Computation of Green’s Function of the Bounded Solutions Problem

AbstractIt is well known that the equation {x^{\prime}(t)=Ax(t)+f(t)}, where A is a square matrix, has a unique bounded solution x for any bounded continuous free term f, provided the coefficient A has no eigenvalues on the imaginary axis. This solution can be represented in the formx(t)=\int_{-\infty}^{\infty}\mathcal{G}(t-s)f(s)\,ds.The kernel {\mathcal{G}} is called Green’s function. In this paper, for approximate calculation of {\mathcal{G}}, the Newton interpolating polynomial of a special function {g_{t}} is used. An estimate of the sensitivity of the problem is given. The results of numerical experiments are presented.

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The Gelfand–Shilov Type Estimate for Green’s Function of the Bounded Solutions Problem

Qualitative Theory of Dynamical Systems ◽

10.1007/s12346-017-0262-z ◽

2017 ◽

Vol 17 (3) ◽

pp. 619-629

Author(s):

V. G. Kurbatov ◽

I. V. Kurbatova

Keyword(s):

Green’S Function ◽

Green's Function ◽

Bounded Solutions ◽

Type Estimate

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Green's function of the problem of bounded solutions in the case of a block triangular coefficient

Operators and Matrices ◽

10.7153/oam-2019-13-69 ◽

2019 ◽

pp. 981-1001

Author(s):

Vitalii G. Kurbatov ◽

Irina V. Kurbatova

Keyword(s):

Green’S Function ◽

Green's Function ◽

Bounded Solutions

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Remarks on bounded solutions of linear systems

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700017226 ◽

1996 ◽

Vol 53 (3) ◽

pp. 459-467

Author(s):

Elena Topuzu ◽

Paul Topuzu

Keyword(s):

Linear Systems ◽

Differential System ◽

Continuous Time ◽

Imaginary Axis ◽

Bounded Solution ◽

Bounded Solutions ◽

Bounded Operators ◽

Continuous Time Systems ◽

Linear Differential ◽

Time Systems

In the case of continuous time systems with bounded operators (coefficients) the following result, of Perron type is well known: “The linear differential system ẋ = Ax + f(t) has, for every function f continuous and bounded on ℝ, a unique bounded solution on ℝ, if and only if the spectrum of the operator A has no points on the imaginary axis”.

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FURTHER COMMENTS ON THE DYNAMICS OF EXTENDED OBJECTS

Modern Physics Letters A ◽

10.1142/s0217732398002928 ◽

1998 ◽

Vol 13 (34) ◽

pp. 2757-2761 ◽

Cited By ~ 1

Author(s):

U. KHANAL

Keyword(s):

Wave Equation ◽

Green’S Function ◽

Green's Function ◽

Imaginary Axis ◽

Momentum Tensor ◽

Positive Pressure ◽

Positive Energy ◽

Constant Density ◽

Vacuum Interface ◽

Hilbert Action

The equation of motion of the internal coordinates of a p-brane, arising from the stationarity of the world space Einstein–Hilbert action, is discussed. Assuming the Einstein equation of general relativity, with the energy–momentum tensor of a perfect fluid having positive energy density, it is shown that the governing equation, in matter dominated regions with positive pressure is a (p+1)-dimensional elliptic differential equation which reduces to the (p+1)-dimensional Laplace equation for constant density and pressure. The EOM becomes a hyperbolic p-dimensional wave equation in conducto-dispersive medium only in regions of world space, like those dominated by vacuum, where the total pressure is negative. The Green's function of the matter dominated, elliptic potential problem presented here for p=3, can be analytically continued into the complexified time domain to make contact on the imaginary axis with the Green's function of the wave equation for vacuum domination. Such considerations will allow the study of the matter–vacuum interface using complex time, whence matter domination with positive pressure would be represented on the real axis, vacuum domination with negative pressure on the imaginary axis, and the transition represented by the complex region.

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