scholarly journals Uniform Approximation to the Solution of a Singularly Perturbed Boundary Value Problem with an Initial Jump

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2216
Author(s):  
Assiya Zhumanazarova ◽  
Young Im Cho
Keyword(s):  
Boundary Layer ◽  
Boundary Value Problem ◽  
Initial Conditions ◽  
Boundary Value ◽  
Regular Part ◽  
Zeroth Order ◽  
Third Order ◽  
Original Boundary ◽  
Higher Derivatives ◽  
Initial Jump

In this study, a third-order linear integro-differential equation with a small parameter at two higher derivatives was considered. An asymptotic expansion of the solution to the boundary value problem for the considered equation is constructed by considering the phenomenon of an initial jump of the second degree zeroth order on the left end of a given segment. The asymptotics of the solution has been sought in the form of a sum of the regular part and the part of the boundary layer. The terms of the regular part are defined as solutions of integro-differential boundary value problems, in which the equations and boundary conditions contain additional terms, called the initial jumps of the integral terms and solutions. Boundary layer terms are defined as solutions of third-order differential equations with initial conditions. A theorem on the existence, uniqueness, and asymptotic representation of a solution is presented along with an asymptotic estimate of the remainder term of the asymptotics. The purpose of this study is to construct a uniform asymptotic approximation to the solution to the original boundary value problem over the entire considered segment.

scholarly journals On Solving the Nonlinear Falkner–Skan Boundary-Value Problem: A Review

Fluids ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 153
Author(s):  
Asai Asaithambi
Keyword(s):  
Boundary Layer ◽  
Mixed Methods ◽  
Boundary Value Problem ◽  
Velocity Distribution ◽  
Laminar Boundary Layer ◽  
Boundary Value ◽  
Third Order ◽  
Ongoing Research ◽  
The Third

This article is a review of ongoing research on analytical, numerical, and mixed methods for the solution of the third-order nonlinear Falkner–Skan boundary-value problem, which models the non-dimensional velocity distribution in the laminar boundary layer.

scholarly journals Solving nonlinear third-order three-point boundary value problems by boundary shape functions methods

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ji Lin ◽  
Yuhui Zhang ◽  
Chein-Shan Liu
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Initial Conditions ◽  
Boundary Value ◽  
Accurate Solution ◽  
Point Boundary ◽  
Third Order ◽  
Boundary Shape ◽  
Free Function ◽  
New Algorithms

AbstractFor nonlinear third-order three-point boundary value problems (BVPs), we develop two algorithms to find solutions, which automatically satisfy the specified three-point boundary conditions. We construct a boundary shape function (BSF), which is designed to automatically satisfy the boundary conditions and can be employed to develop new algorithms by assigning two different roles of free function in the BSF. In the first algorithm, we let the free functions be complete functions and the BSFs be the new bases of the solution, which not only satisfy the boundary conditions automatically, but also can be used to find solution by a collocation technique. In the second algorithm, we let the BSF be the solution of the BVP and the free function be another new variable, such that we can transform the BVP to a corresponding initial value problem for the new variable, whose initial conditions are given arbitrarily and terminal values are determined by iterations; hence, we can quickly find very accurate solution of nonlinear third-order three-point BVP through a few iterations. Numerical examples confirm the performance of the new algorithms.

Numerical Method for Singularly Perturbed Parabolic Equations in Unbounded Domains in the Case of Solutions Growing at Infinity

2009 ◽  
Vol 9 (1) ◽  
pp. 100-110
Author(s):  
G. I. Shishkin
Keyword(s):  
Boundary Layer ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Maximum Norm ◽  
Lateral Part ◽  
Initial Boundary Value

AbstractAn initial-boundary value problem is considered in an unbounded do- main on the x-axis for a singularly perturbed parabolic reaction-diffusion equation. For small values of the parameter ε, a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm grows without bound even for fixed values of the parameter ε. In the present paper, the proximity of solutions of the initial-boundary value problem and of its numerical approximations is considered. Using the method of special grids condensing in a neighbourhood of the boundary layer, a special finite difference scheme converging ε-uniformly in the weight maximum norm has been constructed.

scholarly journals On a third order initial boundary value problem in a plane domain

2012 ◽  
Vol 17 (3) ◽  
pp. 312-326
Author(s):  
Neringa Klovienė
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Auxiliary Problem ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Plane Domain ◽  
Third Order ◽  
Initial Boundary Value

Third order initial boundary value problem is studied in a bounded plane domain σ with C4 smooth boundary ∂σ. The existence and uniqueness of the solution is proved using Galerkin approximations and a priory estimates. The problem under consideration appear as an auxiliary problem by studying a second grade fluid motion in an infinite three-dimensional pipe with noncircular cross-section.

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