scholarly journals Numerical Solution for Third-Order Two-Point Boundary Value Problems with the Barycentric Rational Interpolation Collocation Method

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Qian Ge ◽  
Xiaoping Zhang
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Collocation Method ◽  
Boundary Value ◽  
Rational Interpolation ◽  
Point Boundary ◽  
Third Order ◽  
The Third ◽  
The Matrix ◽  
Barycentric Rational Interpolation

The numerical solution for a kind of third-order boundary value problems is discussed. With the barycentric rational interpolation collocation method, the matrix form of the third-order two-point boundary value problem is obtained, and the convergence and error analysis are obtained. In addition, some numerical examples are reported to confirm the theoretical analysis.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

scholarly journals The barycentric rational interpolation collocation method for boundary value problems

2018 ◽  
Vol 22 (4) ◽  
pp. 1773-1779 ◽  
Author(s):  
Dan Tian ◽  
Ji-Huan He
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Analytical Methods ◽  
Boundary Value ◽  
Rational Interpolation ◽  
Higher Order ◽  
Barycentric Rational Interpolation

Higher-order boundary value problems have been widely studied in thermal science, though there are some analytical methods available for such problems, the barycentric rational interpolation collocation method is proved in this paper to be the most effective as shown in three examples.

Applications of the Method of Barriers II. Some Singularly Perturbed Problems

1995 ◽  
Vol 2 (3) ◽  
pp. 323-335
Author(s):  
V. G. Sushko ◽  
N. Kh. Rozov
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Point Boundary ◽  
Singularly Perturbed Problems ◽  
Quasilinear Equations ◽  
Third Order ◽  
The Third ◽  
Asymptotic Representations

Abstract The method of barriers is used to justify asymptotic representations of solutions of two-point boundary value problems for singularly perturbed quasilinear equations of the second and the third order. This paper is a continuation of [Rozov and Sushko, Georgian Math. J. 2: 99-110, 1995].

The spectral representation of two-point boundary-value problems for third-order linear evolution partial differential equations

2005 ◽  
Vol 461 (2061) ◽  
pp. 2965-2984 ◽  
Author(s):  
Beatrice Pelloni
Keyword(s):  
Integral Representation ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Series Representation ◽  
Point Boundary ◽  
Third Order ◽  
Linear Evolution ◽  
The Third ◽  
Order Case

We use a spectral transform method to study general boundary-value problems for third-order, linear, evolution partial differential equations with constant coefficients, posed on a finite space domain. We show how this method yields a simple characterization of the discrete spectrum of the associated spatial differential operator, and discuss the obstructions that arise when trying to represent the solution of such a problem as a series of exponential functions. We first review the theory for second-order two-point boundary-value problems, and present an alternative way to derive the classical series representation, as well as an equivalent integral representation, which generally involves complex contours. We illustrate the advantages of the integral representation by studying in some detail the case where Robin-type boundary conditions are prescribed. We then consider the third-order case and show that the integral representation is in general not equivalent to a discrete series representation, justifying a posteriori the failure of some of the classical approaches. We illustrate the third-order case in detail, using the example of the equation q t + q xxx =0 for various types of boundary conditions. In contrast with the second-order case, the qualitative properties of the spectrum of the associated spatial differential operator depend in this case not only on the equation but also on the type of boundary conditions. In particular, the solution appears to admit a series representation only when the prescribed boundary conditions couple the two endpoints of the interval.

scholarly journals Numerical Solution of Two-Point Boundary Value Problems by Interpolating Subdivision Schemes

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Ghulam Mustafa ◽  
Syeda Tehmina Ejaz
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Point Boundary ◽  
Approximation Properties ◽  
Third Order ◽  
Continuous Solutions ◽  
Subdivision Schemes ◽  
Numerical Examples

A numerical interpolating algorithm of collocation is formulated, based on 8-point binary interpolating subdivision schemes for the generation of curves, to solve the two-point third order boundary value problems. It is observed that the algorithm produces smooth continuous solutions of the problems. Numerical examples are given to illustrate the algorithm and its convergence. Moreover, the approximation properties of the collocation algorithm have also been discussed.

A Collocation Method for Global Approximation of General Second Order BVPs

2007 ◽  
Vol 3 (1) ◽  
pp. 23-34 ◽  
Author(s):  
F. Costabile ◽  
A. Napoli
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Collocation Method ◽  
Boundary Value ◽  
Second Order ◽  
Point Boundary ◽  
Global Approximation ◽  
Numerical Examples

For the numerical solution of the second order nonlinear two-point boundary value problems a family of polynomial global methods is derived.Numerical examples provide favorable comparisons with other existing methods.

scholarly journals Linear Barycentric Rational Method for Two-Point Boundary Value Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Qian Ge ◽  
Xiaoping Zhang
Keyword(s):  
Theoretical Analysis ◽  
Convergence Rate ◽  
Boundary Value Problems ◽  
Collocation Method ◽  
Boundary Value ◽  
Rational Method ◽  
Matrix Form ◽  
Point Boundary ◽  
Numerical Examples ◽  
The Matrix

Linear barycentric rational method for solving two-point boundary value equations is presented. The matrix form of the collocation method is also obtained. With the help of the convergence rate of the interpolation, the convergence rate of linear barycentric rational collocation method for solving two-point boundary value problems is proved. Several numerical examples are provided to validate the theoretical analysis.

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