Evaluation of generalized Mittag–Leffler functions on the real line

Garrappa, Roberto; Popolizio, Marina

doi:10.1007/s10444-012-9274-z

Cited by 82 publications

(66 citation statements)

References 25 publications

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“…In the case of the gamma function, however, f (s) = s −z can have large or small magnitude when z is far in the left or right halfplanes and the contour parameters (15.13) can become less reliable [145]. The same thing was pointed out in the case of the generalized Mittag-Leffler function in [55].…”

Section: Laplace Transforms and Hankel Contours The Laplace Transformentioning

confidence: 70%

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The Exponentially Convergent Trapezoidal Rule

Trefethen¹,

Weideman²

2014

SIAM Rev.

470

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: 70%

“…in the special cases α = 1 and α = 1 2 , respectively; see [55] and [118]. By taking a Laplace transform of (16.10), one can derive a contour integral representation and its approximation identical to (16.2) and (16.3), but the transform is now [118] …”

Section: Laplace Transforms and Hankel Contours The Laplace Transformentioning

confidence: 99%

The Exponentially Convergent Trapezoidal Rule

Trefethen¹,

Weideman²

2014

SIAM Rev.

470

362

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“…However, for a long time, this has been considered only as a theoretical tool because of the lack of effective methods to numerically approximate this function. Only recently have many advances been made for the numerical evaluation of the scalar ML function [26][27][28][29]; the case of matrix arguments has since been analyzed [30,31], and finally a numerical algorithm has been accomplished, which reaches very high accuracies [32]. In this paper, we show the effectiveness of the matrix approach when solving MTFDEs, both in terms of execution time and in terms of accuracy, and also in comparison with some well-established numerical methods.…”

Section: Introductionmentioning

confidence: 93%

Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions

Popolizio

2018

Mathematics

Self Cite

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Abstract:Multiterm fractional differential equations (MTFDEs) nowadays represent a widely used tool to model many important processes, particularly for multirate systems. Their numerical solution is then a compelling subject that deserves great attention, not least because of the difficulties to apply general purpose methods for fractional differential equations (FDEs) to this case. In this paper, we first transform the MTFDEs into equivalent systems of FDEs, as done by Diethelm and Ford; in this way, the solution can be expressed in terms of Mittag-Leffler (ML) functions evaluated at matrix arguments. We then propose to compute it by resorting to the matrix approach proposed by Garrappa and Popolizio. Several numerical tests are presented that clearly show that this matrix approach is very accurate and fast, also in comparison with other numerical methods.

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“…An efficient algorithm relies on partitioning the complex plane C into different regions, where different approximations, i.e., power series, integral representation and exponential asymptotic for small values of the argument, intermediate values and large values, respectively, are used for efficient numerical computation; see [94] for the some partition and error estimates. The special case of the Mittag-Leffler function E α,β (z) with a real argument z ∈ R, which plays a predominant role in time-fractional diffusion, can also be efficiently computed with the Laplace transform and suitable quadrature rules [27].…”

mentioning

confidence: 99%

An inverse problem for a one-dimensional time-fractional diffusion problem

Jin¹,

Rundell²

2012

Inverse Problems

150

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Abstract. Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weakly singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest.In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of ill-posedness of several "classical" inverse problems for fractional differential equations involving a Djrbashian-Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm-Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of "fractional" inverse problems, and a partial list of surprising observations is given below.(a) Classical backward diffusion is exponentially ill-posed, whereas time fractional backward diffusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more effective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm-Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suffice. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the preci...

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Evaluation of generalized Mittag–Leffler functions on the real line

Cited by 82 publications

References 25 publications

The Exponentially Convergent Trapezoidal Rule

The Exponentially Convergent Trapezoidal Rule

Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions

An inverse problem for a one-dimensional time-fractional diffusion problem

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