The Neumann Boundary Value for the Singularly Perturbed Problem with Degenerate Equation Having Double Root

2018 ◽  
Vol 08 (03) ◽  
pp. 296-303
Author(s):  
玉娇 孙
Keyword(s):  
Boundary Value ◽  
Neumann Boundary ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Degenerate Equation ◽  
Double Root

scholarly journals Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 213 ◽  
Author(s):  
Assiya Zhumanazarova ◽  
Young Im Cho
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Singularly Perturbed Problems ◽  
Cauchy Function ◽  
Higher Derivatives ◽  
Singularly Perturbed Differential Equation ◽  
Perturbed Differential Equation

In this study, the asymptotic behavior of the solutions to a boundary value problem for a third-order linear integro-differential equation with a small parameter at the two higher derivatives has been examined, under the condition that the roots of the additional characteristic equation are negative. Via the scheme of methods and algorithms pertaining to the qualitative study of singularly perturbed problems with initial jumps, a fundamental system of solutions, the Cauchy function, and the boundary functions of a homogeneous singularly perturbed differential equation are constructed. Analytical formulae for the solutions and asymptotic estimates of the singularly perturbed problem are obtained. Furthermore, a modified degenerate boundary value problem has been constructed, and it was stated that the solution of the original singularly perturbed boundary value problem tends to this modified problem’s solution.

Multiresolution wavelet bases with augmentation method for solving singularly perturbed reaction–diffusion Neumann problem

2019 ◽  
Vol 17 (01) ◽  
pp. 1850064
Author(s):  
Somlak Utudee ◽  
Montri Maleewong
Keyword(s):  
Boundary Conditions ◽  
Variational Approach ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Original System ◽  
Neumann Boundary Conditions ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Wavelet Bases ◽  
Numerical Examples

This paper developed the anti-derivative wavelet bases to handle the more general types of boundary conditions: Dirichlet, mixed and Neumann boundary conditions. The boundary value problem can be formulated by the variational approach, resulting in a system involving unknown wavelet coefficients. The wavelet bases are constructed to solve the unknown solutions corresponding to the types of solution spaces. The augmentation method is presented to reduce the dimension of the original system, while the convergence rate is in the same order as the multiresolution method. Some numerical examples have been shown to confirm the rate of convergence. The examples of the singularly perturbed problem with Neumann boundary conditions are also demonstrated, including highly oscillating cases.

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