Fully numerical Laplace transform methods
The role of the Laplace transform in scientific computing has been predominantly that of a semi-numerical tool. That is, typically only the inverse transform is computed numerically, with all steps leading up to that executed by analytical manipulations or table look-up. Here, we consider fully nume...
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| Published in | Numerical algorithms Vol. 92; no. 1; pp. 985 - 1006 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.01.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1017-1398 1572-9265 |
| DOI | 10.1007/s11075-022-01368-x |
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| Abstract | The role of the Laplace transform in scientific computing has been predominantly that of a semi-numerical tool. That is, typically only the inverse transform is computed numerically, with all steps leading up to that executed by analytical manipulations or table look-up. Here, we consider fully numerical methods, where both forward and inverse transforms are computed numerically. Because the computation of the inverse transform has been studied extensively, this paper focus mainly on the forward transform. Existing methods for computing the forward transform based on exponential sums are considered along with a new method based on the formulas of Weeks. Numerical examples include a nonlinear integral equation of convolution type, a fractional ordinary differential equation, and a partial differential equation with an inhomogeneous boundary condition. |
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| AbstractList | The role of the Laplace transform in scientific computing has been predominantly that of a semi-numerical tool. That is, typically only the inverse transform is computed numerically, with all steps leading up to that executed by analytical manipulations or table look-up. Here, we consider fully numerical methods, where both forward and inverse transforms are computed numerically. Because the computation of the inverse transform has been studied extensively, this paper focus mainly on the forward transform. Existing methods for computing the forward transform based on exponential sums are considered along with a new method based on the formulas of Weeks. Numerical examples include a nonlinear integral equation of convolution type, a fractional ordinary differential equation, and a partial differential equation with an inhomogeneous boundary condition. |
| Author | Weideman, J. A. C. Fornberg, Bengt |
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| Keywords | Weeks method Laplace transform Exponential sums Padé approximation |
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Anal.1974557458035927310.1137/05050580253.44002 LynessJNGiuntaGA modification of the Weeks method for numerical inversion of the Laplace transformMath Comp.19864717531332284213810.1090/S0025-5718-1986-0842138-10611.65088 TalbotAThe accurate numerical inversion of Laplace transformsJ. Inst. Math. Appl.19792319712052628610.1093/imamat/23.1.970406.65054 WeeksWTNumerical inversion of Laplace transforms using Laguerre functionsJ. Assoc. Comput. Mach.19661341942919524110.1145/321341.3213510141.33401 Potts, D., Tasche, M., Volkmer, T.: Efficient spectral estimation by MUSIC and ESPRIT with application to sparse FFT. Front. Appl. Math. Stat., 2 (2016) DubnerHAbateJNumerical inversion of Laplace transforms by relating them to the finite Fourier cosine transformJ. Assoc Comput. Mach.19681511512323572610.1145/321439.3214460165.51403 Butcher, J.C.: On the numerical inversion of Laplace and Mellin transforms. In: Conference on Data Processing and Automatic Computing Machines. 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| References_xml | – reference: WeidemanJACReddySCA MATLAB differentiation matrix suiteACM Trans. Math. Software2000264465519193996210.1145/365723.365727 – reference: LynessJNGiuntaGA modification of the Weeks method for numerical inversion of the Laplace transformMath Comp.19864717531332284213810.1090/S0025-5718-1986-0842138-10611.65088 – reference: Butcher, J.C.: On the numerical inversion of Laplace and Mellin transforms. In: Conference on Data Processing and Automatic Computing Machines. Salisbury (1957) – reference: CohenAMNumerical methods for Laplace transform inversion, Volume 5 of Numerical Methods and Algorithms2007New YorkSpringer – reference: GiuntaGLaccettiGRizzardiMRMore on the Weeks method for the numerical inversion of the Laplace transformNumer. Math.198854219320096592110.1007/BF013969740659.65138 – reference: DingfelderBWeidemanJACAn improved Talbot method for numerical Laplace transform inversionNumer Algorithms2015681167183329670510.1007/s11075-014-9895-z1432.65190 – reference: TalbotAThe accurate numerical inversion of Laplace transformsJ. Inst. Math. Appl.19792319712052628610.1093/imamat/23.1.970406.65054 – reference: DubnerHAbateJNumerical inversion of Laplace transforms by relating them to the finite Fourier cosine transformJ. Assoc Comput. Mach.19681511512323572610.1145/321439.3214460165.51403 – reference: SheenDSloanIHThoméeVA parallel method for time discretization of parabolic equations based on Laplace transformation and quadratureIMA J. Numer. Anal.2003232269299197526710.1093/imanum/23.2.2691022.65108 – reference: Benner, P., Mehrmann, V., Sima, V., Van Huffel, S., Varga, A.: SLICOT—a subroutine library in systems and control theory. In: Applied and Computational Control, Signals, and Circuits, volume 1 of Appl. Comput. Control Signals Circuits, pp 499–539. Boston, Birkhäuser (1999) – reference: BeylkinGMonzónLApproximation by exponential sums revisitedAppl. Comput. Harmon. Anal.2010282131149259588110.1016/j.acha.2009.08.0111189.65035 – reference: LongmanIMBest rational function approximation for Laplace transform inversionSIAM J. Math. Anal.1974557458035927310.1137/05050580253.44002 – reference: CarrierGFKrookMPearsonCEFunctions of a complex variable: Theory and Technique1966New YorkMcGraw-Hill Book Co.0146.29801 – reference: Driscoll, T.A., Hale, N., Trefethen, L.N.: Chebfun Guide. Pafnuty Publications (2014) – reference: PottsDTascheMParameter estimation for nonincreasing exponential sums by Prony-like methodsLinear Algebra Appl.2013439410241039306175310.1016/j.laa.2012.10.0361281.65021 – reference: BeylkinGMonzónLOn approximation of functions by exponential sumsAppl. Comput. Harmon. Anal.20051911748214706010.1016/j.acha.2005.01.0031075.65022 – reference: TrefethenLNWeidemanJACThe exponentially convergent trapezoidal ruleSIAM Rev.2014563385458324585810.1137/1309321321307.65031 – reference: WeidemanJACTrefethenLNParabolic and hyperbolic contours for computing the Bromwich integralMath. Comp.20077625913411356229977710.1090/S0025-5718-07-01945-X1113.65119 – reference: Derevianko, N., Plonka, G., Petz, M.: From ESPRIT to ESPIRA: estimation of signal parameters by iterative rational approximation (2022) – reference: PodlubnyIFractional differential equations, Volume 198 of Mathematics in Science and Engineering1999San DiegoAcademic Press, Inc. – reference: GilASeguraJTemmeNMEfficient computation of Laguerre polynomialsComput. Phys. Commun.2017210124131357634910.1016/j.cpc.2016.09.0021378.65066 – reference: HaleNWeidemanJACContour integral solution of elliptic PDEs in cylindrical domainsSIAM J. Sci Comput.2015376A2630A2655342162510.1137/15M10327641327.65042 – reference: LukeYLThe special functions and their approximations. Vol. II Mathematics in Science and Engineering, vol. 531969New YorkAcademic Press – reference: ElliottDTuanPDAsymptotic estimates of Fourier coefficientsSIAM J. Math Anal.1974511034093710.1137/05050010238.42008 – reference: Potts, D., Tasche, M., Volkmer, T.: Efficient spectral estimation by MUSIC and ESPRIT with application to sparse FFT. Front. Appl. Math. Stat., 2 (2016) – reference: SpiegelMRTheory and problems of Laplace transforms1965New YorkSchaum Publishing Co. – reference: WeidemanJACAlgorithms for parameter selection in the Weeks method for inverting the Laplace transformSIAM J. Sci. Comput.1999211111128172208710.1137/S10648275963124320944.65137 – reference: DiethelmKThe analysis of fractional differential equations, Volume 2004 of Lecture Notes in Mathematics2010BerlinSpringer – reference: XiangSAsymptotics on Laguerre or Hermite polynomial expansions and their applications in Gauss quadratureJ. Math. Anal. Appl.20123932434444292168610.1016/j.jmaa.2012.03.0561259.65058 – reference: GottliebDOrszagSANumerical analysis of spectral methods: theory and applications. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 261977PhiladelphiaSociety for Industrial and Applied Mathematics10.1137/1.9781611970425 – reference: WeeksWTNumerical inversion of Laplace transforms using Laguerre functionsJ. Assoc. Comput. Mach.19661341942919524110.1145/321341.3213510141.33401 – reference: TerekhovAVGenerating the Laguerre expansion coefficients by solving a one-dimensional transport equationNumer. Algorithms2022891303322435830810.1007/s11075-021-01115-807456978 – reference: NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.17 of 2017-12-22. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds – volume-title: The analysis of fractional differential equations, Volume 2004 of Lecture Notes in Mathematics year: 2010 ident: 1368_CR8 – volume: 54 start-page: 193 issue: 2 year: 1988 ident: 1368_CR15 publication-title: Numer. Math. doi: 10.1007/BF01396974 – volume-title: Theory and problems of Laplace transforms year: 1965 ident: 1368_CR25 – volume: 393 start-page: 434 issue: 2 year: 2012 ident: 1368_CR33 publication-title: J. Math. Anal. 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| Snippet | The role of the Laplace transform in scientific computing has been predominantly that of a semi-numerical tool. That is, typically only the inverse transform... |
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| SubjectTerms | Algebra Algorithms Boundary conditions Computation Computer Science Integral equations Laplace transforms Methods Numeric Computing Numerical Analysis Numerical methods Original Paper Partial differential equations Theory of Computation |
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| Title | Fully numerical Laplace transform methods |
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