On the stability of the one-step exact collocation method for the second kind volterra integral equation with degenerate kernel

Computing ◽  
1988 ◽  
Vol 40 (4) ◽  
pp. 315-328 ◽  
Author(s):  
Maria Rosaria Crisci ◽  
Elvira Russo ◽  
Antonia Vecchio
Keyword(s):  
Integral Equation ◽  
Collocation Method ◽  
Volterra Integral Equation ◽  
Degenerate Kernel ◽  
One Step ◽  
The Stability ◽  
The One

scholarly journals On a nonlinear inverse problem in viscoelasticity

2009 ◽  
Vol 31 (3-4) ◽  
Author(s):  
H. D. Bui ◽  
S. Chaillat
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Volterra Integral Equation ◽  
Boundary Data ◽  
Viscoelastic Body ◽  
Low Frequency ◽  
Cyclic Loads ◽  
Nonlinear Inverse Problem ◽  
Method Of Solution ◽  
The One

We consider an inverse problem for determining an inhomogeneity in a viscoelastic body of the Zener type, using Cauchy boundary data, under cyclic loads at low frequency. We show that the inverse problem reduces to the one for the Helmholtz equation and to the same nonlinear Calderon equation given for the harmonic case. A method of solution is proposed which consists in two steps: solution of a source inverse problem, then solution of a linear Volterra integral equation.

scholarly journals Efficient Numerical Algorithm for the Solution of Nonlinear Two-Dimensional Volterra Integral Equation Arising from Torsion Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
A. M. Al-Bugami
Keyword(s):  
Integral Equation ◽  
Cross Section ◽  
Volterra Integral Equation ◽  
Kernel Method ◽  
Degenerate Kernel ◽  
Two Dimensional ◽  
Torsion Problem ◽  
Block Method ◽  
Elliptical Cross ◽  
Nonlinear Viscoelastic

In this article, an effective method is given to solve nonlinear two-dimensional Volterra integral equations of the second kind, which is arising from torsion problem for a long bar that consists of the nonlinear viscoelastic material type with a fixed elliptical cross section. First, the existence of a unique solution of this problem is discussed, and then, we find the solution of a nonlinear two-dimensional Volterra integral equation (NT-DVIE) using block-by-block method (B-by-BM) and degenerate kernel method (DKM). Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the method.

scholarly journals A New Efficient Method for the Numerical Solution of Linear Time-Dependent Partial Differential Equations

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 70
Author(s):  
Mina Torabi ◽  
Mohammad-Mehdi Hosseini
Keyword(s):  
Numerical Solution ◽  
Collocation Method ◽  
Efficient Method ◽  
Linear Time ◽  
Time Dependent ◽  
Time Step ◽  
Partial Differential ◽  
Wavelet Collocation Method ◽  
One Step ◽  
The One

This paper presents a new efficient method for the numerical solution of a linear time-dependent partial differential equation. The proposed technique includes the collocation method with Legendre wavelets for spatial discretization and the three-step Taylor method for time discretization. This procedure is third-order accurate in time. A comparative study between the proposed method and the one-step wavelet collocation method is provided. In order to verify the stability of these methods, asymptotic stability analysis is employed. Numerical illustrations are investigated to show the reliability and efficiency of the proposed method. An important property of the presented method is that unlike the one-step wavelet collocation method, it is not necessary to choose a small time step to achieve stability.

scholarly journals Application of Lyapunov matrix inequality based unsymmetrical saturated control to a multi-vectored propeller airship

2017 ◽  
Vol 232 (5) ◽  
pp. 884-901 ◽  
Author(s):  
Li Chen ◽  
James F Whidborne ◽  
Qi Dong ◽  
Deng Ping Duan
Keyword(s):  
Controller Design ◽  
Design Method ◽  
Coefficient Matrix ◽  
Matrix Inequality ◽  
Step Method ◽  
Saturated Control ◽  
Exogenous Disturbances ◽  
One Step ◽  
The Stability ◽  
The One

The problem of the design of a controller for a multi-vectored propeller airship is addressed. The controller includes anti-windup that takes into account unsymmetrical actuator constraints. First, a linear transformation is applied to transform the unsymmetrical constraints into symmetric constraints with an amplitude-bounded exogenous disturbance. Then, a stability condition based on a quadratic Lyapunov function for the saturated closed-loop system is proposed. The condition considers both amplitude-bounded and energy-bounded exogenous disturbances. Thus, the controller design problem is transformed into a convex optimization problem expressed in a bilinear matrix inequality form. Two controller design methods were applied: one-step controller and traditional anti-windup controller. The one-step method obtains the controller and the anti-windup compensator in one step while the anti-windup controller method separates this process into the linear controller design and the compensator design. Simulation results showed that both controllers enlarge the stability zone of the saturation system and have good tracking performance. It is shown that the anti-windup controller design method not only has a larger region of stability, but the demanded actuator output exceeds the constraints less and has a smaller anti-windup coefficient matrix compared to the one-step method.

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