Third-order Boundary Value Problem Research Articles

Existence of a unique solution for a third-order boundary value problem with nonlocal conditions of integral type

The existence of a unique solution for a third-order boundary value problem with integral condition is proved in several ways. The main tools in the proofs are the Banach fixed point theorem and the Rus’s fixed point theorem. To compare the applicability of the obtained results, some examples are considered.

Existence Theory for Positive Iterative Solutions to a Type of Boundary Value Problem

We introduce some research results on a type of third-order boundary value problem for positive iterative solutions. The existence of solutions to these problems was proved using the monotone iterative technique. As an examination of the proposed method, an example to illustrate the effectiveness of our results was presented.

Positive solutions of nonlinear third-order boundary value problems involving Stieltjes integral conditions

Positive solutions of nonlinear third-order boundary value problems involving Stieltjes integral conditions

On the Solvability of a Resonant Third-Order Integral m-Point Boundary Value Problem on the Half-Line

In this work, the existence of at least one solution for the following third-order integral and m -point boundary value problem on the half-line at resonance ρ t u ′ t ″ = w t , u t , u ′ t , u ″ t , t ∈ 0 , ∞ , u 0 = ∑ j = 1 m α j ∫ 0 η j u t d t , u ′ 0 = 0 , lim t ⟶ ∞ ρ t u ′ t ′ = 0 , will be investigated. The Mawhin’s coincidence degree theory will be used to obtain existence results while an example will be used to validate the result obatined.

Existence and Uniqueness of Solutions to Third-Order Boundary Value Problems

The purpose of this research is to connect fixed point methods with certain third-order boundary value problems in new and interesting ways. Our strategy involves an analysis of the problem under consideration within closed and bounded sets. We develop sufficient conditions under which the associated mappings will be contractive and invariant in these sets, which generates new advances concerning the existence, uniqueness and approximation of solutions.

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

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.

On a Third-Order P-Laplacian Boundary Value Problem at Resonance on the Half-Line

On a Third-Order P-Laplacian Boundary Value Problem at Resonance on the Half-Line

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

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

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.

A two-step hybrid block method with fourth derivatives for solving third-order boundary value problems

This manuscript proposes an implicit two-step hybrid block method which incorporates fourth derivatives, for solving linear and non-linear third-order boundary value problems in ODEs. The derivation of the present method is based on collocation and interpolation techniques, and the convergence analysis of the new strategy is proved to be seventh-order convergent. The proposed approach produces discrete approximations at the grid points, obtained after solving an algebraic system of equations. Numerical experiments are studied to show the performance and viability of the proposed approach. The numerical results demonstrated that the new technique gives accurate approximations, which are better than some existing strategies in the available literature and also found to be in good agreement with known analytical solutions.

Solution of third-order Emden–Fowler-type equations using wavelet methods

PurposeThe numerical solution of third-order boundary value problems (BVPs) has a great importance because of their applications in fluid dynamics, aerodynamics, astrophysics, nuclear reactions, rocket science etc. The purpose of this paper is to develop two computational methods based on Hermite wavelet and Bernoulli wavelet for the solution of third-order initial/BVPs.Design/methodology/approachBecause of the presence of singularity and the strong nonlinear nature, most of third-order BVPs do not occupy exact solution. Therefore, numerical techniques play an important role for the solution of such type of third-order BVPs. The proposed methods convert third-order BVPs into a system of algebraic equations, and on solving them, approximate solution is obtained. Finally, the numerical simulation has been done to validate the reliability and accuracy of developed methods.FindingsThis paper discussed the solution of linear, nonlinear, nonlinear singular (Emden–Fowler type) and self-adjoint singularly perturbed singular (generalized Emden–Fowler type) third-order BVPs using wavelets. A comparison of the results of proposed methods with the results of existing methods has been given. The proposed methods give the accuracy up to 19 decimal places as the resolution level is increased.Originality/valueThis paper is one of the first in the literature that investigates the solution of third-order Emden–Fowler-type equations using Bernoulli and Hermite wavelets. This paper also discusses the error bounds of the proposed methods for the stability of approximate solutions.

A class of nonlinear third-order boundary value problem with integral condition at resonance

A class of nonlinear third-order boundary value problem with integral condition at resonance

Existence results for a resonant third-order problem with two dimensional kernel on the half-line

The purpose of this research is to establish conditions that guarantee the solvability of a resonant third-order boundary value problem (BVP) on the half-line. The resonant problem is subject to integral and m-point boundary conditions while, the kernel of the differential operator is equal to two. The coincidence degree theory will be applied and an illustrative example will be considered to demonstrate the result.

一类含一阶导数的三阶边值问题正解的存在性

一类含一阶导数的三阶边值问题正解的存在性

Quartic Non-Polynomial Spline for Solving the Third-Order Dispersive Partial Differential Equation

In the present paper, we introduce a non-polynomial quadratic spline method for solving third-order boundary value problems. Third-order singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, quantum mechanics, chemical reactor theory, Newtonian fluid mechanics, optimal control, convection-diffusion processes, hydrodynamics, aerodynamics, etc. These problems have various important applications in fluid dynamics. The procedure involves a reduction of a third-order partial differential equation to a first-order ordinary differential equation. Truncation errors are given. The unconditional stability of the method is analysed by the Von-Neumann stability analysis. The developed method is tested with an illustrated example, and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, a graphical comparison between analytical and approximate solutions is also shown for the illustrated example.

一类带积分边界条件的三阶边值问题正解的存在性

一类带积分边界条件的三阶边值问题正解的存在性

Exponential spline method for singularly perturbed third-order boundary value problems

Abstract The exponential spline function is presented to find the numerical solution of third-order singularly perturbed boundary value problems. Convergence analysis of the method is briefly discussed, and it is shown to be sixth order convergence. To validate the applicability of the method, some model problems are solved for different values of the perturbation parameter, and the numerical results are presented both in tables and graphs. Furthermore, the present method gives more accurate solution than some methods existing in the literature.

Existence of positive solutions to multi-point third order problems with sign changing nonlinearities

In this paper, the authors examine the existence of positive solutions to a third-order boundary value problem having a sign changing nonlinearity. The proof makes use of fixed point index theory. An example is included to illustrate the applicability of the results.

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

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.

A geometrical criterion for nonexistence of constant-sign solutions for some third-order two-point boundary value problems

We give a simple geometrical criterion for the nonexistence of constant-sign solutions for a certain type of third-order two-point boundary value problem in terms of the behavior of nonlinearity in the equation. We also provide examples to illustrate the applicability of our results.