Rational Solutions for the Time-Fractional Diffusion Equation

Atkinson, C.; Osseiran, Adel

doi:10.1137/100799307

Cited by 46 publications

(32 citation statements)

References 13 publications

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“…It should be pointed out that this special case was studied by [1]. When 0 < α = β < 1, the generalized Mittag-Leffler function E α,α (−x) admits the two series…”

Section: Global Padé Approximations Of E αβ (−X)mentioning

confidence: 99%

“…The authors applied it to find good approximations of several functions including the elliptic function, the error function of real and imaginary arguments, the Bessel functions, and the Airy function. Also, Atkinson and Osseiran [1] applied this method to find a uniform rational approximation of the Mittag-Leffler function E α (−z) with 0 < α < 1 and z ∈ (0, ∞).…”

Section: Introductionmentioning

confidence: 99%

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Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse

Zeng

Chen

2015

FCAA

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This paper proposes a global Padé approximation of the generalized Mittag-Leffler function E α,β (−x) with x ∈ [0, +∞). This uniform approximation can account for both the Taylor series for small arguments and asymptotic series for large arguments. Based on the complete monotonicity of the function E α,β (−x), we work out the global Padé approximation [1/2] for the particular cases {0 < α < 1, β > α}, {0 < α = β < 1}, and {α = 1, β > 1}, respectively. Moreover, these approximations are inverted to yield a global Padé approximation of the inverse generalized Mittag-Leffler function −L α,β (x) with x ∈ (0, 1/Γ(β)]. We also provide several examples with selected values α and β to compute the relative error from the approximations. Finally, we point out the possible applications using our established approximations in the ordinary and partial time-fractional differential equations in the sense of Riemann-Liouville.MSC 2010 : Primary 26A33; Secondary 33E12, 35S10, 45K05

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“…It should be pointed out that this special case was studied by [1]. When 0 < α = β < 1, the generalized Mittag-Leffler function E α,α (−x) admits the two series…”

Section: Global Padé Approximations Of E αβ (−X)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse

Zeng

Chen

2015

FCAA

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“…In this technique, rational approximations are constructed by matching them with selected combinations of the series definition and the asymptotic expansion. Atkinson and Osseiran [1] used this technique to construct a second-order rational approximation for E α . Later, Ingo et al [14] showed that the rational approximant in [1] is not satisfactory for α close to one.…”

Section: Introductionmentioning

confidence: 99%

“…Atkinson and Osseiran [1] used this technique to construct a second-order rational approximation for E α . Later, Ingo et al [14] showed that the rational approximant in [1] is not satisfactory for α close to one. Alternatively, they constructed a fourth-order global approximation for E α that behaves reasonably well for all values of α ∈ (0, 1).…”

Section: Introductionmentioning

confidence: 99%

Highly Accurate Global Padé Approximations of Generalized Mittag–Leffler Function and Its Inverse

Sarumi

Furati

2020

J Sci Comput

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The two-parametric Mittag-Leffler function (MLF), E α,β , is fundamental to the study and simulation of fractional differential and integral equations. However, these functions are computationally expensive and their numerical implementations are challenging. In this paper, we present a unified framework for developing global rationalThis framework is based on the series definition and the asymptotic expansion at infinity. In particular, we develop three types of fourth-order global rational approximations and discuss how they could be used to approximate the inverse function. Unlike existing approximations which are either limited to MLF of one parameter or of low accuracy for the two-parametric MLF, our rational approximants are of fourth order accuracy and have low percentage error globally. For efficient utilization, we study the partial fraction decomposition and use them to approximate the two-parametric MLF with a matrix argument which arise in the solutions of fractional evolution differential and integral equations.

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“…As mentioned previously, it aims to match asymptotical points of f using (m, n) Padé approximant. In addition, it has been successfully applied to both the Mittag-Leffler function (Atkinson & Osseiran, 2011) and generalized Mittag-Leffler function (Zeng & Chen, 2015). (Gatheral & Radoicic, 2019) demonstrated the use of global rational approximations for the fractional Riccati equation and found the excellent performance of the multipoint Padé approximation (3, 3) especially when H is close to 0.…”

Section: Introductionmentioning

confidence: 99%

Fractional Riccati Equation and Its Applications to Rough Heston Model

Jeng¹,

Kılıçman²

2020

Preprint

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Rough volatility models are popularized by \cite{gatheral2018volatility}, where they have shown that the empirical volatility in the financial market is extremely consistent with rough volatility. Fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form as of now and therefore, we must rely on numerical methods to obtain a solution. In this paper, we give a short introduction to option pricing theory and an overview of the current advancements on the rough Heston model.

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Rational Solutions for the Time-Fractional Diffusion Equation

Cited by 46 publications

References 13 publications

Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse

Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse

Highly Accurate Global Padé Approximations of Generalized Mittag–Leffler Function and Its Inverse

Fractional Riccati Equation and Its Applications to Rough Heston Model

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