Doppler effect described by the solutions of the Cattaneo telegraph equation

Povstenko, Yuriy; Ostoja–Starzewski, Martin

doi:10.1007/s00707-020-02860-y

Cited by 11 publications

(6 citation statements)

References 26 publications

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“…We expect that further analysis of wave phenomena seen in solutions to the Cattaneo equation, like the Doppler studied recently in Ref. [72], will appear helpful to understand better wave properties of solutions to the generalized Cattaneo equation.…”

Section: Discussionmentioning

confidence: 86%

“…(x, ξ) = L −1 [F (x, z); ξ]. This inverse Laplace transform can be calculated by using the formulas placed in [71, Appendix A] and/or[72, Appendix B]. Namely,L −1 [(z 2 − λ 2 ) −1/2 exp[− |x| a (z 2 − λ 2 ) 1/2 ]; ξ} = Θ(ξ − |x| a )I 0 [ 1 2τ (ξ 2 − x 2 a 2 ) 1/2 ] (C2) and L −1 [z(z 2 − λ 2 ) −1/2 exp[− |x| a (z 2 − λ 2 ) 1/2 ]; ξ} = δ(ξ − |x| a ) + Θ(ξ − |x| a )…”

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confidence: 99%

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Integral decomposition for the solutions of the generalized Cattaneo equation

Górska

2021

Preprint

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We present the integral decomposition for the fundamental solution of the generalized Cattaneo equation with both time derivatives smeared through convoluting them with some memory kernels. For power-law kernels t −α , α ∈ (0, 1] this equation becomes the time fractional one governed by the Caputo derivatives which highest order is 2. To invert the solutions from the Fourier-Laplace domain to the space-time domain we use analytic methods based on the Efross theorem and find out that solutions looked for are represented by integral decompositions which tangle the fundamental solution of the standard Cattaneo equation with non-negative and normalizable functions being uniquely dependent on the memory kernels. Furthermore, the use of methodology arising from the theory of complete Bernstein functions allows us to assign such constructed integral decompositions the interpretation of subordination. This fact is preserved in two limit cases built into the generalized Cattaneo equations, i.e., either the diffusion or the wave equations. We point out that applying the Efross theorem enables us to go beyond the standard approach which usually leads to the integral decompositions involving the Gaussian distribution describing the Brownian motion. Our approach clarifies puzzling situation which takes place for the power-law kernels t −α for which the subordination based on the Brownian motion does not work if α ∈ (1/2, 1].

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Section: Discussionmentioning

confidence: 86%

mentioning

confidence: 99%

Integral decomposition for the solutions of the generalized Cattaneo equation

Górska

2021

Preprint

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“…(vi) When n 0 0, n 1 1, χ 0 > 1 and 0 < χ 1 < 1, we obtain the half-space solution of the modified hyperbolic DPL. It is found that for copper the ratio χ 1 0.246 [12], which yields that χ 2 1 0.06 not 0.5 as in the original hyperbolic DPL, refer to (22).…”

Section: Numerical Results and Discussionmentioning

confidence: 95%

“…Furthermore, the generalized variational principles for the Fourier, MCV, parabolic DPL, two-temperature, and Guyer-Krumhansl heat conduction models have been obtained using the Laplace transform and employed to approximate analyses based on the Rayleigh-Ritz method by Li and Cao [18,19]. We refer the reader to the recent works on the MCV telegraph equation and its generalizations which have investigated its connection with continuous-time random walk scheme and its conditional solutions [20,21], the Doppler effect [22], and the integral decomposition [23].…”

Section: Introductionmentioning

confidence: 99%

A comparative numerical study of a semi-infinite heat conductor subject to double strip heating under non-Fourier models

2022

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The present work considers a two-dimensional (2D) heat conduction problem in the semi-infinite domain based on the classical Fourier model and other non-Fourier models, e.g., the Maxwell–Cattaneo–Vernotte (MCV) equation, parabolic, hyperbolic, and modified hyperbolic dual-phase-lag (DPL) equations. Using the integral transform technique, Laplace, and Fourier transforms, we provide a solution of the problem (Green’s function) in Laplace domain. The thermal double-strip problem, allowing the wave interference within the heat conductor, is considered. A numerical technique, based on the Durbin series for inverting Laplace transform and the trapezoidal rule for calculating an integral form of the solution in the double-strip case, is adopted to recover the solution in the physical domain. Finally, discussions for different non-Fourier heat transfer situations are presented. We compare among the speeds of hyperbolic heat transfer models and shed light on the concepts of flux-precedence and temperature-gradient-precedence, hallmarks of the lagging response idea. Otherwise, we emphasize the existence of a relationship between the waves speed and the time instant of interference onset, underlying the five employed heat transfer models.

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“…Effectively, since analytical solutions are impossible, the study method is of the Monte Carlo type. Note that the analytical solution elucidating the Doppler effect in the subsonic range of the second sound in a homogeneous 1d medium has recently been obtained [15].…”

Section: Problem Formulationmentioning

confidence: 99%

Mach Fronts in Random Media with Fractal and Hurst Effects

2021

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An investigation of transient second sound phenomena due to moving heat sources on planar random media is conducted. The spatial material randomness of the relaxation time is modeled by Cauchy or Dagum random fields allowing for decoupling of fractal and Hurst effects. The Maxwell–Cattaneo model is solved by a second-order central differencing. The resulting stochastic fluctuations of Mach wedges are examined and compared to unperturbed Mach wedges resulting from the heat source traveling in a homogeneous domain. All the examined cases are illustrated by simulation movies linked to this paper.

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Doppler effect described by the solutions of the Cattaneo telegraph equation

Cited by 11 publications

References 26 publications

Integral decomposition for the solutions of the generalized Cattaneo equation

Integral decomposition for the solutions of the generalized Cattaneo equation

A comparative numerical study of a semi-infinite heat conductor subject to double strip heating under non-Fourier models

Mach Fronts in Random Media with Fractal and Hurst Effects

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