scholarly journals Solution of Singularly Perturbed Differential-Difference Equations with Mixed Shifts Using Galerkin Method with Exponential Fitting

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
D. Kumara Swamy ◽  
K. Phaneendra ◽  
Y. N. Reddy
Keyword(s):  
Boundary Layer ◽  
Galerkin Method ◽  
Difference Equations ◽  
Singularly Perturbed ◽  
Exponential Fitting ◽  
Thomas Algorithm ◽  
Galerkin Scheme ◽  
Rapid Changes ◽  
Equations With Delay ◽  
Layer Solution

Galerkin method is presented to solve singularly perturbed differential-difference equations with delay and advanced shifts using fitting factor. In the numerical treatment of such type of problems, Taylor’s approximation is used to tackle the terms containing small shifts. A fitting factor in the Galerkin scheme is introduced which takes care of the rapid changes that occur in the boundary layer. This fitting factor is obtained from the asymptotic solution of singular perturbations. Thomas algorithm is used to solve the tridiagonal system of the fitted Galerkin method. The method is analysed for convergence. Several numerical examples are solved and compared to demonstrate the applicability of the method. Graphs are plotted for the solutions of these problems to illustrate the effect of small shifts on the boundary layer solution.

Fitted Mesh Method for a Class of Singularly Perturbed Differential-Difference Equations

2015 ◽  
Vol 8 (4) ◽  
pp. 496-514 ◽  
Author(s):  
Devendra Kumar
Keyword(s):  
Boundary Layer ◽  
Small Parameter ◽  
Uniform Convergence ◽  
Difference Equations ◽  
Numerical Study ◽  
Second Order ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Small Parameters ◽  
Spline Basis

AbstractThis paper deals with a more general class of singularly perturbed boundary value problem for a differential-difference equations with small shifts. In particular, the numerical study for the problems where second order derivative is multiplied by a small parameter ε and the shifts depend on the small parameter ε has been considered. The fitted-mesh technique is employed to generate a piecewise-uniform mesh, condensed in the neighborhood of the boundary layer. The cubic B-spline basis functions with fitted-mesh are considered in the procedure which yield a tridiagonal system which can be solved efficiently by using any well-known algorithm. The stability and parameter-uniform convergence analysis of the proposed method have been discussed. The method has been shown to have almost second-order parameter-uniform convergence. The effect of small parameters on the boundary layer has also been discussed. To demonstrate the performance of the proposed scheme, several numerical experiments have been carried out.

scholarly journals Solving Singularly Perturbed Differential-Difference Equations with Dual Layer using Fourth Order Numerical Method

2019 ◽  
Vol 9 (2) ◽  
pp. 3770-3773
Keyword(s):  
Boundary Layer ◽  
Finite Difference ◽  
Numerical Method ◽  
Fourth Order ◽  
Difference Equations ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Layer Behavior ◽  
Maximum Absolute Errors

In this paper, we presented a fourth-order numerical method to solve SPDDE with the dual-layer. The answer to the problem shows dual-layer behavior. A fourth-order finite difference plan on a uniform mesh is developed. The result of the delay and also advance parameters on the boundary layer(s) has likewise been evaluated as well as represented in charts. The applicability of the planned plan is actually confirmed through executing it on model examples. To show the accuracy of the method, the results are presented in terms of maximum absolute errors.

Non-Newtonian oscillatory layer flow, approximate solution

1979 ◽  
Vol 44 (10) ◽  
pp. 2908-2914 ◽  
Author(s):  
Ondřej Wein
Keyword(s):  
Boundary Layer ◽  
Approximate Solution ◽  
Galerkin Method ◽  
Horizontal Plane ◽  
Oscillatory Flow ◽  
Oscillatory Boundary Layer ◽  
Layer Flow ◽  
Pseudoplastic Liquid ◽  
The Galerkin Method

The problem of the oscillatory flow of pseudoplastic liquid in vicinity of the infinitely long horizontal plane is formulated in stresses. For Re i.e. for conditions of oscillatory boundary layer the problem is solved approximately by the Galerkin method.

Convective diffusion to a slowly rotating spherical electrode; Basic model for Re → O, Pe → ∞

1983 ◽  
Vol 48 (6) ◽  
pp. 1571-1578 ◽  
Author(s):  
Ondřej Wein
Keyword(s):  
Boundary Layer ◽  
Flow Regime ◽  
Peclet Number ◽  
Convective Diffusion ◽  
Creeping Flow ◽  
Péclet Number ◽  
Spherical Electrode ◽  
Basic Model ◽  
Boundary Layer Solution ◽  
Layer Solution

Theory has been formulated of a convective rotating spherical electrode in the creeping flow regime (Re → 0). The currently available boundary layer solution for Pe → ∞ has been confronted with an improved similarity description applicable in the whole range of the Peclet number.

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