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2000
Volume 18, Issue 3
  • ISSN: 2352-0965
  • E-ISSN: 2352-0973

Abstract

Introduction

Linear fractional differential equations of the commensurable order, specifically those related to fractional trigonometry, pose challenges in terms of analytical solutions.

Methods

The purpose of this article is to present a novel approximate analytical technique to solve linear fractional differential equations of the commensurable order, which are related to fractional trigonometry. This method is used to obtain approximate solutions by forming linear combinations of appropriate fractional basic functions, for which Laplace transforms are irrational functions. This approximation technique enables us to represent these functions using time-invariant linear system models. Also, the implementation of analog circuits of these basic fractional order systems can be obtained using approximation of rational functions.

Results

The research demonstrates the efficacy and precision of this method through illustrative examples, showcasing its effectiveness in solving linear fractional systems.

Conclusion

The newly introduced approximate analytical method presents a promising approach to solving linear fractional differential equations of the commensurable order, providing accurate solutions and possibly offering applications in various fields.

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References

  1. MillerK.S. RossB. An introduction to the fractional calculus and fractional differential equations.John Wiley & Sons Inc.New-York1993
     [Google Scholar]
  2. PodlubnyI. Fractional differential equations.San DiegoAcademic Press1999
     [Google Scholar]
  3. KilbasA.A. SrivastavaH.M. TrujilloJ.J. Theory and applications of fractional differential equations.Amsterdam, NetherlandsElsevier2006
     [Google Scholar]
  4. MonjeC.A. ChenY.Q. VinagreB.M. XueD. FeliuV. Fractional-order systems and controls fundamentals and applications.England, UKSpringer-Verlag201010.1007/978‑1‑84996‑335‑0
     [Google Scholar]
  5. CaponettoR. DongolaG. FortunaL. PetrášI. Fractional order systems: Modeling and control applications.SingaporeWorld Scientific Publishing Co. Pte. Ltd201010.1142/7709
     [Google Scholar]
  6. HilferR. Applications of calculus in physics.SingaporeWorld Scientific200010.1142/3779
     [Google Scholar]
  7. SabatierJ. Advances in fractional calculus: Theoretical development and applications in physics and engineering.the NetherlandsSpringer200710.1007/978‑1‑4020‑6042‑7
     [Google Scholar]
  8. CharefA. Modeling and analog realization of the fundamental linear fractional order differential equation.Nonlinear Dyn.2006461-219521010.1007/s11071‑006‑9023‑2
     [Google Scholar]
  9. BonillaB. RiveroM. TrujilloJ.J. On systems of linear fractional differential equations with constant coefficients.Appl. Math. Comput.2007a1871687810.1016/j.amc.2006.08.104
     [Google Scholar]
  10. BonillaB. RiveroM. Rodríguez-GermáL. TrujilloJ.J. Fractional differential equations as alternative models to nonlinear differential equations.Appl. Math. Comput.2007b1871798810.1016/j.amc.2006.08.105
     [Google Scholar]
  11. HartleyT. T. LorenzoC.F. A solution of the fundamental linear fractional order differential equation.NASA TP-1998-2086931998
     [Google Scholar]
  12. OturançG. KurnazA. KeskinY. A new analytical approximate method for the solution of fractional differential equations.Int. J. Comput. Math.200885113114210.1080/00207160701405477
     [Google Scholar]
  13. HuY. LuoY. LuZ. Analytical solution of the linear fractional differential equation by Adomian decomposition method.J. Comput. Appl. Math.2008215122022910.1016/j.cam.2007.04.005
     [Google Scholar]
  14. ArikogluA. OzkolI. Solution of fractional Differential equations by using differential transform method.Chaos. Soliton&Frac200940252152910.1016/j.chaos.2006.09.004
     [Google Scholar]
  15. OdibatZ.M. Analytic study on linear systems of fractional differential equations.Comput. Math. Appl.20105931171118310.1016/j.camwa.2009.06.035
     [Google Scholar]
  16. CharefA. NezzariH. On the fundamental linear fractional order differential equation.Nonlinear Dyn.201165333534810.1007/s11071‑010‑9895‑z
     [Google Scholar]
  17. BushV. Operational circuit analysis.New York, NYWiley& Sons1937
     [Google Scholar]
  18. DavisH.T. The theory of linear operators.Bloomington, INPrincipia Press1936
     [Google Scholar]
  19. OldhamK.B. SpanierJ. The fractional calculus.San Diego, CAAcademic Press1974
     [Google Scholar]
  20. MillerK.S. RossB. An introduction to the fractional calculus and fractional differential equations.New York, NYWiley1993
     [Google Scholar]
  21. SamkoS.G. KilbasA.A. MarichevO.I. Fractional integrals and derivatives: Theory and applicatons. London.UKGordon and Breach Science Publishers1993
     [Google Scholar]
  22. WestB.J. BolognaM. GrigoliniP. Physics of fractal operators.Berlin, GermanySpringer200310.1007/978‑0‑387‑21746‑8
     [Google Scholar]
  23. OustaloupA. The non-integer derivation.Paris, FranceHermes1995
     [Google Scholar]
  24. MaginR.L. Fractional calculus in bioengineering.Redding, CTBegell House Publishers2006
     [Google Scholar]
  25. LiJ. GuY. QinQ.H. ZhangL. The rapid assessment for three-dimensional potential model of large-scale particle system by a modified multilevel fast multipole algorithm.Comput. Math. Appl.20218912713810.1016/j.camwa.2021.03.003
     [Google Scholar]
  26. LiJ. Rapid calculation of large-scale acoustic scattering from complex targets by a dual-level fast direct solver.Computers & Mathematics with Applications20231301910.1016/j.camwa.2022.11.007
     [Google Scholar]
  27. WesselingP. An introduction to multigrid methods.ChichesterJohn Wiley & Sons1992https://core.ac.uk/download/pdf/301647122.pdf
     [Google Scholar]
  28. ShaidurovV.V. Multigrid methods for finite elements.NetherlandsSpringer199510.1007/978‑94‑015‑8527‑9
     [Google Scholar]
  29. RuiP.L. ChenR.S. An efficient sparse approximate inverse preconditioning for FMM implementation.Microw. Opt. Technol. Lett.20074971746175010.1002/mop.22538
     [Google Scholar]
  30. LiJ. FuZ. GuY. QinQ.H. Recent advances and emerging applications of the singular boundary method for large-scale and high-frequency computational acoustics.Adv. Appl. Math. Mech.202214231534310.4208/aamm.OA‑2020‑0356
     [Google Scholar]
  31. LiJ.P. QinQ.H. Radial basis function method for large-scale wave propagation.SharjahBentham Science Publishers202110.2174/97816810889831210101
     [Google Scholar]
  32. LiJ. ZhangL. CaiS. LiN. Regularized singular boundary method for calculating wave forces on three-dimensional large offshore structure.Appl. Math. Lett.202414910893110.1016/j.aml.2023.108931
     [Google Scholar]
  33. LiJ. ZhangL. QinQ.H. A regularized method of moments for three-dimensional time-harmonic electromagnetic scattering.Appl. Math. Lett.202111210674610674610.1016/j.aml.2020.106746
     [Google Scholar]
  34. LiJ. FuZ. ChenW. QinQ.H. A regularized approach evaluating origin intensity factor of singular boundary method for Helmholtz equation with high wavenumbers.Eng. Anal. Bound. Elem.201910116517210.1016/j.enganabound.2019.01.008
     [Google Scholar]
  35. BouchermaD. CharefA. NezzariH. Rational function approximation of a fundamental fractional order transfer function.Lect. Notes Electr. Eng.201741125927510.1007/978‑3‑319‑48929‑2_20
     [Google Scholar]
  36. CharefA. BouchermaD. Analytical solution of the linear fractional system of commensurate order.Comput. Math. Appl.201162124415442810.1016/j.camwa.2011.10.017
     [Google Scholar]
  37. NezzariH. CharefA. Analog realization of generalized fractional order damped sine and cosine functions.Proceedings of the Symposium on Fractional Signals and Systems, FSS452011
     [Google Scholar]
  38. BouchermaD. CharefA. NezzariH. The solution of state space linear fractional system of commensurate order with complex eigenvalues using regular exponential and trigonometric functions.Int. J. Dyn. Control.201751799410.1007/s40435‑015‑0185‑y
     [Google Scholar]
  39. LorenzoC.F. The fractional meta-trigonometry based on the R-function. I. Background, definitions, and complexity function graphics.ASME Int. Design Engineering Technical Conf.2009 SanDiego, CA30 August–2 September 200910.1115/DETC2009‑86731
     [Google Scholar]
  40. LorenzoC.F. The fractional meta-trigonometry based on the R-function. II. Parity function graphics, Laplace transforms, fractional derivatives, meta-properties.ASME Int. Design Engineering Technical Conf2009 San Diego, CA30 August–2 September 200910.1115/DETC2009‑86733
     [Google Scholar]
  41. LorenzoC.F. HartleyT.T. Applications of the fractional trigonometry: Fractional systems and fractional differential equations.IFAC Workshop on Fractional Differentiation and its Applications2004 Bordeaux, France10.1098/rsta.2012.0151
     [Google Scholar]
  42. LorenzoC.F. HartleyT.T. Fractional trigonometry and the spiral functions.Nonlinear Dyn.2004381-4236010.1007/s11071‑004‑3745‑9
     [Google Scholar]
  43. LorenzoCF. HartleyTT. Generalized functions for the fractional calculus.NASATP-1999-20942410.1615/CritRevBiomedEng.v36.i1.40
     [Google Scholar]
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Fractional Trigonometric Function Approximations in a Regular System
Recent Advances in Electrical & Electronic Engineering 18, 316 (2025); https://doi.org/10.2174/0123520965279598240124102952
/content/journals/raeeng/10.2174/0123520965279598240124102952
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  • Article Type:     Research Article
Keyword(s): analog circuit; calculus fractional derivatives and integrals; linear ordinary system; R-function; transfer irrational function; Trigonometric fractional functions

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