Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy

2007 ◽  
Vol 188 (2) ◽  
pp. 1816-1831 ◽  
Author(s):  
Mohan K. Kadalbajoo ◽  
V.P. Ramesh
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Adaptation Strategy ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Grid Adaptation ◽  
Delay Differential

scholarly journals Comparative Study on Numerical Methods for Singularly Perturbed Advanced-Delay Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
P. Hammachukiattikul ◽  
E. Sekar ◽  
A. Tamilselvan ◽  
R. Vadivel ◽  
N. Gunasekaran ◽  
...  
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Diffusion Type ◽  
Convection Diffusion ◽  
Discrete Norm ◽  
Difference Methods ◽  
Second Order Convergence ◽  
Delay Differential

In this paper, we consider a class of singularly perturbed advanced-delay differential equations of convection-diffusion type. We use finite and hybrid difference schemes to solve the problem on piecewise Shishkin mesh. We have established almost first- and second-order convergence with respect to finite difference and hybrid difference methods. An error estimate is derived with the discrete norm. In the end, numerical examples are given to show the advantages of the proposed results (Mathematics Subject Classification: 65L11, 65L12, and 65L20).

scholarly journals An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Süleyman Cengizci
Keyword(s):  
Differential Equations ◽  
Hybrid Method ◽  
Delay Differential Equations ◽  
Numerical Procedure ◽  
Taylor Series Expansion ◽  
Expansion Method ◽  
Singularly Perturbed ◽  
Convection Term ◽  
Linear Delay ◽  
Delay Differential

In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.

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