References

The primary sources for Yonderdrake’s numerical methods, software infrastructure, and gallery examples are collected here. See Method map for the method-to-source summary.

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

  • C. Lubich, Discretized fractional calculus, SIAM Journal on Mathematical Analysis 17(3) (1986), 704-719, doi:10.1137/0517050.

  • C. Lubich, Convolution quadrature and discretized operational calculus. I, Numerische Mathematik 52 (1988), 129-145, doi:10.1007/BF01398686.

  • C. Lubich, Convolution quadrature and discretized operational calculus. II, Numerische Mathematik 52 (1988), 413-425, doi:10.1007/BF01398687.

  • A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, Journal of Computational Physics 280 (2015), 424-438, doi:10.1016/j.jcp.2014.09.031.

  • A. Schädle, M. López-Fernández, and C. Lubich, Fast and oblivious convolution quadrature, SIAM Journal on Scientific Computing 28(2) (2006), 421-438, doi:10.1137/050623139.

  • 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, Diffusive representations for the numerical evaluation of fractional integrals, Proceedings of the 2023 International Conference on Fractional Differentiation and its Applications (2023), doi:10.1109/ICFDA58234.2023.10153228, arXiv:2301.11931.

  • S. Jiang, J. Zhang, Q. Zhang, and Zhimin Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Communications in Computational Physics 21(3) (2017), 650-678, doi:10.4208/cicp.OA-2016-0136, arXiv:1511.03453.

  • H. Khosravian-Arab and M. Dehghan, The sine and cosine diffusive representations for the Caputo fractional derivative, Applied Numerical Mathematics 204 (2024), 265-290, doi:10.1016/j.apnum.2024.06.017, open-access companion.

  • T. S. Gutleb and J. A. Carrillo, A static memory sparse spectral method for time-fractional PDEs, Journal of Computational Physics 494 (2023), 112522, doi:10.1016/j.jcp.2023.112522.

  • 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.

  • R. Chaudhary and K. Diethelm, Novel variants of diffusive representation of fractional integrals: Construction and numerical computation, IFAC-PapersOnLine 58(12) (2024), 412-417, doi:10.1016/j.ifacol.2024.08.226.

  • R. Chaudhary and K. Diethelm, Revisiting diffusive representations for enhanced numerical approximation of fractional integrals, IFAC-PapersOnLine 58(12) (2024), 418-423, doi:10.1016/j.ifacol.2024.08.227.

  • 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. Kaltenbacher, B. Kaltenbacher, and I. Sim, A modified and stable version of a perfectly matched layer technique for the 3-d second order wave equation in time domain with an application to aeroacoustics, Journal of Computational Physics 235 (2013), 407-422, doi:10.1016/j.jcp.2012.10.016.

  • 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.