scholarly journals Numerical dynamics of a nonstandard finite difference method for a class of delay differential equations

2013 ◽  
Vol 400 (1) ◽  
pp. 25-34 ◽  
Author(s):  
Huan Su ◽  
Wenxue Li ◽  
Xiaohua Ding
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Delay Differential Equations ◽  
Delay Differential ◽  
Nonstandard Finite Difference ◽  
Numerical Dynamics

Fitted Finite Difference Method for Third Order Singularly Perturbed Delay Differential Equations of Convection Diffusion Type

2019 ◽  
Vol 16 (05) ◽  
pp. 1840007 ◽  
Author(s):  
R. Mahendran ◽  
V. Subburayan
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Delay Differential Equations ◽  
Singularly Perturbed ◽  
Third Order ◽  
Diffusion Type ◽  
Convection Diffusion ◽  
Delay Differential

In this paper, a fitted finite difference method on Shishkin mesh is suggested to solve a class of third order singularly perturbed boundary value problems for ordinary delay differential equations of convection-diffusion type. Numerical solution converges uniformly to the exact solution. The order of convergence of the numerical method is almost first order. Numerical results are provided to illustrate the theoretical results.

Numerical Dynamics of Nonstandard Finite Difference Method for Nonlinear Delay Differential Equation

2018 ◽  
Vol 28 (11) ◽  
pp. 1850133 ◽  
Author(s):  
Xiaolan Zhuang ◽  
Qi Wang ◽  
Jiechang Wen
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Delay Differential Equation ◽  
Difference Method ◽  
Discrete System ◽  
Delay Differential ◽  
Nonlinear Delay Differential Equation ◽  
Nonlinear Delay ◽  
Nonstandard Finite Difference

In this paper, we study the dynamics of a nonlinear delay differential equation applied in a nonstandard finite difference method. By analyzing the numerical discrete system, we show that a sequence of Neimark–Sacker bifurcations occur at the equilibrium as the delay increases. Moreover, the existence of local Neimark–Sacker bifurcations is considered, and the direction and stability of periodic solutions bifurcating from the Neimark–Sacker bifurcation of the discrete model are determined by the Neimark–Sacker bifurcation theory of discrete system. Finally, some numerical simulations are adopted to illustrate the corresponding theoretical results.

scholarly journals A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations

2020 ◽  
Vol 5 (1) ◽  
pp. 425-436 ◽  
Author(s):  
Fevzi Erdogan ◽  
Mehmet Giyas Sakar ◽  
Onur Saldır
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Delay Differential Equations ◽  
Perturbation Parameter ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Initial Value ◽  
Uniformly Convergent ◽  
Delay Differential

AbstractThe purpose of this paper is to present a uniform finite difference method for numerical solution of a initial value problem for semilinear second order singularly perturbed delay differential equation. A numerical method is constructed for this problem which involves appropriate piecewise-uniform Shishkin mesh on each time subinterval. The method is shown to uniformly convergent with respect to the perturbation parameter. A numerical experiment illustrate in practice the result of convergence proved theoretically.

scholarly journals Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1038 ◽  
Author(s):  
María Ángeles Castro ◽  
Miguel Antonio García ◽  
José Antonio Martín ◽  
Francisco Rodríguez
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Linear Delay ◽  
Delay Differential Systems ◽  
Delay Differential ◽  
Consistency Properties ◽  
Nonstandard Finite Difference

In recent works, exact and nonstandard finite difference schemes for scalar first order linear delay differential equations have been proposed. The aim of the present work is to extend these previous results to systems of coupled delay differential equations X ′ ( t ) = A X ( t ) + B X ( t - τ ) , where X is a vector, and A and B are commuting real matrices, in general not simultaneously diagonalizable. Based on a constructive expression for the exact solution of the vector equation, an exact scheme is obtained, and different nonstandard numerical schemes of increasing order are proposed. Dynamic consistency properties of the new nonstandard schemes are illustrated with numerical examples, and proved for a class of methods.

scholarly journals On the Distributed Order Fractional Multi-Strain Tuberculosis Model: a Numerical Study

2020 ◽  
Vol 8 (1) ◽  
pp. 175-186
Author(s):  
Nasser Sweilam ◽  
S. M. AL-Mekhlafi ◽  
A. O. Albalawi
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Difference Method ◽  
Numerical Study ◽  
Proposed Model ◽  
Fractional Differential ◽  
Distributed Order ◽  
Nonstandard Finite Difference

In this paper, a novel mathematical distributed order fractional model of multistrain Tuberculosis is presented. The proposed model is governed by a system of distributed order fractional differential equations, where the distributed order fractional derivative is defined in the sense of the Grünwald-Letinkov definition. A nonstandard finite difference method is proposed to study the resulting system. The stability analysis of the proposed model is discussed. Numerical simulations show that the nonstandard finite difference method can be applied to solve such distributed order fractional differential equations simply and eectively.

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