scholarly journals Directly Solving Special Second Order Delay Differential Equations Using Runge-Kutta-Nyström Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
M. Mechee ◽  
F. Ismail ◽  
N. Senu ◽  
Z. Siri
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Second Order ◽  
Computational Time ◽  
Test Problems ◽  
Numerical Comparison ◽  
Standard Set ◽  
Runge Kutta ◽  
The Stability ◽  
Delay Differential

Runge-Kutta-Nyström (RKN) method is adapted for solving the special second order delay differential equations (DDEs). The stability polynomial is obtained when this method is used for solving linear second order delay differential equation. A standard set of test problems is solved using the method together with a cubic interpolation for evaluating the delay terms. The same set of problems is reduced to a system of first order delay differential equations and then solved using the existing Runge-Kutta (RK) method. Numerical results show that the RKN method is more efficient in terms of accuracy and computational time when compared to RK method. The methods are applied to a well-known problem involving delay differential equations, that is, the Mathieu problem. The numerical comparison shows that both methods are in a good agreement.

scholarly journals Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Neutral Delay Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Haiyan Yuan ◽  
Cheng Song ◽  
Peichen Wang
Keyword(s):  
Differential Equations ◽  
Nonlinear Stability ◽  
Delay Differential Equations ◽  
Stability And Convergence ◽  
Neutral Delay ◽  
Neutral Delay Differential Equations ◽  
Restricted Type ◽  
Runge Kutta ◽  
The Stability ◽  
Delay Differential

This paper is devoted to the stability and convergence analysis of the two-step Runge-Kutta (TSRK) methods with the Lagrange interpolation of the numerical solution for nonlinear neutral delay differential equations. Nonlinear stability and D-convergence are introduced and proved. We discuss theGR(l)-stability,GAR(l)-stability, and the weakGAR(l)-stability on the basis of(k,l)-algebraically stable of the TSRK methods; we also discuss the D-convergence properties of TSRK methods with a restricted type of interpolation procedure.

Stability Aspects Of Explicit Runge-Kutta Method For Delay Differential Equations

2012 ◽  
Author(s):  
Fudziah Ismail ◽  
San Lwin Aung ◽  
Mohamed Suleiman
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Linear Delay ◽  
Different Types ◽  
Regions Of Stability ◽  
Runge Kutta ◽  
The Stability ◽  
Delay Differential

Persamaan pembezaan lengah linear (PPL) diselesaikan dengan kaedah Runge–Kutta menggunakan interpolasi yang berbeza bagi penghampiran sebutan lengahnya. Polinomial kestabilannya diterbitkan dan rantau kestabilannya dipersembahkan. Kata kunci: Runge-Kutta, persamaan pembezaan lengah, kestabilan, interpolasi The linear delay differential equations (DDEs) are solved by Runge–Kutta method using different types of interpolation to approximate the delay terms. The stability polynomials are derived and the respective regions of stability are presented. Key words: Runge-Kutta, delay differential equations, stability, interpolation

scholarly journals Delay differential equations for the spatially-resolved simulation of epidemics with specific application to COVID-19

2021 ◽  
Author(s):  
Nicola Guglielmi ◽  
Elisa Iacomini ◽  
Alexander Viguerie
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Computational Time ◽  
Partial Differential ◽  
Predictive Capacity ◽  
Spatially Resolved ◽  
The Stability ◽  
Delay Differential

In the wake of the 2020 COVID-19 epidemic, much work has been performed on the development of mathematical models for the simulation of the epidemic, and of disease models generally. Most works follow the susceptible-infected-removed (SIR) compartmental framework, modeling the epidemic with a system of ordinary differential equations. Alternative formulations using a partial differential equation (PDE) to incorporate both spatial and temporal resolution have also been introduced, with their numerical results showing potentially powerful descriptive and predictive capacity. In the present work, we introduce a new variation to such models by using delay differential equations (DDEs). The dynamics of many infectious diseases, including COVID-19, exhibit delays due to incubation periods and related phenomena. Accordingly, DDE models allow for a natural representation of the problem dynamics, in addition to offering advantages in terms of computational time and modeling, as they eliminate the need for additional, difficult-to-estimate, compartments (such as exposed individuals) to incorporate time delays. In the present work, we introduce a DDE epidemic model in both an ordinary- and partial differential equation framework. We present a series of mathematical results assessing the stability of the formulation. We then perform several numerical experiments, validating both the mathematical results and establishing model’s ability to reproduce measured data on realistic problems.

scholarly journals Second-Order Non-Canonical Neutral Differential Equations with Mixed Type: Oscillatory Behavior

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 318
Author(s):  
Osama Moaaz ◽  
Amany Nabih ◽  
Hammad Alotaibi ◽  
Y. S. Hamed
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Mixed Type ◽  
Relevant Literature ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Neutral Differential Equations ◽  
Delay Differential ◽  
Oscillation Of Solutions

In this paper, we establish new sufficient conditions for the oscillation of solutions of a class of second-order delay differential equations with a mixed neutral term, which are under the non-canonical condition. The results obtained complement and simplify some known results in the relevant literature. Example illustrating the results is included.

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