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arxiv: 2512.04737 · v2 · submitted 2025-12-04 · 🧮 math.NA · cs.NA

A Multi-Order Extension of Fractional HBVMs (FHBVMs)

Luigi Brugnano , Gianmarco Gurioli , Felice Iavernaro , Mikk Vikerpuur This is my paper

Pith reviewed 2026-05-17 01:31 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords fractional differential equationsmulti-order problemsHBVMsRunge-Kutta methodsnumerical solutionMatlab code
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The pith

Fractional HBVMs extend to solve fractional differential equations with multiple different derivative orders.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to adapt Fractional HBVMs, a class of Runge-Kutta type methods for fractional differential equations, to multi-order problems where distinct fractional orders appear together in the same system. Earlier versions handled only cases with a single shared order. The work supplies the full construction details for the extension and releases a Matlab implementation specialized to two orders, which it reports performs effectively on such problems. Readers would care because multi-order fractional equations appear in applications and require reliable numerical tools that scale from the single-order case.

Core claim

Fractional HBVMs have so far been defined for systems with the same fractional derivative order. This paper proposes an extension of FHBVMs for addressing fractional multi-order problems, providing full details for such an approach. A corresponding Matlab code, handling the case of two different fractional orders, is also made available, proving very effective for numerically solving these problems.

What carries the argument

The direct adaptation of the single-order FHBVM collocation structure to accommodate distinct fractional orders within one system of equations.

If this is right

  • The extended methods produce numerical solutions for fractional systems containing two or more distinct derivative orders.
  • The supplied Matlab code implements the case of exactly two orders and can be run directly on such problems.
  • The approach remains competitive with other existing numerical schemes for multi-order fractional equations.
  • The detailed construction allows straightforward generalization to problems with more than two orders.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same adaptation pattern might apply to variable-order fractional derivatives without major redesign.
  • Applications in viscoelasticity or anomalous diffusion that naturally produce multi-order models could use this solver as a starting point.
  • Stability proofs for arbitrary numbers of orders would strengthen the method for general use.

Load-bearing premise

The single-order FHBVM framework can be directly adapted to multiple orders while preserving stability, accuracy, and computational efficiency without requiring substantial new theoretical analysis.

What would settle it

A concrete multi-order test problem on which the extended method loses its expected convergence rate or exhibits instability would show that the direct adaptation fails to carry over the desired properties.

Figures

Figures reproduced from arXiv: 2512.04737 by Felice Iavernaro, Gianmarco Gurioli, Luigi Brugnano, Mikk Vikerpuur.

Figure 1
Figure 1. Figure 1: WPD for Problem (65) [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: WPD for Problem (66). 22 [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Solution (upper-plot) and vector field (lower-plot) for Problem (67). [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: WPD for Problem (67) [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: WPD for Problem (68), α1 = 0.7, α2 = 0.8. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Solution in the phase space (upper-plot) and versus time (lower-plot) for Problem (68). [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Solution of Problem (68), α1 = α2 = 0.7 (circles) and α1 = 0.7, α2 = 0.7+ 10−4 (crosses). As is clear, in the first case only one fractional derivative order occurs (i.e., ν = 1 in (1)), whereas they are two in the second case (i.e., ν = 2 in (1)). Despite this fact, the computed solutions are very similar, as is shown in [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Solution of Problem (69) in the phase space (upper plot) and versus time (lower plot). [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗
read the original abstract

The efficient numerical solution of fractional differential equations has been recently tackled through the definition of Fractional HBVMs (FHBVMs), a class of Runge-Kutta type methods. Corresponding Matlab (c) codes have been also made available on the internet, proving to be very competitive w.r.t. existing ones. However, so far, FHBVMs have been given for solving systems of fractional differential equations with the same order of fractional derivative, whereas the numerical solution of multi-order problems (i.e., problems in which different orders of fractional derivatives occur) has not been handled, yet. Due to their relevance in applications, in this paper we propose an extension of FHBVMs for addressing fractional multi-order problems, providing full details for such an approach. A corresponding Matlab (c) code, handling the case of two different fractional orders, is also made available, proving very effective for numerically solving these problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends Fractional HBVMs (FHBVMs) from single-order to multi-order fractional differential equations. It adapts the quadrature and collocation framework by applying separate discrete fractional integral operators for each order, supplies implementation details, and provides a Matlab code for the two-order case, claiming the resulting scheme remains competitive for solving such problems.

Significance. If the extension preserves the accuracy, stability, and efficiency of the original FHBVMs for coupled multi-order systems, it would fill a practical gap, since multi-order fractional models appear in applications. The public code is a positive feature for reproducibility.

major comments (2)
  1. The central construction in the multi-order extension section adapts the single-order FHBVM quadrature and collocation conditions by applying separate discrete fractional integral operators for each order α and β. However, no derivation or verification is given of the modified order conditions that arise from the interaction of the two distinct fractional kernels when the orders are incommensurate and the equations are coupled; this is load-bearing for the claim that the scheme retains the original accuracy and stability properties.
  2. Numerical experiments section: the reported tests use commensurate or simple cases and do not include a direct check (e.g., via order tables or stability plots) that the multi-order scheme satisfies the same convergence rates and stability bounds as the single-order FHBVM when the fractional orders differ and the system is coupled.
minor comments (2)
  1. Notation for the multi-order discrete integral operators should be introduced with an explicit comparison table to the single-order operators to prevent reader confusion.
  2. The abstract and introduction could add a one-sentence statement of the precise order conditions that are being preserved (or modified) under the extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address the major comments point by point below, indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: The central construction in the multi-order extension section adapts the single-order FHBVM quadrature and collocation conditions by applying separate discrete fractional integral operators for each order α and β. However, no derivation or verification is given of the modified order conditions that arise from the interaction of the two distinct fractional kernels when the orders are incommensurate and the equations are coupled; this is load-bearing for the claim that the scheme retains the original accuracy and stability properties.

    Authors: We agree that an explicit derivation of the order conditions for the multi-order case would clarify the extension. The construction applies independent discrete fractional integral operators to each term according to its order, so that the local truncation error analysis proceeds componentwise even when the orders are incommensurate and the right-hand side is coupled. We will add a short subsection deriving the resulting order conditions and confirming that the accuracy and stability properties carry over from the single-order setting. revision: yes

  2. Referee: Numerical experiments section: the reported tests use commensurate or simple cases and do not include a direct check (e.g., via order tables or stability plots) that the multi-order scheme satisfies the same convergence rates and stability bounds as the single-order FHBVM when the fractional orders differ and the system is coupled.

    Authors: The existing experiments illustrate the method on multi-order problems, some of which already involve distinct orders. To provide the requested direct verification, we will augment the numerical section with additional tests on coupled systems with incommensurate orders, including convergence tables for successive step-size reductions and a brief discussion of observed stability behavior. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extension provides independent implementation details

full rationale

The paper explicitly states it provides full details for adapting FHBVMs to multi-order fractional problems and releases corresponding Matlab code for two orders. This constitutes new implementation steps rather than a reduction of the central claim to prior inputs by construction, fitted parameters, or unverified self-citation chains. The derivation chain is self-contained against the stated goal of handling incommensurate orders via separate operators, with no quoted equations showing the multi-order scheme equaling its single-order inputs without additional content. This is the normal honest outcome for an extension paper that supplies concrete new material.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are described in the abstract; the work relies on the existing FHBVM framework as background.

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Works this paper leans on

54 extracted references · 54 canonical work pages

  1. [1]

    Amodio, L

    P. Amodio, L. Brugnano, F. Iavernaro. Analysis of Spectral Hamiltonian Boundary Value Meth- ods (SHBVMs) for the numerical solution of ODE problems.Numer. Algorithms83(2020) 1489–1508.https://doi.org/10.1007/s11075-019-00733-7

    work page doi:10.1007/s11075-019-00733-7 2020

  2. [2]

    Amodio, L

    P. Amodio, L. Brugnano, F. Iavernaro. P.Amodio, L.Brugnano, F.Iavernaro. (Spectral) Cheby- shev collocation methods for solving differential equations.Numer. Algorithms93(2023) 1613– 1638.https://doi.org/10.1007/s11075-022-01482-w

    work page doi:10.1007/s11075-022-01482-w 2023

  3. [3]

    Amodio, L

    P. Amodio, L. Brugnano, F. Iavernaro. A note on a stable algorithm for computing the fractional integrals of orthogonal polynomials.Appl. Math. Lett.134(2022) 108338.https://doi.org/ 10.1016/j.aml.2022.108338

    work page doi:10.1016/j.aml.2022.108338 2022

  4. [4]

    Arciga-Alejandre, J

    M.P. Arciga-Alejandre, J. Sanchez-Ortiz, F.J. Ariza-Hernandez, G. Catelan-Angeles. A multi- stage homotopy perturbation method for the fractional Lotka-Volterra model.Symmetry11 (2019) 1330.https://doi.org/10.3390/sym11111330

    work page doi:10.3390/sym11111330 2019

  5. [5]

    Baleanu, S.S

    D. Baleanu, S.S. Sajjadi, J.H. Asad, A. Jajarmi, E. Estiri. Hyperchaotic behaviors, optimal con- trol, and synchronization of a nonautonomous cardiac conduction system.Adv. Difference Equ. (2021) paper no. 157.https://doi.org/10.1186/s13662-021-03320-0

    work page doi:10.1186/s13662-021-03320-0 2021

  6. [6]

    C.F. Borges. On a class of Gauss-like quadrature rules.Numer. Math.67(1994) 271–288. https://doi.org/10.1007/s002110050028

    work page doi:10.1007/s002110050028 1994

  7. [7]

    Brugnano

    L. Brugnano. Blended Block BVMs (B3VMs): A Family of Economical Implicit Meth- ods for ODEs.J. Comput. Appl. Math.116(2000) 41–62.https://doi.org/10.1016/ S0377-0427(99)00280-0

    work page 2000

  8. [8]

    Brugnano, K

    L. Brugnano, K. Burrage, P. Burrage, F. Iavernaro. A spectrally accurate step-by-step method for the numerical solution of fractional differential equations.J. Sci. Comput.99(2024) 48. https://doi.org/10.1007/s10915-024-02517-1

    work page doi:10.1007/s10915-024-02517-1 2024

  9. [9]

    Brugnano, G

    L. Brugnano, G. Gurioli, F. Iavernaro. Numerical solution of FDE-IVPs by using Frac- tional HBVMs: thefhbvmcode.Numer. Algor.(2024).https://doi.org/10.1007/ s11075-024-01884-y

    work page 2024

  10. [10]

    Brugnano, G

    L. Brugnano, G. Gurioli, F. Iavernaro. Solving FDE-IVPs by using Fractional HBVMs: Some experiments with thefhbvmcode.J. Comput. Methods Sci. Eng.25, No. 1 (2025) 1030-1038. https://doi.org/10.1177/14727978251321328

    work page doi:10.1177/14727978251321328 2025

  11. [11]

    Brugnano, G

    L. Brugnano, G. Gurioli, F. Iavernaro, M. Vikerpuur. Analysis and implementation of collo- cation methods for fractional differential equations.J. Sci. Comput.104(2025) 92.https: //doi.org/10.1007/s10915-025-03006-9 32

    work page doi:10.1007/s10915-025-03006-9 2025

  12. [12]

    Brugnano, G

    L. Brugnano, G. Gurioli, F. Iavernaro, M. Vikerpuur. FDE-Testset: comparing Matlab codes for solving fractional differential equations of Caputo type.Fractal Fract.9(5) (2025) 312. https://doi.org/10.3390/fractalfract9050312

    work page doi:10.3390/fractalfract9050312 2025

  13. [13]

    Brugnano, F

    L. Brugnano, F. Iavernaro.Line Integral Methods for Conservative Problems.CRC Press, Boca Raton, FL, 2016

    work page 2016

  14. [14]

    Brugnano, F

    L. Brugnano, F. Iavernaro. A general framework for solving differential equations.Ann. Univ. Ferrara68(2022) 243–258.https://doi.org/10.1007/s11565-022-00409-6

    work page doi:10.1007/s11565-022-00409-6 2022

  15. [15]

    Finite element analysis for the axisymmetric laplace operator on polygonal domains,

    L. Brugnano, F. Iavernaro, D. Trigiante. A note on the efficient implementation of Hamilto- nian BVMs.J. Comput. Appl. Math.236(2011) 375–383.https://doi.org/10.1016/j.cam. 2011.07.022

    work page doi:10.1016/j.cam 2011

  16. [16]

    Brugnano, C

    L. Brugnano, C. Magherini. Blended Implementation of Block Implicit Methods for ODEs. Appl. Numer. Math.42(2002) 29–45.https://doi.org/10.1016/S0168-9274(01)00140-4

    work page doi:10.1016/s0168-9274(01)00140-4 2002

  17. [18]

    Y. Chen, F. Liu, Q. Yu, T. Li. Review of fractional epidemic models.Appl. Math. Model.97 (2021) 281–307.https://doi.org/10.1016/j.apm.2021.03.044

    work page doi:10.1016/j.apm.2021.03.044 2021

  18. [19]

    Coussement, W

    J. Coussement, W. Van Assche. Gaussian quadrature for multiple orthogonal polynomials.J. Compt. Appl. Math.178(2005) 131–145.https://doi.org/10.1016/j.cam.2004.04.016

    work page doi:10.1016/j.cam.2004.04.016 2005

  19. [20]

    Springer, Berlin (2010)

    K. Diethelm.The Analysis of Fractional Differential Equations. An Application-oriented Expo- sition using Differential Operators of Caputo Type. Lecture Notes in Math., Springer, Berlin, 2010.https://doi.org/10.1007/978-3-642-14574-2

    work page doi:10.1007/978-3-642-14574-2 2010

  20. [21]

    Diethelm, N.J

    K. Diethelm, N.J. Ford, A.D. Freed. Detailed error analysis for a fractional Adams method.Nu- mer. Algorithms36(2004) 31–52.https://doi.org/10.1023/B:NUMA.0000027736.85078.be

    work page doi:10.1023/b:numa.0000027736.85078.be 2004

  21. [22]

    Diethelm, N.J

    K. Diethelm, N.J. Ford, A.D. Freed, Y. Luchko. Algorithms for the fractional calculus: a se- lection of numerical methods.Comput. Methods Appl. Mech. Eng.194(6–8) (2005) 743–773. https://doi.org/10.1016/j.cma.2004.06.006

    work page doi:10.1016/j.cma.2004.06.006 2005

  22. [23]

    Diethelm, R

    K. Diethelm, R. Garrappa, M. Stynes. Good (and Not So Good) Practices in Computational Methods for Fractional Calculus.Mathematics8(2020) 324.https://doi.org/10.3390/ math8030324

    work page 2020

  23. [24]

    N.J. Ford, A. Pedas, M. Vikerpuur. High order approximations of solutions to initial value prob- lems for linear fractional integro-differential equations.Fract. Calc. Appl. Anal.26(2023) 2069–2100.https://doi.org/10.1007/s13540-023-00186-9

    work page doi:10.1007/s13540-023-00186-9 2023

  24. [25]

    Masset, R

    R. Garrappa. Trapezoidal methods for fractional differential equations: theoretical and com- putational aspects.Math. Comput. Simul.110(2015) 96–112https://doi.org/10.1016/j. matcom.2013.09.012

    work page doi:10.1016/j 2015

  25. [26]

    Garrappa

    R. Garrappa. Numerical evaluation of two and three parameter Mittag-Leffler functions.SIAM Jour. Numer. Anal.53(3) (2015) 1350–1369.https://doi.org/10.1137/140971191

    work page doi:10.1137/140971191 2015

  26. [27]

    Mathematics6, 16 (2018)

    R. Garrappa. Numerical solution of fractional differential equations: a survey and a software tutorial.Mathematics6(2) (2018) 16.http://doi.org/10.3390/math6020016 33

    work page doi:10.3390/math6020016 2018

  27. [28]

    Gautschi, Orthogonal polynomials: Applications and computation , Acta Numerica, 5 (1996), pp

    W. Gautschi. Orthogonal polynomials: applications and computations.Acta Numerica5(1996) 45–119.https://doi.org/10.1017/S0962492900002622

    work page doi:10.1017/s0962492900002622 1996

  28. [29]

    Gautschi.Orthogonal Polynomials Computation and Approximation

    W. Gautschi.Orthogonal Polynomials Computation and Approximation. Oxford University Press, 2004

    work page 2004

  29. [30]

    Grigorenko, E

    I. Grigorenko, E. Grigorenko. Chaotic dynamics of the fractional Lorenz system.Phys. Rev. Letters91(2003) 034101.https://doi.org/10.1103/PhysRevLett.91.034101

    work page doi:10.1103/physrevlett.91.034101 2003

  30. [31]

    Henry, T

    B.I. Henry, T. A. M. Langlands. Fractional cable models for spiny neuronal dendrites.Phys. Rev. Letts.100(12) (2008) 128103.https://doi.org/10.1103/PhysRevLett.100.128103

    work page doi:10.1103/physrevlett.100.128103 2008

  31. [32]

    M. Hori, I. Fukunaga, V. Masutani, T. Taoka, K. Kamagata, Y. Suzuki, et al. Visualizing non- Gaussian diffusion: clinical application of q-space imaging and diffusional kurtosis imaging of the brain, and spine.Magn. Reson. Med. Sc.11(2012) 221–233.https://doi.org/10.2463/ mrms.11.221

    work page 2012

  32. [33]

    Koh, J.M

    C.G. Koh, J.M. Kelly. Application of fractional derivatives to seismic analysis of base-isolated models.Earthquake engineering&structural dynamics19(2) (1990) 229–241.https://doi. org/10.1002/eqe.4290190207

    work page doi:10.1002/eqe.4290190207 1990

  33. [34]

    Laudadio, N

    T. Laudadio, N. Mastronardi, W. Van Assche, P. Van Dooren. A Matlab package computing simultaneous Gaussian quadrature rules for multiple orthogonal polynomials.J. Compt. Appl. Math.451(2024) 116109.https://doi.org/10.1016/j.cam.2024.116109

    work page doi:10.1016/j.cam.2024.116109 2024

  34. [35]

    Laudadio, N

    T. Laudadio, N. Mastronardi, P. Van Dooren. Computational aspects of simultaneous Gaus- sian quadrature.Numer. Algorithms100(2025) 621–643.https://doi.org/10.1007/ s11075-024-01785-0

    work page 2025

  35. [36]

    C. Li, Q. Yi, A. Chen. Finite difference methods with non-uniform meshes for nonlinear frac- tional differential equations.J. Comput. Phys.316(2016) 614–631.https://doi.org/10. 1016/j.jcp.2016.04.039

    work page 2016

  36. [37]

    T. Li, Y. Wang, F. Liu, I. Turner. Novel parameter estimation techniques for a multi-order fractional dynamical epidemic model of dengue fever.Numer. Algorithms82(2019) 1467–1495. https://doi.org/10.1007/s11075-019-00665-2

    work page doi:10.1007/s11075-019-00665-2 2019

  37. [38]

    Z. Li, D. Chen, J. Zhu, Y. Liu. Nonlinear dynamics of fractional order Duffing system.Chaos Solitons Fractals81(2015) 111–116.http://doi.org/10.1016/j.chaos.2015.09.012

    work page doi:10.1016/j.chaos.2015.09.012 2015

  38. [39]

    Liang, M

    H. Liang, M. Stynes. Collocation methods for general Caputo two-point boundary value prob- lems.J. Sci. Comput.76(2018) 390–425.http://doi.org/10.1007/s10915-017-0622-5

    work page doi:10.1007/s10915-017-0622-5 2018

  39. [40]

    Ch. Lubich. Fractional linear multistep methods for Abel-Volterra integral equations of the second kind.Math. Comput.45(172), (1985) 463–469.https://doi.org/10.1090/ S0025-5718-1985-0804935-7

    work page 1985

  40. [41]

    R.L. Magin. Fractional calculus models of complex dynamics in biological tissues.Comput. Math. Appl.59(5) (2010) 1586–1593.https://doi.org/10.1016/j.camwa.2009.08.039

    work page doi:10.1016/j.camwa.2009.08.039 2010

  41. [42]

    Milovanovi´ c, M

    G.V. Milovanovi´ c, M. Stani´ c. Construction of multiple orthogonal polynomials by discretized Stieltjes-Gautschi procedure and corresponding Gaussian quadratures.Facta Universitatis (Niˇ s), Ser. Math. Inform.18(2003) 9–29. 34

    work page 2003

  42. [43]

    Nigmatullin, A.A

    R.R. Nigmatullin, A.A. Arbuzov, F. Salehli, A. Giz, I. Bayrak, H. Catalgil-Giz. The first ex- perimental confirmation of the fractional kinetics containing the complexpower-law expo- nents: Dielectric measurements of polymerization reactions.Phys. B388(2007) 418–434. https://doi.org/10.1016/j.physb.2006.06.153

    work page doi:10.1016/j.physb.2006.06.153 2007

  43. [44]

    P. Panja. Dynamics of a fractional order predator-prey model with intraguild predation.Int. J. Math. Model. Simul. Appl.39(4) (2019) 256–268.https://doi.org/10.1080/02286203. 2019.1611311

    work page doi:10.1080/02286203 2019

  44. [45]

    Petr´ a˘ s.Fractional-order nonlinear systems

    I. Petr´ a˘ s.Fractional-order nonlinear systems. Modeling, analysis and simulation. Nonlinear Physical Science Ser., Springer, Heidelberg, 2011

    work page 2011

  45. [46]

    Pedas, E

    A. Pedas, E. Tamme. Numerical solution of nonlinear fractional differential equations by spline collocation methods.J. Comput. Appl. Math.255(2014) 216–230.https://doi.org/10. 1016/j.cam.2013.04.049

    work page 2014

  46. [47]

    Pedas, E

    A. Pedas, E. Tamme, M. Vikerpuur. Smoothing transformation and spline collocation for non- linear fractional initial and boundary value problems.J. Comput. Appl. Math.317(2017) 1–16. https://doi.org/10.1016/j.cam.2016.11.022

    work page doi:10.1016/j.cam.2016.11.022 2017

  47. [48]

    I. Podlubny.Fractional differential equations: an introduction to fractional derivatives, frac- tional differential equations, to methods of their solution and some of their applications. Aca- demic Press, Inc., San Diego, CA, 1999

    work page 1999

  48. [49]

    Sheu, H.-K

    L.-J. Sheu, H.-K. Chen, J.-H. Chen, L.-M. Tam, W.-C. Chen, K.-T. Lin, Y. Kang. Chaos in the Newton-Leipnik system with fractional order.Chaos Solitons Fractals36(2008) 98–103. https:/doi.org/10.1016/j.chaos.2006.06.013

    work page doi:10.1016/j.chaos.2006.06.013 2008

  49. [50]

    H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen. A new collection of real world applications of fractional calculus in science and engineering.Commun. Nonlinear Sci. Numer. Simul.64 (2018) 213–231.https://doi.org/10.1016/j.cnsns.2018.04.019

    work page doi:10.1016/j.cnsns.2018.04.019 2018

  50. [51]

    Uchaikin, E

    V. Uchaikin, E. Kozhemiakina. Non-local seismo-dynamics: A Fractional Approach.Fractal and Fractional6(9) (2022) 513.https://doi.org/10.3390/fractalfract6090513

    work page doi:10.3390/fractalfract6090513 2022

  51. [52]

    Journal of Applied Mechanics51(2), 294–298 (1984) https://doi

    P.J. Torvik, R.L. Bagley. On the appearance of the fractional derivative in the behavior of real materials.J. Appl. Mech.51(2) (1984) 294–298.https://doi.org/10.1115/1.3167615

    work page doi:10.1115/1.3167615 1984

  52. [53]

    Van Assche

    W. Van Assche. A Golub-Welsch version for simultaneous Gaussian quadrature.Numer. Algo- rithms100(2025) 599–620.https://doi.org/10.1007/s11075-024-01767-2

    work page doi:10.1007/s11075-024-01767-2 2025

  53. [54]

    Van Assche, J

    W. Van Assche, J. Coussement. Some classical multiple orthogonal polynomials.J. Compt. Appl. Math.127(2001) 317–347.https://doi.org/10.1016/S0377-0427(00)00503-3

    work page doi:10.1016/s0377-0427(00)00503-3 2001

  54. [55]

    D. Verotta. Fractional compartmental models and multi-order Mittag-Leffler response func- tions.J. Pharmacokinet. Pharmacodyn.37(2) (2010) 209–215.https://doi.org/10.1007/ s10928-010-9155-3 [56]https://people.dimai.unifi.it/brugnano/fhbvm/ [57]https://people.dimai.unifi.it/brugnano/FDEtestset/ [58]https://www.dm.uniba.it/it/members/garrappa/software 35

    work page 2010