10,735 research outputs found
Stability of an integro-differential equation
AbstractIn this work we study a scalar integro-differential equation and give some new conditions ensuring that the zero solution is asymptotically stable by means of the fixed-point theory. Our work extends and improves the results in the literature
Conformal Dynamics of Precursors to Fracture
An exact integro-differential equation for the conformal map from the unit
circle to the boundary of an evolving cavity in a stressed 2-dimensional solid
is derived. This equation provides an accurate description of the dynamics of
precursors to fracture when surface diffusion is important. The solution
predicts the creation of sharp grooves that eventually lead to material failure
via rapid fracture. Solutions of the new equation are demonstrated for the
dynamics of an elliptical cavity and the stability of a circular cavity under
biaxial stress, including the effects of surface stress.Comment: 4 pages, 3 figure
Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity
An integro-differential equation, modeling dynamic fractional order
viscoelasticity, with a Mittag-Leffler type convolution kernel is considered. A
discontinuous Galerkin method, based on piecewise constant polynomials is
formulated for temporal semidiscretization of the problem. Stability estimates
of the discrete problem are proved, that are used to prove optimal order a
priori error estimates. The theory is illustrated by a numerical example.Comment: 16 pages, 2 figure
Stability in Functional Integro-differential Equations of Second Order with Variable Delay
In this paper, we investigate the stability of the zero solution of an integro-differential equation of the second order with variable delay. By means of the fixed point theory and an exponential weighted metric, we find sufficient conditions under which the zero solution of the equation considered is stable
Can distributed delays perfectly stabilize dynamical networks?
Signal transmission delays tend to destabilize dynamical networks leading to
oscillation, but their dispersion contributes oppositely toward stabilization.
We analyze an integro-differential equation that describes the collective
dynamics of a neural network with distributed signal delays. With the gamma
distributed delays less dispersed than exponential distribution, the system
exhibits reentrant phenomena, in which the stability is once lost but then
recovered as the mean delay is increased. With delays dispersed more highly
than exponential, the system never destabilizes.Comment: 4pages 5figure
Existence of approximate current-vortex sheets near the onset of instability
The paper is concerned with the free boundary problem for 2D current-vortex
sheets in ideal incompressible magneto-hydrodynamics near the transition point
between the linearized stability and instability. In order to study the
dynamics of the discontinuity near the onset of the instability, Hunter and
Thoo have introduced an asymptotic quadratically nonlinear integro-differential
equation for the amplitude of small perturbations of the planar discontinuity.
The local-in-time existence of smooth solutions to the Cauchy problem for such
amplitude equation was already proven, under a suitable stability condition.
However, the solution found there has a loss of regularity (of order two) from
the initial data. In the present paper, we are able to obtain an existence
result of solutions with optimal regularity, in the sense that the regularity
of the initial data is preserved in the motion for positive times
Evaluation of numerical integration schemes for a partial integro-differential equation
Numerical methods are important in computational neuroscience where complex
nonlinear systems are studied. This report evaluates two methodologies,
finite differences and Fourier series, for numerically integrating a nonlinear
neural model based on a partial integro-differential equation. The stability
and convergence criteria of four finite difference methods is investigated and
their efficiency determined. Various ODE solvers in Matlab are used with the
Fourier series method to solve the neural model, with an evaluation of the
accuracy of the approximation versus the efficiency of the method. The two
methodologies are then compared
Refined Analytical Approximations to Limit Cycles for Non-Linear Multi-Degree-of-Freedom Systems
This paper presents analytical higher order approximations to limit cycles of an autonomous multi-degree-of-freedom system based on an integro-differential equation method for obtaining periodic solutions to nonlinear differential equations. The stability of the limit cycles obtained was then investigated using a method for carrying out Floquet analysis based on developments to extensions of the method for solving Hill's Determinant arising in analysing the Mathieu equation, which have previously been reported in the literature. The results of the Floquet analysis, together with the limit cycle predictions, have then been used to provide some estimates of points on the boundary of the domain of attraction of stable equilibrium points arising from a sub-critical Hopf bifurcation. This was achieved by producing a local approximation to the stable manifold of the unstable limit cycle that occurs. The integro-differential equation to be solved for the limit cycles involves no approximations. These only arise through the iterative approach adopted for its solution in which the first approximation is that which would be obtained from the harmonic balance method using only fundamental frequency terms. The higher order approximations are shown to give significantly improved predictions for the limit cycles for the cases considered. The Floquet analysis based approach to predicting the boundary of domains of attraction met with some success for conditions just following a sub-critical Hopf bifurcation. Although this study has focussed on cubic non-linearities, the method presented here could equally be used to refine limit cycle predictions for other non-linearity types.Peer reviewedFinal Accepted Versio
Stability analysis of an implicit and explicit numerical method for Volterra integro-differential equations with kernel K(x,y(t),t)
We present implicit and explicit versions of a numerical algorithm for
solving a Volterra integro-differential equation. These algorithms are an
extension of our previous work, and cater for a kernel of general form. We use
an appropriate test equation to study the stability of both algorithms,,
numerically deriving stability regions. The region for the implicit method
appears to be unbounded, while the explicit has a bounded region close to the
origin. We perform a few calculations to demonstrate our results.Comment: 10 pages, 1 Figur
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