An improved Talbot method for numerical Laplace transform inversion

Dingfelder, Benedict; Weideman, J. A. C.

doi:10.1007/s11075-014-9895-z

Cited by 85 publications

(83 citation statements)

References 14 publications

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“…The version with 0 < β < 1 is known as the modified Talbot contour [40]. The optimal choices of parameters in the modified contour appear in the code and in Table 15.1, where the expected rate of convergence is also listed.…”

Section: Laplace Transforms and Hankel Contours The Laplace Transformentioning

confidence: 99%

“…This decay is too fast, and the reason for considering 0 < β < 1 in (15.7) is precisely to control it better [40,180]. Figure 15.2 shows most of the contours we have discussed.…”

Section: Laplace Transforms and Hankel Contours The Laplace Transformentioning

confidence: 99%

“…The result can be destructive cancellation errors on a computer even when a formula should mathematically be accurate. Roundoff control in the case of parabolas, hyperbolas, and Talbot contours is discussed in [181], [106], and [40], respectively.…”

Section: Laplace Transforms and Hankel Contours The Laplace Transformentioning

confidence: 99%

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The Exponentially Convergent Trapezoidal Rule

Trefethen¹,

Weideman²

2014

SIAM Rev.

Self Cite

469

362

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Abstract.It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.

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Section: Laplace Transforms and Hankel Contours The Laplace Transformentioning

confidence: 99%

“…This decay is too fast, and the reason for considering 0 < β < 1 in (15.7) is precisely to control it better [40,180]. Figure 15.2 shows most of the contours we have discussed.…”

Section: Laplace Transforms and Hankel Contours The Laplace Transformentioning

confidence: 99%

See 1 more Smart Citation

The Exponentially Convergent Trapezoidal Rule

Trefethen¹,

Weideman²

2014

SIAM Rev.

Self Cite

469

362

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“…6 R. GARRAPPA Several families of contours have been proposed so far. After the original work of Talbot [34] on contours of cotangent shape (see also [7,28,38]), a special attention has been paid to parabolic [2,14,39,40] and hyperbolic contours [21,14,33,40].The convergence rates of the N -points trapezoidal rule on cotangent, hyperbolic and parabolic contours have been studied in [36]; the respective rates of O 3.89 −N , O 3.20 −N and O 2.85 −N indicate a fast convergence with all these contours. Although the convergence with cotangent and hyperbolic contours is slightly faster, the simpler representation of parabolic contours makes them much more easy to handle; therefore, parabolas appear to be preferable especially when the presence of a certain number of singularities demands the fulfillment of tightened constraints.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Numerical Evaluation of Two and Three Parameter Mittag-Leffler Functions

Garrappa¹

2015

SIAM J. Numer. Anal.

242

172

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Abstract. The Mittag-Leffler (ML) function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation. In this work we present a method for the efficient computation of the ML function based on the numerical inversion of its Laplace transform (LT): an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the LT, with the aim of minimizing the computational effort and reduce the propagation of errors. Numerical experiments are presented to show accuracy and efficiency of the proposed approach. The application to the three parameter ML (also known as Prabhakar) function is also presented.Key words. Mittag-Leffler function, Laplace transform, trapezoidal rule, fractional calculus, Prabhakar function, special function.AMS subject classifications. 33E12, 44A10, 65D30, 33F05, 26A33, 1. Introduction. The Mittag-Leffler (ML) function was introduced, at the beginning of the twentieth century, by the Swedish mathematician Magnus Gustaf Mittag-Leffler [24,25] while studying summation of divergent series. Extensions to two [41] and three [31] parameters of the original single parameter function were successively considered; all these functions can be regarded as special instances of the generalized hypergeometric function investigated by Fox [9] and Wright [43].Until the 1960s, few authors (e.g., [19]) recognized the importance of the ML function in fractional calculus and, in particular, in describing anomalous processes with hereditary effects [1,4,5,8]. For an historical outline and a review of the main properties of the ML function we refer to [17,23] and to the recent monograph [15].For any argument z ∈ C, the ML function with two parameters α, β ∈ C, with ℜ(α) > 0, is defined by means of the series expansion where Γ(z) = ∞ 0 t z−1 e −t dt is the Euler's gamma function; since the integral representation of Γ(z) holds only for ℜ(z) > 0, the extension to the half-plane ℜ(z) ≤ 0, with z ∈ {0, −1, −2, . . .}, is accomplished by means of the relationship Γ(z + n) = z(z + 1) · · · (z + n − 1)Γ(z), n ∈ N, [20,22].As a special case, the ML function with one parameter is obtained for β = 1, i.e. E α (z) = E α,1 (z), whilst the generalization to a third parameter γ E γ α,β (z) = 1 Γ(γ)is recently receiving an increasing attention due to the applications in modeling polarization processes in anomalous or inhomogeneous materials [3].In this work we restrict our attention to real parameters α, β and γ, with α > 0 and γ > 0, since they are of more practical interest. * This work has been supported under the GNCS -INdAM Project 2014.† Università degli Studi di Bari "Aldo Moro", Dipartimento di Matematica Via E. Orabona n.4, 70125 Bari, Italy (roberto.garrappa@uniba.it) 1 2 R. GARRAPPAWith the exception of a few special cases in which the ML function can be represented in terms of other elementary and special functions, as for instance E 1,1 (z) = e z , E 2,1 (z 2 ) = cosh(z), E 2,1 (−z 2 ) = cos(z) and E 1 2 ,1 (±z 1/2 ) = e z ...

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Hybrid Electrospun Nanofibers as Electrocatalyst for Vanadium Redox Flow Batteries: Theory and Experiment

et al. 2021

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Selective modification of the morphology and intrinsic electrocatalytic activity of porous electrodes is urgently required to improve the performance of vanadium redox flow batteries (VRFBs). For this purpose, electrospinning was exploited to prepare high‐performance nanofiber‐based composites. Blends of polyacrylonitrile, polyacrylic acid, and polyaniline with carbon black were electrospun into a 3D free‐standing nanofibrous web, which was utilized as a novel electrode. By extending the recent theory of cyclic voltammetry at porous electrodes to account for interfacial double‐layer capacities, nonlinear effects of ohmic resistances, and parasitic reactions, we could quantitatively investigate non‐faradaic as well as desired and undesired faradaic current contributions. Combination of experimental and theoretical studies allowed a unique quantitative assessment of the intrinsic catalytic activity of selected electrode materials concerning the VO2+/VO2+ redox reaction.

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An improved Talbot method for numerical Laplace transform inversion

Cited by 85 publications

References 14 publications

The Exponentially Convergent Trapezoidal Rule

The Exponentially Convergent Trapezoidal Rule

Numerical Evaluation of Two and Three Parameter Mittag-Leffler Functions

Hybrid Electrospun Nanofibers as Electrocatalyst for Vanadium Redox Flow Batteries: Theory and Experiment

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