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 Mathematics , 2014, Abstract: We consider an abstract equation with memory of the form $$\partial_t \boldsymbol{x}(t)+\int_{0}^\infty k(s) \boldsymbol{A}\boldsymbol{x}(t-s){\rm d} s+\boldsymbol{B}\boldsymbol{x}(t)=0$$ where $\boldsymbol{A},\boldsymbol{B}$ are operators acting on some Banach space, and the convolution kernel $k$ is a nonnegative convex summable function of unit mass. The system is translated into an ordinary differential equation on a Banach space accounting for the presence of memory, both in the so-called history space framework and in the minimal state one. The main theoretical result is a theorem providing sufficient conditions in order for the related solution semigroups to possess finite-dimensional exponential attractors. As an application, we prove the existence of exponential attractors for the integrodifferential equation $$\partial_{tt} u - h(0)\Delta u - \int_{0}^\infty h'(s) \Delta u(t-s){\rm d} s+ f(u) = g$$ arising in the theory of isothermal viscoelasticity, which is just a particular concrete realization of the abstract model, having defined the new kernel $h(s)=k(s)+1$.
 Fardin Saedpanah Mathematics , 2012, Abstract: A hyperbolic type integro-differential equation with two weakly singular kernels is considered together with mixed homogeneous Dirichlet and non-homogeneous Neumann boundary conditions. Existence and uniqueness of the solution is proved by means of Galerkin's method. Regularity estimates are proved and the limitations of the regularity are discussed. The approach presented here is also used to prove regularity of any order for models with smooth kernels, that arise in the theory of linear viscoelasticity, under the appropriate assumptions on data.
 Mathematics , 2014, Abstract: A new nonparametric approach for system identification has been recently proposed where the impulse response is modeled as the realization of a zero-mean Gaussian process whose covariance (kernel) has to be estimated from data. In this scheme, quality of the estimates crucially depends on the parametrization of the covariance of the Gaussian process. A family of kernels that have been shown to be particularly effective in the system identification framework is the family of Diagonal/Correlated (DC) kernels. Maximum entropy properties of a related family of kernels, the Tuned/Correlated (TC) kernels, have been recently pointed out in the literature. In this paper we show that maximum entropy properties indeed extends to the whole family of DC kernels. The maximum entropy interpretation can be exploited in conjunction with results on matrix completion problems in the graphical models literature to shed light on the structure of the DC kernel. In particular, we prove that DC kernels admit a closed-form inverse, determinant and factorization. Maximum likelihood properties of the DC kernel are also highlighted. These results can be exploited both to improve the stability and to reduce the computational complexity associated with the computation of DC estimators, as detailed in the paper.
 力学与实践 , 2008, Abstract: With the aid of the integral transformation, the symplectic system is introduced into the problem of two-dimensional thermo-viscoelasticity and the dual equations of the fundamental problem are constructed. All solutions of Saint-Venant problems can be obtained directly via zero eigenvalue eigensolutions, which satisfy the conjugated relationships of the symplectic orthogonality. Meanwhile, an effective method for boundary problems is provided by the technologies of variable substitution and eigensolution expansion. Numerical examples show that the symplectic method is effective for some typical boundary problems with creep and relaxation characteristics of thermo-viscoelasticity.
 Andrzej Hanyga Physics , 2014, Abstract: A method of eliminating the memory from the equations of motion of linear viscoelasticity is presented. Replacing the unbounded memory by a quadrature over a finite or semi-finite interval leads to considerable reduction of computational effort and storage. The method applies to viscoelastic media with separable completely monotonic relaxation moduli with an explicitly known retardation spectrum. In the seismological Strick-Mainardi model the quadrature is a Gauss-Jacobi quaddrature. The relation to fractional-order viscoelasticity is shown
 M. A. Shcherbakov Radioengineering , 1997, Abstract: A method for identification of discrete nonlinear systems in terms of the Volterra-Wiener series is presented. It is shown that use of a special, composite-frequency input signal as approximation to Gaussian noise provides a computational efficiency of this method, especially for high order kernels. Orthogonal functionals and consistent estimations for Wiener kernels in the frequency domains are derived for this class of noise input. A basis of the proposed computational procedure for practical identification is the fast Fourier transform (FFT) algorithm which is used both for a generating of system stimuluses and for an analysis of system reactions.
 Mathematics , 2006, Abstract: We are concerned with the problem of recovering the radial kernel $k$, depending also on time, in the parabolic integro-differential equation $$D_{t}u(t,x)={\cal A}u(t,x)+\int_0^t k(t-s,|x|){\cal B}u(s,x)ds +\int_0^t D_{|x|}k(t-s,|x|){\cal C}u(s,x)ds+f(t,x),$$ ${\cal A}$ being a uniformly elliptic second-order linear operator in divergence form. We single out a special class of operators ${\cal A}$ and two pieces of suitable additional information for which the problem of identifying $k$ can be uniquely solved locally in time when the domain under consideration is a ball or a disk.
 Mathematics , 2006, Abstract: We are concerned with the problem of recovering the radial kernel $k$, depending also on time, in a parabolic integro-differential equation $$D_{t}u(t,x)={\cal A}u(t,x)+\int_0^t k(t-s,|x|){\cal B}u(s,x)ds +\int_0^t D_{|x|}k(t-s,|x|){\cal C}u(s,x)ds+f(t,x),$$ ${\cal A}$ being a uniformly elliptic second-order linear operator in divergence form. We single out a special class of operators ${\cal A}$ and two pieces of suitable additional information for which the problem of identifying $k$ can be uniquely solved locally in time when the domain under consideration is a spherical corona or an annulus.