Improving the two-stage numerical integration in stability identification of oscillation with distributed delay

Chigbogu Godwin Ozoegwu

doi:10.1177/1687814018819905

Abstract

The vibration of the engineering systems with distributed delay is governed by delay integro-differential equations. Two-stage numerical integration approach was recently proposed for stability identification of such oscillators. This work improves the approach by handling the distributed delay—that is, the first-stage numerical integration—with tensor-based higher order numerical integration rules. The second-stage numerical integration of the arising methods remains the trapezoidal rule as in the original method. It is shown that local discretization error is of order [Formula: see text] irrespective of the order of the numerical integration rule used to handle the distributed delay. But [Formula: see text] is less weighted when higher order numerical integration rules are used to handle the distributed delay, suggesting higher accuracy. Results from theoretical error analyses, various numerical rate of convergence analyses, and stability computations were combined to conclude that—from application point of view—it is not necessary to increase the first-stage numerical integration rule beyond the first order (trapezoidal rule) though the best results are expected at the second order (Simpson’s 1/3 rule).

Highlights

Oscillations with distributed delay are governed by delay integro-differential equations (DIDEs)
A numerical scheme for reliably computing the stability-determining characteristic roots of DIDEs has been developed based on linear multistep method and a quadrature method based on Lagrange interpolation and a Gauss–Legendre quadrature rule
These results agree with the theoretical error analysis that from application point of view, it is not necessary to increase the first-stage numerical integration (NI) rule beyond the first order though the best results are expected at second order (Simpson’s 1/3 rule)

Summary

Introduction

Oscillations with distributed delay are governed by delay integro-differential equations (DIDEs). Baker and Ford applied a set of linear multistep methods mixed with quadrature rules to numerically solve Volterra DIDEs. A numerical scheme for reliably computing the stability-determining characteristic roots of DIDEs has been developed based on linear multistep method and a quadrature method based on Lagrange interpolation and a Gauss–Legendre quadrature rule.. Numerical methods based on backward differentiation formulae and repeated quadrature formulae for the solution of Volterra DIDEs were suggested by Zhang and Vandewalle.. Yenixcerioglu established several theorems on the behavior of solutions of scalar linear second-order-delay DIDEs. Like most of the aforecited works, Huang considered the preservation of the stability properties of DIDEs after some form of discretization based on linear multistep methods. Linear multistep quadrature rules were used by Wu and Gan to Department of Mechanical Engineering, University of Nigeria, Nsukka, Nigeria

Methods

Conclusion