On random time and on the relation between wave and telegraph equations

Janaswamy, Ramakrishna

doi:10.1109/tap.2013.2237739

Cited by 9 publications

(4 citation statements)

References 34 publications

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“…But the classical telegraph equations cannot adequately describe the abnormal diffusion phenomena during the long‐range transmission, where the voltage or current waves possibly occur . As the memory and hereditary properties of different substances can be described using the fractional derivatives and integrals, it is necessary to study the fractional telegraph equations . They are more suitable for characterizing transmission and propagation of electrical signals than the classical ones .…”

Section: The Application Of Fourth‐order Fractional‐compact Numericalmentioning

confidence: 99%

High‐order algorithms for Riesz derivative and their applications (V)

Ding

2017

Numerical Methods Partial

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In this article, based on the idea of combing symmetrical fractional centred difference operator with compact technique, a series of even‐order numerical differential formulas (named the fractional‐compact formulas) are established for the Riesz derivatives with order α ∈ ( 1 , 2 ) . Properties of coefficients in the derived formulas are studied in details. Then applying the constructed fourth‐order formula, a difference scheme is proposed to solve the Riesz spatial telegraph equation. By the energy method, the constructed numerical algorithm is proved to be stable and convergent with order O ( τ 4 + h 4 ) , where τ and h are the temporal and spatial stepsizes, respectively. Finally, several numerical examples are presented to verify the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1754–1794, 2017

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Section: The Application Of Fourth‐order Fractional‐compact Numericalmentioning

confidence: 99%

High‐order algorithms for Riesz derivative and their applications (V)

Ding

2017

Numerical Methods Partial

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“…Due to the memory and hereditary properties of different substances can be described by using the fractional derivatives and integrals, so, it is necessary to study the fractional telegraph equations. In fact, the fractional telegraph equations are the mixture models between diffusion and wave propagation [16,21], hence, they are more suitable for characterizing transmission and propagation of electrical signals than the classical ones [27,29].…”

Section: The Fractional-compact Numerical Formulas For Riesz Derivativesmentioning

confidence: 99%

High-order fractional-compact finite difference method for Riesz spatial telegraph equation

Ding,

2015

Preprint

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In this paper, we establish even order fractional-compact numerical differential formulas (4th-order, . . . , 10th-order) for Riesz derivatives by using the symmetrical fractional centred difference operator. Then we apply the derived 4th-order algorithm to the Riesz spatial telegraph equation. We carefully study the stability and convergence by the energy method, and show that convergence orders in temporal and spatial directions are both 4th order. Numerical experiments are displayed which support the fractional-compact numerical differentiation formulas for Riesz derivatives and the Riesz spatial telegraph equation.

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“…Goldstein [8] and Kac [13] were the first to construct stochastic solutions of the one-dimensional telegrapher's equation under some special initial conditions, using "persistent random walk". More general results are derived later, see [15,9,19,7,11]. Since most of those constructions require the solution of the wave equation associated to the telegrapher's equation, they cannot deal with the wave equation itself.…”

Section: Introductionmentioning

confidence: 99%

Stochastic representations for the wave equation on graphs and their scaling limits

Wang

2017

Journal of Mathematical Analysis and Applications

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This paper is devoted to an interacting particle system that provides probabilistic interpretation of the wave equation on graphs. A Feynman-Kac-type formula is established, connecting the expectation of the process with the wave equation on graphs. Non-asymptotic L 2 estimates are presented. It is then shown that the high-density hydrodynamic limit of the system is given by the wave equation in Euclidean space. The sharpness of scaling limit result is demonstrated by a phase transition phenomenon.

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On random time and on the relation between wave and telegraph equations

Cited by 9 publications

References 34 publications

High‐order algorithms for Riesz derivative and their applications (V)

High‐order algorithms for Riesz derivative and their applications (V)

High-order fractional-compact finite difference method for Riesz spatial telegraph equation

Stochastic representations for the wave equation on graphs and their scaling limits

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