The Exponentially Convergent Trapezoidal Rule

Trefethen, Lloyd N.; Weideman, J. A. C.

doi:10.1137/130932132

Cited by 459 publications

(395 citation statements)

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“…There is no pleasant way to simplify the evaluation of the area integral (3.31) and produce an algebraic expression for the area (analogous to (2.43)), so we resort to using the trapezium rule to enforce (3.31) which is well known to converge geometrically for analytic functions on periodic intervals [38].…”

Section: (C) Non-zero-surface-tension Solutionsmentioning

confidence: 99%

The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell

2017

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New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.

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Section: (C) Non-zero-surface-tension Solutionsmentioning

confidence: 99%

The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell

2017

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“…An extensive analysis of the trapezoidal rule has been recently provided in the remarkable paper by Trefethen and Weideman [35] which not only focuses on the fast convergence of the trapezoidal rule but also discusses its main practical applications. Despite its NUMERICAL EVALUATION OF ML FUNCTIONS 5 simplicity, the trapezoidal rule appears indeed as a powerful tool to perform fast and highly accurate integration in a variety of applications.…”

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confidence: 99%

“…As deeply studied in [35,40], the choice of the contour affects in a significant way the convergence properties of the quadrature rule which depend on the analyticity of the integrand in a region surrounding the path of integration. A satisfactory selection of the deformed contour is therefore not possible without a subtle analysis of the regions in which E α,β (s; λ) is analytic.…”

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confidence: 99%

Numerical Evaluation of Two and Three Parameter Mittag-Leffler Functions

Garrappa¹

2015

SIAM J. Numer. Anal.

232

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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“…Trapezoidal rules such as in (3) generically exhibit low rates of convergence but in the special case of periodic integrands, the rate of convergence is spectacular [40,38], and the error in (3) decreases exponentially with N [43]. With only N = 2 4 , quadrature points in Figure 1 (just eight are needed in our cases of real solutions when symmetry is exploited) we get excellent accuracy at t = 1 ( Figure 2, bottom); using Matlab's expm as a reference solution, the maximum error across all components is ≈ 10 −7 .…”

Section: C36mentioning

confidence: 99%

“…For θ ∈ R and N = 2 4 , this article uses a parabola ( Figure 1) optimised by Weideman and Trefethen [40,43]:…”

Section: Introductionmentioning

confidence: 99%