We consider a problem of reconstructing the shear modulus of an viscoelastic system in a thin cylinder from the measurements of displacements induced by torques applied at the bottom of the cylinder. The viscoelastic system is a mathematical model of a pendulum-type viscoelastic spectrometer (PVS). We first compute in an explicit form the solution of the viscoelastic system, and then derive with an error estimate the leading order term of the average of the solution. This leading order term yields a nonlinear inversion scheme to determine the shear modulus from the measurements of displacements. We apply the inversion scheme to determine the shear modulus using experimental data acquired from a PVS system.
Consider the Cauchy problem for a nonlinear diffusion equationwhereThen the positive solution to problem (P) behaves like a positive solution to ODE ζ ′ = ζ α in (0, ∞) and it tends to +∞ as t → ∞. In this paper we obtain the precise description of the large time behavior of the solution and reveal the relationship between the behavior of the solution and the diffusion effect the nonlinear diffusion equation has.
The usual Helmholtz decomposition gives a decomposition of any vector valued function into a sum of a gradient of a scalar function and a rotation of a vector valued function under some mild condition. In this paper we show that the vector valued function of the second term i.e., the divergence free part of this decomposition can be further decomposed into a sum of a vector valued function polarized in one component and the rotation of a vector valued function also polarized in the same component. Hence the divergence free part only depends on two scalar functions. We refer to this as a special Helmholtz decomposition. Further we show the so-called completeness of representation associated to this decomposition for the stationary wave field of a homogeneous, isotropic viscoelastic medium. That is, this wave field can be expressed as a special Helmholtz decomposition and each of its scalar functions satisfies a Helmholtz equation. Our completeness of representation is useful for solving boundary value problems in a cylindrical domain for several partial differential equations of systems in mathematical physics such as stationary isotropic homogeneous elastic/viscoelastic equations of a system.
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