New solutions of the wave equation by reduction to the heat equation

1995 ◽  
Vol 28 (18) ◽  
pp. 5291-5304 ◽  
Author(s):  
P Basarab-Horwath ◽  
L Barannyk ◽  
W I Fushchych
Keyword(s):  
Wave Equation ◽  
Heat Equation

Rough estimates on the scalar heat kernel

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Wave Equation ◽  
Heat Equation ◽  
Heat Kernel ◽  
Heat Kernels ◽  
Hypoelliptic Operator ◽  
Uniform Bounds ◽  
Standard Heat ◽  
Probabilistic Construction

This chapter establishes rough estimates on the heat kernel rb,tX for the scalar hypoelliptic operator AbX on X defined in the preceding chapter. By rough estimates, this chapter refers to just the uniform bounds on the heat kernel. The chapter also obtains corresponding bounds for the heat kernels associated with operators AbX and another AbX over ̂X. Moreover, it gives a probabilistic construction of the heat kernels. This chapter also explains the relation of the heat equation for the hypoelliptic Laplacian on X to the wave equation on X and proves that as b → 0, the heat kernel rb,tX converges to the standard heat kernel of X.

scholarly journals Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bopeng Rao ◽  
Xu Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Wave Equation ◽  
Heat Equation ◽  
Frequency Domain ◽  
Parabolic System ◽  
Decay Rates ◽  
Fluid Structure Interactions ◽  
Fluid Structure ◽  
Structure Interaction ◽  
Frequency Domain Approach

<p style='text-indent:20px;'>We consider the asymptotic behavior of a linear model arising in fluid-structure interactions. The system is formed by a heat equation and a wave equation in two distinct domains, which are coupled by atransmission condition along the interface of the domains. By means of the frequency domain approach, we establish some decay rates for the whole system. Our results also showthat the decay of the fluid-structure interaction depends not only on the transmission of the damping from the heat equation to the wave equation, but also on the location of the damping for the wave equation.</p>

scholarly journals Uniqueness and numerical reconstruction for inverse problems dealing with interval size search

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jone Apraiz ◽  
Jin Cheng ◽  
Anna Doubova ◽  
Enrique Fernández-Cara ◽  
Masahiro Yamamoto
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Inverse Problems ◽  
Heat Equation ◽  
Numerical Experiments ◽  
Spatial Dimension ◽  
Numerical Reconstruction ◽  
Spatial Interval ◽  
Boundary Information ◽  
Size Estimates

<p style='text-indent:20px;'>We consider a heat equation and a wave equation in one spatial dimension. This article deals with the inverse problem of determining the size of the spatial interval from some extra boundary information on the solution. Under several different circumstances, we prove uniqueness, non-uniqueness and some size estimates. Moreover, we numerically solve the inverse problems and compute accurate approximations of the size. This is illustrated with several satisfactory numerical experiments.</p>

scholarly journals The enclosure method for the heat equation using time-reversal invariance for a wave equation

2020 ◽  
Vol 28 (1) ◽  
pp. 93-104
Author(s):  
Masaru Ikehata
Keyword(s):  
Heat Flux ◽  
Wave Equation ◽  
Heat Equation ◽  
Time Reversal ◽  
The Body ◽  
Time Interval ◽  
Time Reversal Invariance ◽  
Large Parameter ◽  
Reversal Invariance ◽  
Enclosure Method

AbstractThe heat equation does not have time-reversal invariance. However, using a solution of an associated wave equation which has time-reversal invariance, one can establish an explicit extraction formula of the minimum sphere that is centered at an arbitrary given point and encloses an unknown cavity inside a heat conductive body. The data employed in the formula consist of a special heat flux depending on a large parameter prescribed on the surface of the body over an arbitrary fixed finite time interval and the corresponding temperature field. The heat flux never blows up as the parameter tends to infinity. This is different from a previous formula for the heat equation which also yields the minimum sphere. In this sense, the prescribed heat flux is moderate.

scholarly journals Asymptotic properties of solutions of the fractional diffusion-wave equation

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Anatoly Kochubei
Keyword(s):  
Wave Equation ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Parabolic Equations ◽  
Hyperbolic Equations ◽  
Asymptotic Properties ◽  
Fractional Diffusion ◽  
Time Variable ◽  
Diffusion Wave

AbstractFor the fractional diffusion-wave equation with the Caputo-Djrbashian fractional derivative of order α ∈ (1, 2) with respect to the time variable, we prove an analog of the principle of limiting amplitude (well-known for the wave equation and some other hyperbolic equations) and a pointwise stabilization property of solutions (similar to a well-known property of the heat equation and some other parabolic equations).

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