References and provenance

Yonderdrake is released under the MIT License. Its numerical methods are implemented from the primary literature and checked against independently written reference calculations.

Which method comes from which paper is tabulated in Method map. This page is the bibliography, plus the provenance of the parts that are not transcriptions of a published scheme.

Yonderdrake-derived components

Component

Provenance

Recurrence

Exact variation-of-constants update of the cited diffusive modes under Yonderdrake’s linear time interpolant

AuxiliaryODE

Direct backward-Euler or trapezoidal discretization of the cited diffusive mode ODEs

RiemannLiouvilleDerivative initial trace

Exact trace term added to the cited Caputo approximations, following Yuan, Gao, Xiu, and Shi (2020)

ExponentialMemory

Exact internal-state realization of an exponential convolution

CaputoFabrizioOperator

Caputo and Fabrizio (2015), classified as a one-pole non-fractional operator by Ortigueira and Machado (2018)

PeriodicFractionalLaplacian grid mapping

Collective reconstruction and validation of a Firedrake periodic mesh against its logical FFT grid

Triangle-supported Riesz action and edge-sector rule

Yonderdrake derivation from the cited integral operator, independently verified

The Fourier-consistent singular-integral normalization is fixed in Conventions and normalizations and the triangle boundary reduction is derived in Riesz/restricted fractional Laplacian. Both are verified against independently coded direct and polar integrals. SciPy provides the standard hypergeometric special-function evaluation.

Software infrastructure

Yonderdrake is built on Firedrake and UFL. It uses PETSc and petsc4py for parallel linear algebra, solvers, and optimization. Its spatial operators use Firedrake’s external-operator interface.

  • D. A. Ham et al., Firedrake User Manual, first edition (2023), doi:10.25561/104839.

  • M. S. Alnaes, A. Logg, K. B. Ølgaard, M. E. Rognes, and G. N. Wells, Unified Form Language: A domain-specific language for weak formulations of partial differential equations, ACM Transactions on Mathematical Software 40(2) (2014), Article 9, doi:10.1145/2566630.

  • N. Bouziani and D. A. Ham, Escaping the abstraction: A foreign function interface for the Unified Form Language [UFL], Differentiable Programming Workshop at NeurIPS (2021), arXiv:2111.00945.

  • N. Bouziani, D. A. Ham, and A. Farsi, Differentiable programming across the PDE and Machine Learning barrier (2024), arXiv:2409.06085.

  • S. Balay et al., PETSc/TAO Users Manual, Argonne National Laboratory, ANL-21/39, Revision 3.25 (2026), doi:10.2172/3025790.

  • S. Balay, W. D. Gropp, L. Curfman McInnes, and B. F. Smith, Efficient management of parallelism in object-oriented numerical software libraries, in Modern Software Tools in Scientific Computing, Birkhäuser (1997), 163-202.

  • L. D. Dalcin, R. R. Paz, P. A. Kler, and A. Cosimo, Parallel distributed computing using Python, Advances in Water Resources 34(9) (2011), 1124-1139, doi:10.1016/j.advwatres.2011.04.013.

Time-memory methods

  • M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications 1(2) (2015), 73-85, publisher copy.

  • M. D. Ortigueira and J. Tenreiro Machado, A critical analysis of the Caputo-Fabrizio operator, Communications in Nonlinear Science and Numerical Simulation 59 (2018), 608-611, doi:10.1016/j.cnsns.2017.12.001.

  • K. Diethelm, R. Garrappa, A. Giusti, and M. Stynes, Why fractional derivatives with nonsingular kernels should not be used, Fractional Calculus and Applied Analysis 23(3) (2020), 610-634, doi:10.1515/fca-2020-0032.

  • Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics 225 (2007), 1533-1552, doi:10.1016/j.jcp.2007.02.001.

  • L. Yuan and O. P. Agrawal, A numerical scheme for dynamic systems containing fractional derivatives, Journal of Vibration and Acoustics 124(2) (2002), 321-324, doi:10.1115/1.1448322.

  • K. Diethelm, An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives, Numerical Algorithms 47 (2008), 361-390, doi:10.1007/s11075-008-9193-8.

  • C. Birk and C. Song, An improved non-classical method for the solution of fractional differential equations, Computational Mechanics 46 (2010), 721-734, doi:10.1007/s00466-010-0510-4.

  • K. Diethelm, A new diffusive representation for fractional derivatives, Part I: Construction, implementation and numerical examples, in Fractional Differential Equations, Springer INdAM Series 50 (2023), 1-15, doi:10.1007/978-981-19-7716-9_1.

  • K. Diethelm, A new diffusive representation for fractional derivatives, Part II: Convergence analysis of the numerical scheme, Mathematics 10 (2022), 1245, doi:10.3390/math10081245.

  • J. Yuan, S. Gao, G. Xiu, and B. Shi, Equivalence of initialized Riemann-Liouville and Caputo derivatives, Journal of Applied Analysis & Computation 10(5) (2020), 2008-2023, doi:10.11948/20190317.

Fractional spatial methods

  • A. Bonito and J. E. Pasciak, Numerical approximation of fractional powers of elliptic operators, Mathematics of Computation 84 (2015), 2083-2110, doi:10.1090/S0025-5718-2015-02937-8.

  • A. Bonito, W. Lei, and J. E. Pasciak, On sinc quadrature approximations of fractional powers of regularly accretive operators, Journal of Numerical Mathematics 27 (2019), 57-68, doi:10.1515/jnma-2017-0116.

  • E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques 136 (2012), 521-573, doi:10.1016/j.bulsci.2011.12.004.

  • G. Acosta and J. P. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM Journal on Numerical Analysis 55 (2017), 472-495, doi:10.1137/15M1033952.

  • M. Bebendorf, Approximation of boundary element matrices, Numerische Mathematik 86 (2000), 565-589, doi:10.1007/PL00005410.

Fractionally attenuated acoustics

  • M. G. Wismer, Finite element analysis of broadband acoustic pulses through inhomogeneous media with power law attenuation, Journal of the Acoustical Society of America 120(6) (2006), 3493-3502, doi:10.1121/1.2354032, PubMed.

  • B. Kaltenbacher and A. Schlintl, Fractional time stepping and adjoint based gradient computation in an inverse problem for a fractionally damped wave equation, Journal of Computational Physics 449 (2022), 110789, doi:10.1016/j.jcp.2021.110789.

  • M. J. King, T. S. Gutleb, B. E. Treeby, and B. T. Cox, Modelling power-law ultrasound absorption using a time-fractional, static memory, Fourier pseudo-spectral method, Journal of the Acoustical Society of America 157(3) (2025), 1761-1771, doi:10.1121/10.0035937, arXiv:2408.02541.

  • B. E. Treeby and B. T. Cox, k-Wave: MATLAB toolbox for the simulation and reconstruction of photoacoustic wave fields, Journal of Biomedical Optics 15(2) (2010), 021314, doi:10.1117/1.3360308, PubMed.