Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus

[1]K. Oldham and J. Spanier. The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, INC, San Diego Ca, 1974.OldhamK.SpanierJ.The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary OrderAcademic PressINC, San Diego Ca1974Search in Google Scholar

[2]K. S. Miller and B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, Inc., New York, NY, 1993.MillerK. S.RossB.An Introduction to the Fractional Calculus and Fractional Differential EquationsJohn Wiley & Sons, Inc.New York, NY1993Search in Google Scholar

[3]S. Samko, A. Kilbas, and O. Marichev. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London, 1993.SamkoS.KilbasA.MarichevO.Fractional Integrals and Derivatives: Theory and ApplicationsGordon and BreachLondon1993Search in Google Scholar

[4]I. Podlubny. Fractional Differential Equations. Academic Press, INC, San Diego Ca, 1999.PodlubnyI.Fractional Differential EquationsAcademic Press, INCSan Diego Ca1999Search in Google Scholar

[5]K. Diethlem. The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer-Verlag, Berlin, Heidelberg, 2010.DiethlemK.The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo TypeSpringer-VerlagBerlin, Heidelberg2010Search in Google Scholar

[6]D. Baleanu, K. Diethlem, E. Scalas, and J.J. Trujillo. Fractional Calculus. Models and Numerical Methods. World Scientific, Singapore, 2012.BaleanuD.DiethlemK.ScalasE.TrujilloJ.J.Fractional Calculus. Models and Numerical MethodsWorld ScientificSingapore201210.1142/8180Search in Google Scholar

[7]A. A. Kilbas, M. Srivastava H, and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier Science, 2006.KilbasA. A.Srivastava HM.TrujilloJ.J.Theory and Applications of Fractional Differential EquationsElsevier Science2006Search in Google Scholar

[8]F. Mainardi. Fractional Calculus and Waves in Linear Viscoelasticy. Imperial Collage Press, London, 2010.MainardiF.Fractional Calculus and Waves in Linear ViscoelasticyImperial Collage PressLondon201010.1142/p614Search in Google Scholar

[9]Y. Povstenko. Fractional Thermoelasticy. Springer International Publishing, Cham, Heidelberg, New York, Dodrecht, London, 2015.PovstenkoY.Fractional ThermoelasticySpringer International Publishing, Cham, HeidelbergNew York, Dodrecht, London201510.1007/978-3-319-15335-3Search in Google Scholar

[10]Y. Povstenko. Linear Fractional Diffusion-Wave Equation for Scientists and Engineers. Birkhäuser, Springer, Cham, Heidelberg, New York, Dodrecht, London, 2015.PovstenkoY.Linear Fractional Diffusion-Wave Equation for Scientists and EngineersBirkhäuser, SpringerCham, Heidelberg, New York, Dodrecht, London201510.1007/978-3-319-17954-4Search in Google Scholar

[11]M. O’Flynn and E. Moriarty. Time Domain and Transform Analysis. Wiley & Sons, Inc., 1987.O’FlynnM.MoriartyE.Time Domain and Transform AnalysisWiley & Sons, Inc.1987Search in Google Scholar

[12]A. D. Poularikas and S. Seely. Laplace Transforms in The Transforms and Applications Handbook: Second Edition. Boca Raton: CRC Press LLC, 2000.PoularikasA. D.SeelyS.Laplace Transforms in The Transforms and Applications Handbook: Second EditionBoca Raton: CRC Press LLC200010.1201/9781420036756Search in Google Scholar

[13]J. Abate, G.L. Choudhurry, and W.Whitt. An introduction to numerical inversion and its application to probability models. In W. Grassman, editor, Computational Probability, pages 257–323. Kluwer, Boston, 1999.AbateJ.ChoudhurryG.L.WhittW.An introduction to numerical inversion and its application to probability modelsGrassmanW.Computational Probability257323KluwerBoston199910.1007/978-1-4757-4828-4_8Search in Google Scholar

[14]S. I. Kabanikhin. Definition and examples of inverse and ill-posed problems. a survey paper. J.Inv.Ill-Posed Problems, 16:317–357, 2008.KabanikhinS. I.Definition and examples of inverse and ill-posed problems. a survey paperJ.Inv.Ill-Posed Problems16317357200810.1515/JIIP.2008.019Search in Google Scholar

[15]R. Piessens. A bibliography on numerical inversion of the laplace transform and applications. Journal of Computational and Applied Mathematics, 1(2):115–128, 1975.PiessensR.A bibliography on numerical inversion of the laplace transform and applicationsJournal of Computational and Applied Mathematics12115128197510.1016/0771-050X(75)90029-7Search in Google Scholar

[16]D. V. Widder. The Laplace Transform. Princeton Unviersity Press, 1946.WidderD. V.The Laplace TransformPrinceton Unviersity Press1946Search in Google Scholar

[17]D. W. Brzeziński and P. Ostalczyk. Numerical calculations accuracy comparison of the inverse laplace transform algorithms for solutions of fractional order differential equations. Nonlinear Dynamics, 81(1):65–77, 2016.BrzezińskiD. W.OstalczykP.Numerical calculations accuracy comparison of the inverse laplace transform algorithms for solutions of fractional order differential equationsNonlinear Dynamics8116577201610.1007/s11071-015-2225-8Search in Google Scholar

[18]D. Valerio, J. J. Trujillo, M. Rivero, J. A. T. Machado, and D. Baleanu. Fractional calculus: A survey of useful formulas. The European Physical Journal Special Topics, 222:1827–1846, 2013.ValerioD.TrujilloJ. J.RiveroM.MachadoJ. A. T.BaleanuD.Fractional calculus: A survey of useful formulasThe European Physical Journal Special Topics22218271846201310.1140/epjst/e2013-01967-ySearch in Google Scholar

[19]Y. Q. Chen, I. Petras, and B. Vinagre. A list of laplace and inverse laplace transforms related to fractional order calculus, 2010. http://www.tuke.sk/petras/foc_laplace.pdf.ChenY. Q.PetrasI.VinagreB.A list of laplace and inverse laplace transforms related to fractional order calculus2010http://www.tuke.sk/petras/foc_laplace.pdfSearch in Google Scholar

[20]Ch. L. Epstein and J. Schotland. The bad truth about laplace’s transform. SIAM Review, 3:504–520, 2008.EpsteinCh. L.SchotlandJ.The bad truth about laplace’s transformSIAM Review3504520200810.1137/060657273Search in Google Scholar

[21]H. Villinger. Solving cylindrical geothermal problems using gaver-stehfest inverse laplace transform. Geophysics, 50(10):47–49, 1985.VillingerH.Solving cylindrical geothermal problems using gaver-stehfest inverse laplace transformGeophysics50104749198510.1190/1.1441848Search in Google Scholar

[22]W. Weeks. Numerical inversion of laplace transforms using laguerre functions. Journal of the ACM, 13(3):419–429, 1966.WeeksW.Numerical inversion of laplace transforms using laguerre functionsJournal of the ACM133419429196610.1145/321341.321351Search in Google Scholar

[23]R. Piessens. New quadrature formulas for the numerical inversion of the laplace transform. BIT, 9:351–361, 1969.PiessensR.New quadrature formulas for the numerical inversion of the laplace transformBIT9351361196910.1007/BF01935866Search in Google Scholar

[24]P. D. Iseger. Numerical transform inversion using gaussian quadrature. Probability in the Engineering and Informational Sciences, 20:1–44, 2006.IsegerP. D.Numerical transform inversion using gaussian quadratureProbability in the Engineering and Informational Sciences20144200610.1017/S0269964806060013Search in Google Scholar

[25]B. Davies. Integral transforms inversion using gaussian quadratures. Journal of Computational Physics, 33(1):1–32, 1979.DaviesB.Integral transforms inversion using gaussian quadraturesJournal of Computational Physics331132197910.1016/0021-9991(79)90025-1Search in Google Scholar

[26]H. Stehfest. Algorithm 368: Numerical inversion of laplace transforms. Communications of the ACM, 13(1):47–49, 1970.StehfestH.Algorithm 368: Numerical inversion of laplace transformsCommunications of the ACM1314749197010.1145/361953.361969Search in Google Scholar

[27]H. Dubner and J. Abate. Numerical inversion of laplace transforms by relating them to the finite fourier cosine transform. Journal of the ACM, 15:115–123, 1968.DubnerH.AbateJ.Numerical inversion of laplace transforms by relating them to the finite fourier cosine transformJournal of the ACM15115123196810.1145/321439.321446Search in Google Scholar

[28]J. Abate and W. Whitt. The fourier-series method for inverting transforms of probability distributions. Queueing Systems, 10(5):5–87, 1999.AbateJ.WhittW.The fourier-series method for inverting transforms of probability distributionsQueueing Systems105587199910.1007/BF01158520Search in Google Scholar

[29]C. A. O’Cinneide. Euler summation for fourier series and laplace transform inversion. Stochastic Models, 13(2):315–337, 1997.O’CinneideC. A.Euler summation for fourier series and laplace transform inversionStochastic Models132315337199710.1080/15326349708807429Search in Google Scholar

[30]F. Durbin. Numerical inversion of laplace transforms: an efficient improvment to dubner and abate’s method. The Computer Journal, 17(4):371–376, 1973.DurbinF.Numerical inversion of laplace transforms: an efficient improvment to dubner and abate’s methodThe Computer Journal174371376197310.1093/comjnl/17.4.371Search in Google Scholar

[31]F. R. DeHoog, J. H. Knight, and A. N. Stokes. An improved method for numerical inversion of laplace transforms. SIAM J. Sci. Stat. Comput., 3(3):357–366, 1982.DeHoogF. R.KnightJ. H.StokesA. N.An improved method for numerical inversion of laplace transformsSIAM J. Sci. Stat. Comput.33357366198210.1137/0903022Search in Google Scholar

[32]K. L. Kuhlman. Review of inverse laplace transform algorithms for laplace-space approaches. Numer Algor, pages 1–19, 2012.KuhlmanK. L.Review of inverse laplace transform algorithms for laplace-space approachesNumer Algor119201210.1007/s11075-012-9625-3Search in Google Scholar

[33]A. M. Cohen. Numerical Methods for Laplace Transform Inversion. Springer-Verlag, Berlin, Heidelberg, 2007.CohenA. M.Numerical Methods for Laplace Transform InversionSpringer-VerlagBerlin, Heidelberg2007Search in Google Scholar

[34]J. Vlach and K. Singhai. Computer Methods for Circuit Analysis and Design. Van Nostrand Rheinhold Company, 1983.VlachJ.SinghaiK.Computer Methods for Circuit Analysis and DesignVan Nostrand Rheinhold Company1983Search in Google Scholar

[35]V. Zakian. Solution of homogeneous ordinary linear differential systems by numerical inversion of laplace transforms. Electronic Letters, 7:546–548, 1971.ZakianV.Solution of homogeneous ordinary linear differential systems by numerical inversion of laplace transformsElectronic Letters7546548197110.1049/el:19710369Search in Google Scholar

[36]A. Talbot. The accurate numerical inversion of laplace transforms. IMA Journal of Applied Mathematics, 23(1):97–112, 1979.TalbotA.The accurate numerical inversion of laplace transformsIMA Journal of Applied Mathematics23197112197910.1093/imamat/23.1.97Search in Google Scholar

[37]A. Murli and M. Rizzardi. Algorithm 682: Talbot’s method for the laplace inversion. ACM Transactions on Mathematical Software, 16(2):158–168, 1990.MurliA.RizzardiM.Algorithm 682: Talbot’s method for the laplace inversionACM Transactions on Mathematical Software162158168199010.1145/78928.78932Search in Google Scholar

[38]J. A. C. Weideman. Optimizing talbot’s contours for the inversion of the laplace transform. SIAM J. Anal., 44(6):2342–2362, 2006.WeidemanJ. A. C.Optimizing talbot’s contours for the inversion of the laplace transformSIAM J. Anal.44623422362200610.1137/050625837Search in Google Scholar

[39]Guillaume Hanrot et al. mpfr: The MPFR library for multiple-precision floating-point computations with correct rounding. (version 3.13), 2015. http://www.mpfr.org/.GuillaumeHanrotmpfr: The MPFR library for multiple-precision floating-point computations with correct rounding. (version 3.13)2015http://www.mpfr.org/Search in Google Scholar

[40]Torbjörn Granlund et al. gmp: GMP is a free library for arbitrary precision arithmetic (version 6.0.0a), 2015. https://gmplib.org.TorbjörnGranlundgmp: GMP is a free library for arbitrary precision arithmetic (version 6.0.0a)2015https://gmplib.orgSearch in Google Scholar

[41]Fredrik Johansson et al. mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 0.19), June 2014. http://mpmath.org/.FredrikJohanssonmpmath: a Python library for arbitrary-precision floating-point arithmetic (version 0.19)June2014http://mpmath.org/Search in Google Scholar

[42]Pavel Holoborodko. High-performance C++ interface for MPFR library (version 3.6.2), 2015. http://www.holoborodko.com/pavel/mpfr/.PavelHoloborodkoHigh-performance C++ interface for MPFR library (version 3.6.2)2015http://www.holoborodko.com/pavel/mpfr/Search in Google Scholar

[43]P. Humbert and R. P. Agarwal. Sur la fonction de mittag-leffler et quelques-unes de ses géneéralisations. Bull. Sci. Math. Ser. II, 77:180–185, 1953.HumbertP.AgarwalR. P.Sur la fonction de mittag-leffler et quelques-unes de ses géneéralisationsBull. Sci. Math. Ser. II771801851953Search in Google Scholar

[44]K. M. Kowankar and A. D. Gangal. Fractional differentability of nowhere differentable functions and dimensions. CHAOS, 6(4):180–185, 1996.KowankarK. M.GangalA. D.Fractional differentability of nowhere differentable functions and dimensionsCHAOS64180185199610.1063/1.16619712780280Search in Google Scholar

[45]K. R. Ghazi, V. Lefevre, P. Theveny, and P. Zimmermann. Why and how to use arbitrary precision. IEEE Computer Society, 12(3):1–5, 2001.GhaziK. R.LefevreV.ThevenyP.ZimmermannP.Why and how to use arbitrary precisionIEEE Computer Society12315200110.1109/MCSE.2010.73Search in Google Scholar