Stochastic solutions for fractional wave equations

Meerschaert, Mark M.; Schilling, René L.; Sikorskii, Alla

doi:10.1007/s11071-014-1299-z

Cited by 32 publications

(34 citation statements)

References 22 publications

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“…The analysis of the results presented in Figs. 1,2,3,4,5,6,7,8,9,10 and 11 grant to conclude that the evaluated numerical methods of the Inverse Laplace Transform enable to obtain satisfactory accurate inversions. The methods can be successfully applied as tools for obtaining solutions of fractional order differential equations.…”

Section: Discussionmentioning

confidence: 93%

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

Brzeziński

Ostalczyk

2015

Nonlinear Dyn

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In the paper we present results of a numerical experiment in which we evaluate and compare some numerical algorithms of the Inverse Laplace Transform for inversion accuracy of some fractional order differential equations solutions. The algorithms represent diverse lines of approach to the subject of the numerical inversion 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 Fourier series convergence acceleration. We used C++ and Python languages and applied arbitrary precision mathematical libraries to address some crucial issues of an numerical implementation. Introductory test set includes Laplace transforms which are considered as difficult to compute as well as some others commonly applied in fractional calculus. In the main part of the evaluation, there is assessed accuracy of the numerical Inverse Laplace Transform of some popular fractional differential equations solutions, e.g., the initial value problem in case of the inhomogeneous Bagley-Torvik equation and composite fractional oscillation equation. Evaluation results enable to conclude that the Talbot method which involves deformed Bromwich contour integration, the De Hoog and the Abate and Whitt meth-

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Section: Discussionmentioning

confidence: 93%

“…Advantages of fractional calculus application led to a rapid grow of its popularity among scientists in various areas of computer science, mathematics and physics [6][7][8][9][10][11][12]. An important application of fractional calculus is solutions of differential equations of fractional order [13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

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

Brzeziński

Ostalczyk

2015

Nonlinear Dyn

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“…where Y α/2 (t) = sup 0≤s≤t X α (s), and X α (t) (1 ≤ α ≤ 2) is a càdlàg stable process with characteristic function E exp{iξX α (t)} = exp{−t|ξ| 2/α e −(πi/2)(2−2/α)sgn(ξ) }. Y α/2 (t) can also be regarded as the inverse of an α/2 stable subordinator [24,26]. When α = 2, the expression (1.6) reduces to d'Alembert's formula (1.2).…”

Section: Introductionmentioning

confidence: 99%

The fractional d'Alembert's formulas

Piskarev

et al. 2019

Journal of Functional Analysis

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In this paper we develop generalized d'Alembert's formulas for abstract fractional integro-differential equations and fractional differential equations on Banach spaces. Some examples are given to illustrate our abstract results, and the probability interpretation of these fractional d'Alembert's formulas are also given. Moreover, we also provide d'Alembert's formulas for abstract fractional telegraph equations.1991 Mathematics Subject Classification. Primary 45K05, 45N05, 35R11, 60G52; Secondary 26A33, 35L05, 60G22.

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“…For example, the subordination principle for fractional evolution equations with the Caputo derivative (see [16], Ch. 3) has been successfully applied to inverse problems [17], for asymptotic analysis of diffusion wave equations [18], for the study of stochastic solutions [19], semilinear equations of fractional order [20], systems of fractional order equations [21], nonlocal fragmentation models with the Michaud time derivative [22], etc. This gives the author the motivation to present an analogous principle for Problem (1).…”

Section: Introductionmentioning

confidence: 99%

Subordination Principle for a Class of Fractional Order Differential Equations

Bazhlekova

2015

Mathematics

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The fractional order differential equationis studied, where A is an operator generating a strongly continuous one-parameter semigroup on a Banach space X, D α t is the Riemann-Liouville fractional derivative of order α ∈ (0, 1), γ > 0 and f is an X-valued function. Equations of this type appear in the modeling of unidirectional viscoelastic flows. Well-posedness is proven, and a subordination identity is obtained relating the solution operator of the considered problem and the C 0 -semigroup, generated by the operator A. As an example, the Rayleigh-Stokes problem for a generalized second-grade fluid is considered.

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Stochastic solutions for fractional wave equations

Cited by 32 publications

References 22 publications

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

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

The fractional d'Alembert's formulas

Subordination Principle for a Class of Fractional Order Differential Equations

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