Analytical solution of a system of homogeneous delay differential equations via the Lambert function

2000 ◽  
Author(s):  
F. Maghami Asl ◽  
A. Galip Ulsoy
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Lambert Function ◽  
Delay Differential

Chatter Stability Analysis Using the Matrix Lambert Function and Bifurcation Analysis

2006 ◽  
Author(s):  
Sun Yi ◽  
Patrick W. Nelson ◽  
A. Galip Ulsoy
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Lambert Function ◽  
Chatter Stability ◽  
Approximate Methods ◽  
Linear Delay ◽  
The Matrix ◽  
The Stability ◽  
Delay Differential

We investigate the stability of the regenerative machine tool chatter problem, in a turning process modeled using delay differential equations (DDEs). An approach using the matrix Lambert function for the analytical solution to systems to delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert function, known to be useful for solving scalar first order DDEs, has recently been extended to a matrix Lambert function approach to solve systems of DDEs. The essential advantage of the matrix Lambert approach is not only the similarity to the concept of the state transition matrix in linear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy, and certain other advantages, when compared to traditional graphical, computational and approximate methods.

scholarly journals Approximate analytical solution of linear and nonlinear fractional delay differential equations using new variational iteration method

2020 ◽  
Vol 5 (4) ◽  
Author(s):  
Vineet Srivastava
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Iteration Method ◽  
Delay Differential Equations ◽  
Approximate Analytical Solution ◽  
Variational Iteration Method ◽  
Fractional Delay ◽  
Linear And Nonlinear ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

In this paper, an approximate analytical method, New Variational Iteration Method (NVIM) is introduced in this paper for the approximate analytical solution of Fractional Delay Differential Equations (FDDE). The algorithm is illustrated by studying initial value linear and nonlinear problems. The results obtained are presented and show that only few terms are required to get an approximate solution.

Sign in / Sign up

Export Citation Format

Share Document