Grid approximation of singularly perturbed boundary value problem for the quasi-linear elliptic equation degenerating into the first-order equation

1991 ◽  
Vol 6 (1) ◽  
Author(s):  
G. I. SHISHKIN
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Order Equation ◽  
Singularly Perturbed ◽  
Linear Elliptic Equation ◽  
First Order ◽  
Grid Approximation

scholarly journals NUMERICAL SOLUTION OF A BOUNDARY VALUE PROBLEM INCLUDING BOTH DELAY AND BOUNDARY LAYER

2018 ◽  
Vol 23 (4) ◽  
pp. 568-581 ◽  
Author(s):  
Erkan Cimen
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Boundary Value Problem ◽  
Delay Differential Equation ◽  
Order Parameter ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
First Order ◽  
Delay Differential

Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a nonlinear second order delay differential equation is analyzed. Also, the method is proved that it gives essentially first order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory.

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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