Wavelet-based approximation for two-parameter singularly perturbed problems with Robin boundary conditions

2021 ◽  
Author(s):  
Devendra Kumar ◽  
Komal Deswal
Keyword(s):  
Boundary Conditions ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Singularly Perturbed Problems ◽  
Robin Boundary ◽  
Two Parameter

Boundary shape functions methods for solving the nonlinear singularly perturbed problems with Robin boundary conditions

2020 ◽  
Vol 21 (7-8) ◽  
pp. 797-806
Author(s):  
Chein-Shan Liu ◽  
Jiang-Ren Chang
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Singularly Perturbed Problems ◽  
Boundary Shape ◽  
Type Algorithm ◽  
Free Function ◽  
Robin Boundary

AbstractFor a second-order nonlinear singularly perturbed boundary value problem (SPBVP), we develop two novel algorithms to find the solution, which automatically satisfies the Robin boundary conditions. For the highly singular nonlinear SPBVP the Robin boundary functions are hard to be fulfilled exactly. In the paper we first introduce the new idea of boundary shape function (BSF), whose existence is proven and it can automatically satisfy the Robin boundary conditions. In the BSF, there exists a free function, which leaves us a chance to develop new algorithms by adopting two different roles of the free function. In the first type algorithm we let the free functions be the exponential type bases endowed with fractional powers, which not only satisfy the Robin boundary conditions automatically, but also can capture the singular behavior to find accurate numerical solution by a simple collocation technique. In the second type algorithm we let the BSF be solution and the free function be another variable, such that we can transform the boundary value problem to an initial value problem (IVP) for the new variable, which can quickly find accurate solution for the nonlinear SPBVP through a few iterations.

scholarly journals An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Li-Bin Liu ◽  
Ying Liang ◽  
Xiaobing Bao ◽  
Honglin Fang
Keyword(s):  
Boundary Conditions ◽  
Grid Method ◽  
Posteriori Error ◽  
Adaptive Grid ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Euler Formula ◽  
Convection Diffusion ◽  
Backward Euler ◽  
Robin Boundary

AbstractA system of singularly perturbed convection-diffusion equations with Robin boundary conditions is considered on the interval $[0,1]$ [ 0 , 1 ] . It is shown that any solution of such a problem can be expressed to a system of first-order singularly perturbed initial value problem, which is discretized by the backward Euler formula on an arbitrary nonuniform mesh. An a posteriori error estimation in maximum norm is derived to design an adaptive grid generation algorithm. Besides, in order to establish the initial values of the original problems, we construct a nonlinear optimization problem, which is solved by the Nelder–Mead simplex method. Numerical results are given to demonstrate the performance of the presented method.

A parameter uniform essentially first-order convergent numerical method for a parabolic system of singularly perturbed differential equations of reaction–diffusion type with initial and Robin boundary conditions

2019 ◽  
Vol 12 (01) ◽  
pp. 1950001 ◽  
Author(s):  
R. Ishwariya ◽  
J. J. H. Miller ◽  
S. Valarmathi
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Method ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Diffusion Type ◽  
Parabolic Boundary ◽  
First Order ◽  
Robin Boundary

In this paper, a class of linear parabolic systems of singularly perturbed second-order differential equations of reaction–diffusion type with initial and Robin boundary conditions is considered. The components of the solution [Formula: see text] of this system are smooth, whereas the components of [Formula: see text] exhibit parabolic boundary layers. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first-order convergent in time and essentially first-order convergent in the space variable in the maximum norm uniformly in the perturbation parameters.

scholarly journals On a singularly perturbed semi-linear problem with Robin boundary conditions

2021 ◽  
Vol 26 (1) ◽  
pp. 401-414
Author(s):  
Qianqian Hou ◽  
◽  
Tai-Chia Lin ◽  
Zhi-An Wang ◽  
◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Linear Problem ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Robin Boundary
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