scholarly journals Analysis of the energy decay of a viscoelasticity type equation

2016 ◽  
Vol 24 (3) ◽  
pp. 21-45 ◽  
Author(s):  
Amel Atallah-Baraket ◽  
Maryem Trabelsi
Keyword(s):  
Wave Equation ◽  
Differential Operator ◽  
Energy Density ◽  
Type Equation ◽  
Weak Limit ◽  
Energy Decay ◽  
Pseudo Differential Operator ◽  
Viscosity Term ◽  
Special Assumption

AbstractIn this paper, we study the evolution of the energy density of a sequence of solutions of a problem related to a viscoelasticity model where the viscosity term is a pseudo-differential operator of order 2α with α ∈ (0, 1). We calculate the weak limit of the energy density in terms of microlocal defect measures and under special assumption we prove that the viscosity term prevents propagation of concentration and oscillation effects contrary to what happens in the wave equation.

HIGH FREQUENCY ANALYSIS OF FAMILIES OF SOLUTIONS TO THE EQUATION OF VISCOELASTICITY OF KELVIN–VOIGT

2004 ◽  
Vol 01 (04) ◽  
pp. 789-812 ◽  
Author(s):  
AMEL ATALLAH-BARAKET ◽  
CLOTILDE FERMANIAN KAMMERER
Keyword(s):  
Wave Equation ◽  
Energy Density ◽  
Initial Data ◽  
Lower Bounds ◽  
Frequency Analysis ◽  
High Frequency ◽  
Weak Limit ◽  
Negative Viscosity

In this paper, we study the evolution of the energy density of a sequence of solutions to the Kelvin–Voigt viscoelasticity equation. We do not suppose lower bounds on the non-negative viscosity matrix. We prove that, in the zone where the viscosity matrix is invertible, this term prevents propagation of concentation and oscillation effects contrary to what happens in the wave equation. We calculate precisely the weak limit of the energy density in terms of microlocal defect measures associated with the initial data under the assumption that the oscillations of the data are not microlocally localized on directions which are in the kernel of the viscosity matrix.

Efficient acoustic scalar wave equation modeling in VTI media

Geophysics ◽  
2020 ◽  
pp. 1-93
Author(s):  
Yury Nikonenko ◽  
Marwan Charara
Keyword(s):  
Wave Equation ◽  
Differential Operator ◽  
Acoustic Wave ◽  
Stress Component ◽  
Finite Difference Methods ◽  
P Wave ◽  
Computational Time ◽  
Equation Modeling ◽  
Pseudo Differential Operator ◽  
Spatial Operator

We present a new approach for acoustic wave modeling in transversely isotropic media with a vertical axis of symmetry. This approach is based on using a pure acoustic wave equation derived from the basic physical laws – Hooke’s law and the equation of motion. We show that the conventional equation noted as pure quasi-P wave equation computes only one stress component. In our approach, there is no need to approximate the pseudo-differential operator for decomposition purposes. We make a discrete inverse Fourier transform of the desired frequency response contained in the pseudo-differential operator to build the corresponding spatial operator. We then cut off the operator with a window to reduce edge effects. As a result, the obtained spatial operator is applied locally to the wavefield through a simple convolution. Consequently, we derive an explicit numerical scheme for a pure quasi-P wave mode. The most important advantage of our method lies in its locality, which means that our spatial operator can be applied in any selected region separately. Our approach can be combined with classical fast finite-difference methods when media are isotropic or elliptically anisotropic, therefore avoiding spurious fields and reducing the total computational time and memory. The accuracy, stability, and the absence of the residual S-waves of our approach were demonstrated with several numerical examples.

Time distributed-order diffusion-wave equation. I. Volterra-type equation

2009 ◽  
Vol 465 (2106) ◽  
pp. 1869-1891 ◽  
Author(s):  
Teodor M. Atanackovic ◽  
Stevan Pilipovic ◽  
Dusan Zorica
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Differential Operator ◽  
Fractional Derivative ◽  
Type Equation ◽  
General Linear ◽  
Uniqueness Of Solutions ◽  
Volterra Type ◽  
Diffusion Wave ◽  
Distributed Order

A single-order time-fractional diffusion-wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator. The obtained equation is (in the case of the Laplacian) called a time distributed-order diffusion-wave equation. We analyse a Cauchy problem for such an equation by means of the theory of an abstract Volterra equation. The weight distribution, occurring in the distributed-order fractional derivative, is specified as the sum of the Dirac distributions and the existence and uniqueness of solutions to the Cauchy problem, and the corresponding Volterra-type equation were proven for a general linear spatial differential operator, as well as in the special case when the operator is Laplacian.

The Mehler-Fock-Clifford transform and pseudo-differential operator on function spaces

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2457-2469
Author(s):  
Akhilesh Prasad ◽  
S.K. Verma
Keyword(s):  
Differential Operator ◽  
Function Spaces ◽  
Pseudo Differential Operator ◽  
Test Function ◽  
Basic Properties ◽  
Cone Function ◽  
Translation Operators

In this article, weintroduce a new index transform associated with the cone function Pi ??-1/2 (2?x), named as Mehler-Fock-Clifford transform and study its some basic properties. Convolution and translation operators are defined and obtained their estimates under Lp(I, x-1/2 dx) norm. The test function spaces G? and F? are introduced and discussed the continuity of the differential operator and MFC-transform on these spaces. Moreover, the pseudo-differential operator (p.d.o.) involving MFC-transform is defined and studied its continuity between G? and F?.

SOME COMMENTS ON THE DYNAMICS OF EXTENDED OBJECTS

1998 ◽  
Vol 13 (17) ◽  
pp. 2979-2990 ◽  
Author(s):  
U. KHANAL
Keyword(s):  
Wave Equation ◽  
Energy Density ◽  
Pure Shear ◽  
Equation Of Motion ◽  
Conducting Medium ◽  
Constant Density ◽  
Metric Tensor ◽  
World Volume ◽  
The World ◽  
Hilbert Action

A variational method is used to investigate the dynamics of extended objects. The stationary world volume requires the internal coordinates to propagate as free waves. Stationarity of the action which is the integral of a variable energy density over the world volume leads to the wave equation in a medium, with conductivity given by the gradient of the logarithm of reciprocal energy density, constant density corresponding to free space. The Einstein–Hilbert action for the world curvature gives an equation of motion which, in world space with the Einstein tensor proportional to the metric tensor, reduces to the free wave equation. A similar method applied to the action consisting of the surface area enclosing an incompressible world volume undergoing pure shear again yields the wave equation in a conducting medium. Simultaneous stationarity of the volume can be imposed with a stationary area only in the case of pure shear; stationary Einstein–Hilbert action can also be included and lead to an equation of motion which has a similar interpretation of the wave in the conducting medium. Some Green functions applicable to the medium with constant conductivity are also presented.

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