scholarly journals A New Multi-Step Method for Solving Delay Differential Equations using Lagrange Interpolation

2021 ◽  
pp. 159-164
Author(s):  
V. J. Shaalini ◽  
S. E. Fadugba
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Lagrange Interpolation ◽  
Search Algorithm ◽  
Step Method ◽  
Delay Argument ◽  
State Dependent ◽  
Constant Delay Time ◽  
The Stability ◽  
Delay Differential

This paper presents 2-step p-th order (p = 2,3,4) multi-step methods that are based on the combination of both polynomial and exponential functions for the solution of Delay Differential Equations (DDEs). Furthermore, the delay argument is approximated using the Lagrange interpolation. The local truncation errors and stability polynomials for each order are derived. The Local Grid Search Algorithm (LGSA) is used to determine the stability regions of the method. Moreover, applicability and suitability of the method have been demonstrated by some numerical examples of DDEs with constant delay, time dependent and state dependent delays. The numerical results are compared with the theoretical solution as well as the existing Rational Multi-step Method2 (RMM2). 

scholarly journals Solving Delay Differential Equations Using Reformulated Block Backward Differentiation Formulae Methods

2019 ◽  
pp. 1-15
Author(s):  
U. W. Sirisena ◽  
S. Y. Yakubu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Approximate Solutions ◽  
Delay Argument ◽  
Block Form ◽  
One Step ◽  
Backward Differentiation Formulae ◽  
The Stability ◽  
Delay Differential ◽  
Step Number

In this paper, the conventional backward differentiation formulae methods for step numbers k = 3 and 4 were reformulated by shifting them one-step backward to produce two and three approximate solutions respectively, in a step when implemented in block form. The derivation of the continuous formulations of the reformulated methods were carried out through multistep collocation method by matrix inversion technique. The discrete schemes were deduced from their respective continuous formulations. The convergence analysis of the discrete schemes were discussed. The stability analysis of these schemes were ascertained and the P- and Q-stability were also investigated. When the discrete schemes were implemented in block form to solve some first order delay differential equations together with an accurate and efficient formula for the solution of the delay argument, it was observed that the results obtained from the schemes for step number k = 4 performed slightly better than the schemes for step number k = 3 when compared with the exact solutions. More so, on comparing these methods with some existing ones, it was observed that the methods derived performed better in terms of accuracy.

NUMERICAL BIFURCATION ANALYSIS OF DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

2001 ◽  
Vol 11 (03) ◽  
pp. 737-753 ◽  
Author(s):  
TATYANA LUZYANINA ◽  
KOEN ENGELBORGHS ◽  
DIRK ROOSE
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Constant Delay ◽  
State Dependent ◽  
State Dependent Delay ◽  
Numerical Bifurcation ◽  
Delay Differential ◽  
Steady State Solutions ◽  
Numerical Bifurcation Analysis

In this paper we apply existing numerical methods for bifurcation analysis of delay differential equations with constant delay to equations with state-dependent delay. In particular, we study the computation, continuation and stability analysis of steady state solutions and periodic solutions. We collect the relevant theory and describe open theoretical problems in the context of bifurcation analysis. We present computational results for two examples and compare with analytical results whenever possible.

scholarly journals Stability of delay differential equations in the sense of Ulam on unbounded intervals

2019 ◽  
Vol 9 (2) ◽  
pp. 125-131 ◽  
Author(s):  
Süleyman Öğrekçi
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Stability Problem ◽  
The Stability ◽  
Delay Differential ◽  
Unbounded Intervals

In this paper, we consider the stability problem of delay differential equations in the sense of Hyers-Ulam-Rassias. Recently this problem has been solved for bounded intervals, our result extends and improve the literature by obtaining stability in unbounded intervals. An illustrative example is also given to compare these results and visualize the improvement.

Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Anwar Sadath ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Approximate Model ◽  
Periodic Coefficients ◽  
Galerkin Approximations ◽  
Approximate System ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Least Squares Approach

A numerical method to determine the stability of delay differential equations (DDEs) with time periodic coefficients is proposed. The DDE is converted into an equivalent partial differential equation (PDE) with a time periodic boundary condition (BC). The PDE, along with its BC, is then converted into a system of ordinary differential equations (ODEs) with time periodic coefficients using the Galerkin least squares approach. In the Galerkin approach, shifted Legendre polynomials are used as basis functions, allowing us to obtain explicit expressions for the approximate system of ODEs. We analyze the stability of the discretized ODEs, which represent an approximate model of the DDEs, using Floquet theory. We use numerical examples to show that the stability charts obtained with our method are in excellent agreement with those existing in the literature and those obtained from direct numerical simulation.

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