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New explicit stability conditions are derived for a linear integro-differential equation with periodic operator coefficients. The equation under consideration describes oscillations of thin-walled viscoelastic structural members d...
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New explicit stability conditions are derived for a linear integro-differential equation with periodic operator coefficients. The equation under consideration describes oscillations of thin-walled viscoelastic structural members driven by periodic loads. To develop stability conditions two approaches are combined. The first is based on the direct Lyapunov method of constructing stability functionals. It allows stability conditions to be derived for unbounded operator coefficients, but fails to correctly predict the critical loads for high-frequency excitations. The other approach is based on transforming the equation under consideration in such a way that an appropriate 'differential' part of the new equation would possess some reserve of stability. Stability conditions for the transformed equation are obtained by using a technique of integral estimates. This method provides acceptable estimates of the critical forces for periodic loads, but can be applied to equations with bounded coefficients only. Combining these two approaches, we derive explicit stability conditions which are close to the Floquet criterion when the integral term vanishes. These conditions are applied to the stability problem for a viscoelastic bar compressed by periodic forces. The effect of material and structural parameters on the critical load is studied numerically. (C) 1998 B. G. Teubner Stuttgart-John Wiley & Sons Ltd. [References: 24]
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In order to maintain spectrally accurate solutions, the grids on which a non-linear physical problem is to be solved must also be obtained by spectrally accurate techniques. The purpose of this paper is to describe a pseudospectra...
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In order to maintain spectrally accurate solutions, the grids on which a non-linear physical problem is to be solved must also be obtained by spectrally accurate techniques. The purpose of this paper is to describe a pseudospectral computational method of solving integro-differential systems with quadratic performance index. The proposed method is based on the idea of relating grid points to the structure of orthogonal interpolating polynomials. The optimal control and the trajectory are approximated by the mth degree interpolating polynomial. This interpolating polynomial is spectrally constructed using Legendre-Gauss-Lobatto grid points as the collocation points, and Lagrange polynomials as trial functions. The integrals involved in the formulation of the problem are calculated by Gauss-Lobatto integration rule, thereby reducing the problem to a mathematical programming one to which existing well-developed algorithms may be applied. The method is easy to implement and yields very accurate results. An illustrative example is included to confirm the convergence of the pseudospectral Legendre method, and a comparison is made with an existing result in the literature. (C) 1998 B. G. Teubner Stuttgart-John Wiley & Sons Ltd. [References: 6]
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We consider a class of fully nonlinear integro-differential operators where the nonlocal integral has two components: the non-degenerate one corresponds to the alpha-stable operator and the second one (possibly degenerate) corresp...
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We consider a class of fully nonlinear integro-differential operators where the nonlocal integral has two components: the non-degenerate one corresponds to the alpha-stable operator and the second one (possibly degenerate) corresponds to a class of lower order Levy measures. Such operators do not have a global scaling property. We establish Holder regularity, Harnack inequality and boundary Harnack property of solutions of these operators.
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Inverse problems for identification of the memory kernel in the linear constitutive stress-strain relation of Boltzmann type are reduced to a non-linear Volterra integral equation using Fourier's method for solving the direct prob...
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Inverse problems for identification of the memory kernel in the linear constitutive stress-strain relation of Boltzmann type are reduced to a non-linear Volterra integral equation using Fourier's method for solving the direct problem. To this equation the contraction principle in weighted norms is applied. In this way global existence of a solution to the inverse problem is proved and stability estimates for it are derived. (C) 1997 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd. [References: 26]
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This paper presents the result on existence, uniqueness of mild solutions to neutral stochastic partial functional integrodifferential equations under the Caratheodory-type conditions on the coefficients. The results are obtained ...
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This paper presents the result on existence, uniqueness of mild solutions to neutral stochastic partial functional integrodifferential equations under the Caratheodory-type conditions on the coefficients. The results are obtained by using the method of successive approximation. An example is provided to illustrate the results of this work.
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We consider abstract incomplete linear second-order integrodifferential equations in a Hilbert space. Operator coefficients of the equations are unbounded selfadjoint nonnegative operators. These equations arise naturally in visco...
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We consider abstract incomplete linear second-order integrodifferential equations in a Hilbert space. Operator coefficients of the equations are unbounded selfadjoint nonnegative operators. These equations arise naturally in viscoelasticity and hydroelasticity. We prove a theorem on asymptotic stability of strong solutions of the equations.
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In this paper we investigate local and global existence as well as asymptotic behavior of the solution for a class of abstract; (hyperbolic) quasilinear equations perturbed by bounded delay operators. We assume that the leading op...
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In this paper we investigate local and global existence as well as asymptotic behavior of the solution for a class of abstract; (hyperbolic) quasilinear equations perturbed by bounded delay operators. We assume that the leading operator is of bounded variation in time. In the last section, the abstract results are applied on a heat conduction model.
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The paper is concerned with the applicability of the method of Kantorovich majorants to nonlinear singular integro-differential equations with shift. Also the generalized Kantorovich majorization principle is applied to a class of...
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The paper is concerned with the applicability of the method of Kantorovich majorants to nonlinear singular integro-differential equations with shift. Also the generalized Kantorovich majorization principle is applied to a class of nonlinear singular integral equations with shift. The results are illustrated in the generalized Holder spaces.
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An automated algorithm is presented which enables the harmonic balance equations for any polynomial type pure or cross-product non-linear differential system to be written down directly in terms of the coefficients of the governin...
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An automated algorithm is presented which enables the harmonic balance equations for any polynomial type pure or cross-product non-linear differential system to be written down directly in terms of the coefficients of the governing equation and the complex amplitudes of general harmonic waveform. The system frequency response, in the form of multi-input, amplitude dependent describing function, is therefore readily computed. The method is illustrated by means of an example, and the results validated against detailed numeric simulation. [References: 13]
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In this paper, we consider the illustrative example of generalised logistic equations where the carrying-capacity effect is modelled by a distributed-delay effect (which may be over the infinite past). These distributed delay diff...
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In this paper, we consider the illustrative example of generalised logistic equations where the carrying-capacity effect is modelled by a distributed-delay effect (which may be over the infinite past). These distributed delay differential equations, though simple in structure, possess a rich array of solutions. If the delay is sufficiently large a supercritical Hopf bifurcation occurs, which finally disappears asymptotically when the delay becomes distributed infinitely. This mirrors the situation when there is just a point delay. Similar models with two or more state variables occur in pasture mixtures.
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