Accurate Padé Global Approximations for the Mittag-Leffler Function, Its Inverse, and Its Partial Derivatives to Efficiently Compute Convergent Power Series

Ingo, Carson; Barrick, Thomas R.; Webb, Andrew; Ronen, Itamar

doi:10.1007/s40819-016-0158-7

Cited by 19 publications

(17 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To find the other 2ν unknowns {p i , q i } ν−1 i=0 , we solve the system of linear equations obtained by satisfying the requirement (14) which takes the form…”

Section: 2mentioning

confidence: 99%

“…For β = α, the coefficients satisfy the system [14] is not given, based on their construction, we expect the type to be a special case of one of the approximants above when β = 1.…”

Section: Fourth-order Global Padé Approximantsmentioning

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%

“…Lately, approximations of L α,β have been introduced to overcome the difficulty of solving the functional equation (2). Atkinson and Osseiran [1]; and Ingo et al [14] discussed the approximation of (−L α,1 ) based on the inversion of their global Padé approximants. Similarly, Zeng and Chen [25]; and Iyiola et al [15] inverted their second order approximants of E α,β to obtain an approximation of (−L α,β ).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Sarumi

Furati

2020

J Sci Comput

View full text Add to dashboard Cite

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.

show abstract

“…To find the other 2ν unknowns {p i , q i } ν−1 i=0 , we solve the system of linear equations obtained by satisfying the requirement (14) which takes the form…”

Section: 2mentioning

confidence: 99%

“…For β = α, the coefficients satisfy the system [14] is not given, based on their construction, we expect the type to be a special case of one of the approximants above when β = 1.…”

Section: Fourth-order Global Padé Approximantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Sarumi

Furati

2020

J Sci Comput

View full text Add to dashboard Cite

show abstract

“…An engagement of the community in developing the appropriate numerical tools for the accurate evaluation of these extended (Kilbas-Saigo) Mittag-Leffler functions might further increase the value of the results presented in this paper. One possibility in this regard might be the use of global Padé approximations, which have been proven useful in the accurate and efficient evaluation of other types of Mittag-Leffler functions [37,38]. A related but different approach might involve using peridynamics models with fractional power law kernels in order to model the Bloch-Torrey equation.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

Exact solutions to the fractional time-space Bloch–Torrey equation for magnetic resonance imaging

Bueno‐Orovio

Burrage

2017

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

Highlights• The full fractional Bloch-Torrey setting for magnetic resonance imaging is solved.• Solutions are given by convolution of extended Mittag-Leffler/Lauricella functions.• Analytical solutions replicate super-and sub-diffusive regimes in signal decay.• Residual signal phase shifts due to incomplete spin refocusing are also captured.• This allows an estimation of tissue properties based on exact diffusive processes. AbstractThe quantification of anomalous diffusion is increasingly being recognised as an advanced modality of analysis for the evaluation of tissue microstructure in magnetic resonance imaging (MRI). One powerful framework to account for anomalous diffusion in biological and structurally heterogeneous tissues is the use of diffusion operators based on fractional calculus theory, which generalises the physical principles of standard diffusion in homogeneous media. However, their non-locality makes analytical solutions often unavailable, limiting the applicability of these modelling and analysis techniques. In this paper, we derive compact analytical signal decays for practical MRI sequences in the anisotropic fractional Bloch-Torrey setting, as described by the space fractional Laplacian and importantly the time Caputo derivative. The attained solutions convey relevant characteristics of MRI in biological tissues not replicated by standard diffusion, including super-diffusive and sub-diffusive regimes in signal decay and the diffusion-driven incomplete refocusing of spins at the end of the sequence. These results may therefore have significant implications for advancing the current interpretation of MRI, and for the estimation of tissue properties based on exact solutions to underlying diffusive processes.

show abstract

Optimization of quasi‐diffusion magnetic resonance imaging for quantitative accuracy and time‐efficient acquisition

Spilling

Howe

Barrick

2022

Magnetic Resonance in Med

Self Cite

View full text Add to dashboard Cite

Quasi-diffusion MRI (QDI) is a novel quantitative technique based on the continuous time random walk model of diffusion dynamics. QDI provides estimates of the diffusion coefficient, D 1,2 in mm 2 s −1 and a fractional exponent, α, defining the non-Gaussianity of the diffusion signal decay. Here, the b-value selection for rapid clinical acquisition of QDI tensor imaging (QDTI) data is optimized. Methods: Clinically appropriate QDTI acquisitions were optimized in healthy volunteers with respect to a multi-b-value reference (MbR) dataset comprising 29 diffusion-sensitized images arrayed between b = 0 and 5000 s mm −2 . The effects of varying maximum b-value (b max ), number of b-value shells, and the effects of Rician noise were investigated. Results: QDTI measures showed b max dependence, most significantly for α in white matter, which monotonically decreased with higher b max leading to improved tissue contrast. Optimized 2 b-value shell acquisitions showed small systematic differences in QDTI measures relative to MbR values, with overestimation of D 1,2 and underestimation of α in white matter, and overestimation of D 1,2 and α anisotropies in gray and white matter. Additional shells improved the accuracy, precision, and reliability of QDTI estimates with 3 and 4 shells at b max = 5000 s mm −2 , and 4 b-value shells at b max = 3960 s mm −2 , providing minimal bias in D 1,2 and α compared to the MbR. Conclusion: A highly detailed optimization of non-Gaussian dMRI for in vivo brain imaging was performed. QDI provided robust parameterization of non-Gaussian diffusion signal decay in clinically feasible imaging times with high reliability, accuracy, and precision of QDTI measures.

show abstract

Accurate Padé Global Approximations for the Mittag-Leffler Function, Its Inverse, and Its Partial Derivatives to Efficiently Compute Convergent Power Series

Cited by 19 publications

References 44 publications

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

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

Exact solutions to the fractional time-space Bloch–Torrey equation for magnetic resonance imaging

Optimization of quasi‐diffusion magnetic resonance imaging for quantitative accuracy and time‐efficient acquisition

Contact Info

Product

Resources

About