Numerical calculations accuracy comparison of the Inverse Laplace Transform algorithms for solutions of fractional order differential equations

Brzeziński, Dariusz W.; Ostalczyk, Piotr

doi:10.1007/s11071-015-2225-8

Cited by 16 publications

(5 citation statements)

References 48 publications

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“…For y = 0, the particle distribution on the backbone is obtained, we apply the Fox H-function [22] to describe the particle distribution on the backbone: Finally, we can apply numerical Laplace inversion [23] to calculate MSD on the x axis.…”

Section: Verification Of Numerical Solutionmentioning

confidence: 99%

Anomalous diffusion and heat transfer on comb structure with anisotropic relaxation in fractal porous media

2021

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A kind of anomalous diffusion and heat transfer on a comb structure with anisotropic relaxation are studied, which can be used to model many problems in Biologic and Nature in fractal porous media. The Hausdorff derivative is introduced and new governing equations is formulated in view of fractal dimension. Numerical solutions are obtained and the Fox H-function analytical solutions is given for special cases. The particles spatial-temporal evolution (STE) and the mean square displacement (MSD) versus time are presented. The effects of backbone and finger relaxation parameters, and the time fractal parameter are discussed. Results show that the MSD decreases with the increase of backbone parameter or the decrease of finger relaxation parameter in a short of time, but they have little effect on MSD in a long period. Particularly, the MSD has time dependence in the form of /2 t  ( 01   ) when t >τ, which indicates that the diffusion is an anomalous sub-diffusion and heat transfer.

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Section: Verification Of Numerical Solutionmentioning

confidence: 99%

Anomalous diffusion and heat transfer on comb structure with anisotropic relaxation in fractal porous media

2021

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“…Numerical algorithms of the Laplace inversion can also be used for a transient analysis [20,21]. An excellent discussion of the selected algorithms is given in [22,23].…”

Section: Introductionmentioning

confidence: 99%

Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems

et al. 2023

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In this paper, the possibilities of expressing the natural response of a linear commensurate fractional-order system (FOS) as a linear combination of basis functions are analyzed. For all possible types of sα-domain poles, the corresponding basis functions are found, the kernel of which is the two-parameter Mittag–Leffler function Eα,β, β = α. It is pointed out that there are mutually unambiguous correspondences between the basis functions of FOS and the known basis functions of the integer-order system (IOS) for α = 1. This correspondence can be used to algorithmically find analytical formulas for the impulse responses of FOS when the formulas for the characteristics of IOS are known. It is shown that all basis functions of FOS can be generated with Podlubny‘s function of type εk (t, c; α, α), where c and k are the corresponding pole and its multiplicity, respectively.

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“…In the paper [17] we gave an initial evaluation for the inversion accuracy of several methods representing the main approaches to the NumILPT. Testbed included Laplace transforms of FODE solutions we have found in the most cited papers at those times.…”

Section: Introductionmentioning

confidence: 99%

Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus

Brzeziński

2018

Applied Mathematics and Nonlinear Sciences

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In the paper we present results of accuracy evaluation of numerous numerical algorithms for the numerical approximation of the Inverse Laplace Transform. The selected algorithms represent diverse lines of approach to this problem and include methods by Stehfest, Abate and Whitt, Vlach and Singhai, De Hoog, Talbot, Zakian and a one in which the FFT is applied for the Fourier series convergence acceleration. We use C++ and Python languages with arbitrary precision mathematical libraries to address some crucial issues of numerical implementation. The test set includes Laplace transforms considered as difficult to compute as well as some others commonly applied in fractional calculus. Evaluation results enable to conclude that the Talbot method which involves deformed Bromwich contour integration, the De Hoog and the Abate and Whitt methods using Fourier series expansion with accelerated convergence can be assumed as general purpose high-accuracy algorithms. They can be applied to a wide variety of inversion problems.

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Numerical calculations accuracy comparison of the Inverse Laplace Transform algorithms for solutions of fractional order differential equations

Cited by 16 publications

References 48 publications

Anomalous diffusion and heat transfer on comb structure with anisotropic relaxation in fractal porous media

Anomalous diffusion and heat transfer on comb structure with anisotropic relaxation in fractal porous media

Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems

Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus

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