scholarly journals A Numerical Patching Technique for Singularly Perturbed Nonlinear Differential-Difference Equations with a Negative Shift

2012 ◽  
Vol 2 (2) ◽  
pp. 11-20 ◽  
Author(s):  
R. Nageshwar Rao ◽  
P. Pramod Chakravarthy
Keyword(s):  
Difference Equations ◽  
Singularly Perturbed ◽  
Negative Shift

scholarly journals Numerical Solution of Singularly Perturbed Delay Differential Equations with Layer Behavior

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
F. Ghomanjani ◽  
A. Kılıçman ◽  
F. Akhavan Ghassabzade
Keyword(s):  
Boundary Value Problems ◽  
Difference Equations ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Time Interval ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Approximation Process ◽  
Negative Shift ◽  
Control Functions

We present a numerical method to solve boundary value problems (BVPs) for singularly perturbed differential-difference equations with negative shift. In recent papers, the term negative shift has been used for delay. The Bezier curves method can solve boundary value problems for singularly perturbed differential-difference equations. The approximation process is done in two steps. First we divide the time interval, intoksubintervals; second we approximate the trajectory and control functions in each subinterval by Bezier curves. We have chosen the Bezier curves as piecewise polynomials of degreenand determined Bezier curves on any subinterval byn+1control points. The proposed method is simple and computationally advantageous. Several numerical examples are solved using the presented method; we compared the computed result with exact solution and plotted the graphs of the solution of the problems.

scholarly journals A seventh order numerical method for singular perturbed differential-difference equations with negative shift

2011 ◽  
Vol 16 (2) ◽  
pp. 206-219
Author(s):  
Kolloju Phaneendra ◽  
Y. N. Reddy ◽  
GBSL. Soujanya
Keyword(s):  
Exact Solution ◽  
Boundary Value Problem ◽  
Numerical Method ◽  
Difference Equations ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
First Order ◽  
Negative Shift ◽  
Time Problem ◽  
Seventh Order

In this paper, a seventh order numerical method is presented for solving singularly perturbed differential-difference equations with negative shift. In recent papers the term negative shift has been used for delay. Such problems are associated with expected first exit time problem of the membrane, potential in models for neuron and in variational problems in control theory. In the numerical treatment for such type of boundary value problems, we first use Taylor approximation to tackle terms containing small shifts which converts into a singularly perturbed boundary value problem. This two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a seventh order compact difference scheme is employed for the first order system and solved by using the boundary conditions. Several numerical examples are solved and compared with exact solution. We also present least square errors, maximum errors and observed that the present method approximates the exact solution very well.

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