scholarly journals Convergence and Stability in Collocation Methods of Equationu′(t)=au(t)+bu([t])

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Han Yan ◽  
Shufang Ma ◽  
Yanbin Liu ◽  
Hongquan Sun
Keyword(s):  
Numerical Stability ◽  
Numerical Experiments ◽  
Stability Region ◽  
Collocation Methods ◽  
Stability Regions ◽  
Optimal Convergence ◽  
Analytic Stability ◽  
The Stability ◽  
Local Superconvergence ◽  
Global Superconvergence

This paper is concerned with the convergence, global superconvergence, local superconvergence, and stability of collocation methods foru′(t)=au(t)+bu([t]). The optimal convergence order and superconvergence order are obtained, and the stability regions for the collocation methods are determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained, and some numerical experiments are given.

scholarly journals The stability of collocation methods for higher-order Volterra integro-differential equations

2005 ◽  
Vol 2005 (19) ◽  
pp. 3075-3089 ◽  
Author(s):  
Edris Rawashdeh ◽  
Dave McDowell ◽  
Leela Rakesh
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Numerical Stability ◽  
Polynomial Spline ◽  
Higher Order ◽  
Collocation Methods ◽  
New Method ◽  
Spline Collocation ◽  
Spline Collocation Method ◽  
The Stability

The numerical stability of the polynomial spline collocation method for general Volterra integro-differential equation is being considered. The convergence and stability of the new method are given and the efficiency of the new method is illustrated by examples. We also proved the conjecture suggested by Danciu in 1997 on the stability of the polynomial spline collocation method for the higher-order integro-differential equations.

scholarly journals Stability conditions for some distributed systems: buffered random access systems

1994 ◽  
Vol 26 (02) ◽  
pp. 498-515 ◽  
Author(s):  
Wojciech Szpankowski
Keyword(s):  
Stability Region ◽  
Sufficient Conditions ◽  
Random Access ◽  
Symmetric System ◽  
Slotted Aloha ◽  
Stability Regions ◽  
Novel Technique ◽  
Token Passing ◽  
The Stability ◽  
Necessary And Sufficient

We consider the standard slotted ALOHA system with a finite number of buffered users. Stability analysis of such a system was initiated by Tsybakov and Mikhailov (1979). Since then several bounds on the stability region have been established; however, the exact stability region is known only for the symmetric system and two-user ALOHA. This paper proves necessary and sufficient conditions for stability of the ALOHA system. We accomplish this by means of a novel technique based on three simple observations: applying mathematical induction to a smaller copy of the system, isolating a single queue for which Loynes' stability criteria is adopted, and finally using stochastic dominance to verify the required stationarity assumptions in the Loynes criterion. We also point out that our technique can be used to assess stability regions for other multidimensional systems. We illustrate it by deriving stability regions for buffered systems with conflict resolution algorithms (see also Georgiadis and Szpankowski (1992) for similar approach applied to stability of token-passing rings).

INVESTIGATION OF THE GEOMETRY OF THE D-PARTITION OF ONE-DIMENSIONAL PLANE OF PARAMETER OF THE CHARACTERISTIC EQUATION OF A CONTINUOUS SYSTEM

2021 ◽  
Vol 4 ◽  
pp. 125-136
Author(s):  
Leonid Movchan ◽  
◽  
Sergey Movchan ◽  
Keyword(s):  
Imaginary Part ◽  
Characteristic Equation ◽  
Stability Region ◽  
Practical Interest ◽  
Stability Regions ◽  
Odd Order ◽  
Stability Intervals ◽  
Even Order ◽  
The Stability ◽  
Two Parameters

The paper considers two types of boundaries of the D-partition in the plane of one parameter of linear continuous systems given by the characteristic equation with real coefficients. The number of segments and intervals of stability of the X-partition curve is estimated. The maximum number of stability intervals is determined for different orders of polynomials of the equation of the boundary of the D-partition of the first kind (even order, odd order, one of even order, and the other of odd order). It is proved that the maximum number of stability intervals of a one-parameter family is different for all cases and depends on the ratio of the degrees of the polynomials of the equation of the D-partition curve. The derivative of the imaginary part of the expression of the investigated parameter at the initial point of the D-partition curve is obtained in an analytical form, the sign of which depends on the ratio of the coefficients of the characteristic equation and establishes the stability of the first interval of the real axis of the parameter plane. It is shown that for another type of the boundary of the D-partition in the plane of one parameter, there is only one interval of stability, the location of which, as for the previous type of the boundary of the stability region (BSR), is determined by the sign of the first derivative of the imaginary part of the expression of the parameter under study. Consider an example that illustrates the effectiveness of the proposed approach for constructing a BSR in a space of two parameters without using «Neimark hatching» and constructing special lines. In this case, a machine implementation of the construction of the stability region is provided. Considering that the problem of constructing the boundary of the stability region in the plane of two parameters is reduced to the problem of determining the BSR in the plane of one parameter, then the given estimates of the maximum number of stability regions in the plane of one parameter allow us to conclude about the number of maximum stability regions in the plane of two parameters, which are of practical interest. In this case, one of the parameters can enter nonlinearly into the coefficients of the characteristic equation.

scholarly journals Stability Analysis ofθ-Methods for Neutral Multidelay Integrodifferential System

2007 ◽  
Vol 2007 ◽  
pp. 1-8
Author(s):  
Y. Xu ◽  
J. J. Zhao ◽  
Z. N. Sui
Keyword(s):  
Numerical Stability ◽  
Numerical Experiments ◽  
Analytic Solutions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Neutral Delay ◽  
Integrodifferential System ◽  
System A ◽  
The Stability ◽  
Necessary And Sufficient

This paper studies the stability of a class of neutral delay integrodifferential system. A necessary and sufficient condition of stability for its analytic solutions is considered. The improvedθ-methods are developed. Some numerical stability properties are obtained and numerical experiments are given.

scholarly journals BBDF-Alpha for solving stiff ordinary differential equations with oscillating solutions

2020 ◽  
Vol 51 (2) ◽  
pp. 123-136
Author(s):  
Iskandar Shah Mohd Zawawi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Experiments ◽  
Stability Region ◽  
Newton Iteration ◽  
Computational Time ◽  
First Order ◽  
Stiff Ordinary Differential Equations ◽  
Oscillating Solutions ◽  
The Stability

In this paper, the block backward differentiation α formulas (BBDF-α) is derived for solving first order stiff ordinary differential equations with oscillating solutions. The consistency and zero stability conditions are investigated to prove the convergence of the method. The stability region in the entire negative half plane shows that the derived method is A-stable for certain values of α. The implementation of the method using Newton iteration is also discussed. Several numerical experiments are conducted to demonstrate the performance of the method in terms of accuracy and computational time.

scholarly journals ORDER OF THE RUNGE-KUTTA METHOD AND EVOLUTION OF THE STABILITY REGION

2019 ◽  
Vol 5 (2) ◽  
pp. 64
Author(s):  
Hippolyte Séka ◽  
Kouassi Richard Assui
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Absolute Stability ◽  
Stability Region ◽  
Kutta Method ◽  
Stability Regions ◽  
Runge Kutta Method ◽  
The Absolute ◽  
Runge Kutta ◽  
The Stability

In this article, we demonstrate through specific examples that the evolution of the size of the absolute stability regions of Runge–Kutta methods for ordinary differential equation does not depend on the order of methods.

scholarly journals EXPLICIT GENERAL LINEAR METHODS WITH A LARGE STABILITY REGION FOR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

2019 ◽  
Vol 24 (4) ◽  
pp. 478-493
Author(s):  
Hassan Mahdi ◽  
Gholamreza Hojjati ◽  
Ali Abdi
Keyword(s):  
Differential Equations ◽  
Numerical Experiments ◽  
Stability Region ◽  
Quadrature Rule ◽  
General Linear Methods ◽  
General Linear ◽  
Large Area ◽  
Integral Term ◽  
Free Parameters ◽  
The Stability

In this paper, we describe the construction of a class of methods with a large area of the stability region for solving Volterra integro-differential equations. In the structure of these methods which is based on a subclass of explicit general linear methods with and without Runge-Kutta stability property, we use an adequate quadrature rule to approximate the integral term of the equation. The free parameters of the methods are used to obtain methods with a large stability region. The efficiency of the proposed methods is verified with some numerical experiments and comparisons with other existing methods.

scholarly journals Deterministic Sparse Sublinear FFT with Improved Numerical Stability

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Gerlind Plonka ◽  
Therese von Wulffen
Keyword(s):  
Numerical Stability ◽  
Numerical Experiments ◽  
A Priori ◽  
Condition Numbers ◽  
Vandermonde Matrices ◽  
Numerical Instabilities ◽  
Huge Impact ◽  
Fast Reconstruction ◽  
The Stability ◽  
Modified Algorithm

AbstractIn this paper we extend the deterministic sublinear FFT algorithm in Plonka et al. (Numer Algorithms 78:133–159, 2018. 10.1007/s11075-017-0370-5) for fast reconstruction of M-sparse vectors $${\mathbf{x}}$$ x of length $$N= 2^J$$ N = 2 J , where we assume that all components of the discrete Fourier transform $$\hat{\mathbf{x}}= {\mathbf{F}}_{N} {\mathbf{x}}$$ x ^ = F N x are available. The sparsity of $${\mathbf{x}}$$ x needs not to be known a priori, but is determined by the algorithm. If the sparsity M is larger than $$2^{J/2}$$ 2 J / 2 , then the algorithm turns into a usual FFT algorithm with runtime $${\mathcal O}(N \log N)$$ O ( N log N ) . For $$M^{2} < N$$ M 2 < N , the runtime of the algorithm is $${\mathcal O}(M^2 \, \log N)$$ O ( M 2 log N ) . The proposed modifications of the approach in Plonka et al. (2018) lead to a significant improvement of the condition numbers of the Vandermonde matrices which are employed in the iterative reconstruction. Our numerical experiments show that our modification has a huge impact on the stability of the algorithm. While the algorithm in Plonka et al. (2018) starts to be unreliable for $$M>20$$ M > 20 because of numerical instabilities, the modified algorithm is still numerically stable for $$M=200$$ M = 200 .

scholarly journals Classical and Multisymplectic Schemes for Linearized KdV Equation: Numerical Results and Dispersion Analysis

Fluids ◽  
2021 ◽  
Vol 6 (6) ◽  
pp. 214
Author(s):  
Adebayo Abiodun Aderogba ◽  
Appanah Rao Appadu
Keyword(s):  
Finite Difference ◽  
Numerical Experiments ◽  
Stability Region ◽  
Kdv Equation ◽  
Finite Difference Methods ◽  
Dispersion Analysis ◽  
Explicit Scheme ◽  
Difference Methods ◽  
Relative Errors ◽  
The Stability

We construct three finite difference methods to solve a linearized Korteweg–de-Vries (KdV) equation with advective and dispersive terms and specified initial and boundary conditions. Two numerical experiments are considered; case 1 is when the coefficient of advection is greater than the coefficient of dispersion, while case 2 is when the coefficient of dispersion is greater than the coefficient of advection. The three finite difference methods constructed include classical, multisymplectic and a modified explicit scheme. We obtain the stability region and study the consistency and dispersion properties of the various finite difference methods for the two cases. This is one of the rare papers that analyse dispersive properties of methods for dispersive partial differential equations. The performance of the schemes are gauged over short and long propagation times. Absolute and relative errors are computed at a given time at the spatial nodes used.

scholarly journals Stability conditions for some distributed systems: buffered random access systems

1994 ◽  
Vol 26 (2) ◽  
pp. 498-515 ◽  
Author(s):  
Wojciech Szpankowski
Keyword(s):  
Stability Region ◽  
Sufficient Conditions ◽  
Random Access ◽  
Symmetric System ◽  
Slotted Aloha ◽  
Stability Regions ◽  
Novel Technique ◽  
Token Passing ◽  
The Stability ◽  
Necessary And Sufficient

We consider the standard slotted ALOHA system with a finite number of buffered users. Stability analysis of such a system was initiated by Tsybakov and Mikhailov (1979). Since then several bounds on the stability region have been established; however, the exact stability region is known only for the symmetric system and two-user ALOHA. This paper proves necessary and sufficient conditions for stability of the ALOHA system. We accomplish this by means of a novel technique based on three simple observations: applying mathematical induction to a smaller copy of the system, isolating a single queue for which Loynes' stability criteria is adopted, and finally using stochastic dominance to verify the required stationarity assumptions in the Loynes criterion. We also point out that our technique can be used to assess stability regions for other multidimensional systems. We illustrate it by deriving stability regions for buffered systems with conflict resolution algorithms (see also Georgiadis and Szpankowski (1992) for similar approach applied to stability of token-passing rings).

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