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.

Solving nonlinear third-order boundary value problems based-on boundary shape functions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Chein-Shan Liu ◽  
Jiang-Ren Chang
Keyword(s):  
Boundary Conditions ◽  
Initial Conditions ◽  
Boundary Value ◽  
Numerical Algorithms ◽  
Shape Functions ◽  
Unknown Parameters ◽  
Third Order ◽  
Boundary Shape ◽  
Type Algorithm ◽  
Free Function

Abstract For a third-order nonlinear boundary value problem (BVP), we develop two novel methods to find the solutions, satisfying boundary conditions automatically. A boundary shape function (BSF) is created to automatically satisfy the boundary conditions, which is then employed to develop new numerical algorithms by adopting two different roles of the free function in the BSF. In the first type algorithm, we let the BSF be the solution of the BVP and the free function be a new variable. In doing so, the nonlinear BVP is certainly and exactly transformed to an initial value problem for the new variable with its terminal values as unknown parameters, whereas the initial conditions are given. In the second type algorithm, let the free functions be a set of complete basis functions and the corresponding boundary shape functions be the new bases. Since the solution already satisfies the boundary conditions automatically, we can apply a simple collocation technique inside the domain to determine the expansion coefficients and then the solution is obtained. For the general higher-order boundary conditions, the BSF method (BSFM) can easily and quickly find a very accurate solution. Resorting on the BSFM, the existence of solution is proved, under the Lipschitz condition for the ordinary differential equation system of the new variable. Numerical examples, including the singularly perturbed ones, confirm the high performance of the BSF-based numerical algorithms.

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.

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 On the Solvability of Nonlinear Third-Order Two-Point Boundary Value Problems

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 62
Author(s):  
Ravi P. Agarwal ◽  
Petio S. Kelevedjiev ◽  
Todor Z. Todorov
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Barrier Strips

Under barrier strips type assumptions we study the existence of C 3 [ 0 , 1 ] —solutions to various two-point boundary value problems for the equation x ‴ = f ( t , x , x ′ , x ″ ) . We give also some results guaranteeing positive or non-negative, monotone, convex or concave solutions.

Solving nonlinear boundary value problems by a boundary shape function method and a splitting and linearizing method

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Chein-Shan Liu ◽  
Essam R. El-Zahar ◽  
Chih-Wen Chang
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Shape Function ◽  
Boundary Value ◽  
Linear Equations ◽  
Accurate Solution ◽  
Iteration Step ◽  
Nonlinear Boundary Value Problems ◽  
Boundary Shape ◽  
Nonlinear Boundary Value

Abstract In the paper, we develop two novel iterative methods to determine the solution of a second-order nonlinear boundary value problem (BVP), which precisely satisfies the specified non-separable boundary conditions by taking advantage of the property of the corresponding boundary shape function (BSF). The first method based on the BSF can exactly transform the BVP to an initial value problem for the new variable with two given initial values, while two unknown terminal values are determined iteratively. By using the BSF in the second method, we derive the fractional powers exponential functions as the bases, which automatically satisfy the boundary conditions. A new splitting and linearizing technique is used to transform the nonlinear BVP into linear equations at each iteration step, which are solved to determine the expansion coefficients and then the solution is available. Upon adopting those two novel methods very accurate solution for the nonlinear BVP with non-separable boundary conditions can be found quickly. Several numerical examples are solved to assess the efficiency and accuracy of the proposed iterative algorithms, which are compared to the shooting method.

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