF1 or BDF2 convolution quadrature with starting corrections	Direct first- or second-order method	Lubich (1986), (1988, Part I), (1988, Part II)
	FastObliviousCQ	BDF1 CQ with dyadic Talbot-contour history	Long uniform-grid histories	Schädle, López-Fernández, and Lubich (2006)
	AlikhanovL21Sigma	Uniform-grid quadratic formula at \(t_{n+\sigma}\)	Direct second-order method	Alikhanov (2015)
	FullHistory	Variable-step L1 history integral	Reference method	Lin and Xu (2007)
Exponential memory	ExponentialMemory	One-timescale exponential convolution	Direct fading-memory operator	Standard one-state realization
Spectral fractional Laplacian	SpectralFractionalLaplacian	Power of the homogeneous-Dirichlet or natural-Neumann Laplacian	Spectral realization	Bonito and Pasciak (2015)
Riesz fractional Laplacian	RieszFractionalLaplacian	Whole-space integral on the zero extension	Integral realization	Acosta and Borthagaray (2017)
Periodic fractional Laplacian	PeriodicFractionalLaplacian	Fourier multiplier on a uniform periodic cell	Periodic realization	Standard Fourier-series multiplier

Convenience constructors

Published members of the families above. Each is the general class at a fixed setting and behaves identically to it.

Constructor	Equivalent to	Use	Primary source
BirkSong(n)	Cayley(n, power=4)	Default representation	Birk and Song (2010)
Diethelm2008(n)	Cayley(n, power=2)	Narrow rate spans	Diethelm (2008)
CaputoFabrizioOperator	Rescaled ExponentialMemory	Labelled Caputo-Fabrizio interface	Caputo and Fabrizio (2015), with the classification in Ortigueira and Machado (2018)