Analytical solutions and numerical schemes of certain generalized fractional diffusion models

2019 ◽  
Vol 134 (5) ◽  
Author(s):  
Ndolane Sene
Keyword(s):  
Analytical Solutions ◽  
Fractional Diffusion ◽  
Diffusion Models ◽  
Numerical Schemes

scholarly journals Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization

2014 ◽  
Vol 11 (97) ◽  
pp. 20140352 ◽  
Author(s):  
Alfonso Bueno-Orovio ◽  
David Kay ◽  
Vicente Grau ◽  
Blanca Rodriguez ◽  
Kevin Burrage
Keyword(s):  
Structural Heterogeneity ◽  
Biological Tissues ◽  
Tissue Level ◽  
Fractional Diffusion ◽  
Diffusion Models ◽  
Excitable Media ◽  
Research Fields ◽  
Dispersion Of Repolarization ◽  
Electrical Propagation

Impulse propagation in biological tissues is known to be modulated by structural heterogeneity. In cardiac muscle, improved understanding on how this heterogeneity influences electrical spread is key to advancing our interpretation of dispersion of repolarization. We propose fractional diffusion models as a novel mathematical description of structurally heterogeneous excitable media, as a means of representing the modulation of the total electric field by the secondary electrical sources associated with tissue inhomogeneities. Our results, analysed against in vivo human recordings and experimental data of different animal species, indicate that structural heterogeneity underlies relevant characteristics of cardiac electrical propagation at tissue level. These include conduction effects on action potential (AP) morphology, the shortening of AP duration along the activation pathway and the progressive modulation by premature beats of spatial patterns of dispersion of repolarization. The proposed approach may also have important implications in other research fields involving excitable complex media.

Parameter Estimation in Fractional Diffusion Models

2017 ◽  
Author(s):  
Kęstutis Kubilius ◽  
Yuliya Mishura ◽  
Kostiantyn Ralchenko
Keyword(s):  
Parameter Estimation ◽  
Fractional Diffusion ◽  
Diffusion Models

scholarly journals Algorithms of Finite Difference for Pricing American Options under Fractional Diffusion Models

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Jun Xi ◽  
Yanqing Chen ◽  
Jianwen Cao
Keyword(s):  
Differential Equation ◽  
American Options ◽  
Compound Poisson Process ◽  
Fractional Diffusion ◽  
Diffusion Models ◽  
Singularity Problem ◽  
Order Accuracy ◽  
Fractional Difference ◽  
Finite Moment ◽  
First Order

It is well known that linear complementarity problem (LCP) involving partial integro differential equation (PIDE) arises from pricing American options under Lévy Models. In the case of infinite activity process, the integral part of the PIDE has a singularity, which is generally approximated by a small Brownian component plus a compound Poisson process, in the neighborhood of origin. The PIDE can be reformulated as a fractional partial differential equation (FPDE) under fractional diffusion models, including FMLS (finite moment log stable), CGMY (Carr-Madan-Geman-Yor), and KoBol (Koponen-Boyarchenko-Levendorskii). In this paper, we first present a stable iterative algorithm, which is based on the fractional difference approach and penalty method, to avoid the singularity problem and obtain numerical approximations of first-order accuracy. Then, on the basis of the first-order accurate algorithm, spatial extrapolation is employed to obtain second-order accurate numerical estimates. Numerical tests are performed to demonstrate the effectiveness of the algorithm and the extrapolation method. We believe that this can be used as necessary tools by the engineers in research.

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