Third-order Boundary Value Problem Research Articles

Numerical integration of third-order BVPs using a fourth-order hybrid block method

This research paper introduces a new hybrid block method to solve the third-order boundary value problems (BVPs). The method combines interpolation and collocation and uses a power series polynomial to find an approximate solution to the considered third-order BVPs. Some third-order BVP models are numerically solved to verify the performance and efficiency of the proposed method, and the approximate solution from the proposed method is more efficient when compared to some existing numerical methods. In summary, the proposed method provides reliable and efficient accuracy for solving third-order BVPs, making it a valuable contribution to the fields of numerical analysis and computational mathematics. The advantages of the proposed method include improved computational time efficiency and accuracy in terms of maximum absolute errors for solving third-order BVPs.

Numerical solution of singularly perturbed singular third order boundary value problems with nonclassical sinc method

In this paper, we employ a nonclassical sinc-collocation method to compute numerical solutions for singularly perturbed singular third-order boundary value problems prevalent in various scientific and engineering domains. Utilizing the sinc approximation offers a strategic advantage in navigating singularities, thus enabling an efficient computational strategy. Our method streamlines the solution process by converting singular boundary value problems into sets of linear equations, thereby improving computational efficiency. Moreover, its straightforward implementation adds to its robustness. We explore the convergence properties and error estimation of our proposed methods in detail. Finally, we provide two illustrative examples that demonstrate the effectiveness of our approach.

An efficient fifth-order block method for solving third-order BVPs

This research paper introduces a new block method to solve third-order boundary-value problems (BVPs) directly. The method uses a suitable polynomial and three intermediate points to approximate the theoretical solution of third-order BVPs. An approximate solution is provided within the integration interval, and the basic properties of the proposed method, including convergence and stability, are theoretically analyzed. To assess the effectiveness of the proposed method, several real-life problems, such as boundary layer problems and physical model problems, are solved numerically. After comparison with some existing methods, the results of numerical experiments confirm the practical utility and effectiveness of the proposed method in real-world applications by demonstrating that it is superior and highly efficient in performance.

Numerical solution of third order singular boundary value problems with nonclassical SE-sinc-collocation and nonclassical DE-sinc-collocation

In this paper, we use the nonclassical SE-sinc-collocation and nonclassical DE-sinc-collocation methods for the numerical solution of singular third-order boundary value problems. The novelty of the approach is based on using the new non-classical weight function for sinc method instead of the classic ones. The convergence and error estimation of our methods is discussed. Several examples are solved and numerical results are compared with the existed methods, the obtained results demonstrate the validity of the obtained theoretical results and the efficiency of the our methods.

Existence of sign-changing solutions for a third-order boundary value problem with nonlocal conditions of integral type

We prove the existence of at least one sign-changing solution for a third-order nonlocal boundary value problem by applying Leray-Schauder Continuation Principle. To illustrate the applicability of the obtained results, we consider an example.

Positive Solutions for Third-Order Boundary Value Problems with Indefinite Weight

Positive Solutions for Third-Order Boundary Value Problems with Indefinite Weight

Existence of multiple positive solutions for a third-order boundary value problem with nonlocal conditions

We study the existence of multiple positive solutions for a nonlinear third-order differential equation subject to various nonlocal boundary conditions. The boundary conditions that we study contain Stieltjes integral and include the special cases of m-point conditions and integral conditions. The main tool in the proof of our result is Krasnosel’skii’s fixed point theorem. To illustrate the applicability of the obtained results, we consider examples.

APPROXIMATE SOLUTIONS TO BOUNDARY VALUE PROBLEMS OF HIGHER-ORDER BY THE MODIFIED ADOMAIN DECOMPOSITION METHOD

This paper propose a new modified differential operator for solving third-order boundary value problems into higherorder ordinary differential equation. We found the differential operator for new two inverse operators which can be applied for solving equations at more than one type in different conditions. This paper is given five non-linear examples from a high-order, we got effective and easily approximate solutions to the exact solution.

On Novel Spline Function Method for Solving Third-Order Boundary Value Problems

In this paper, a numerical method is suggested for solving general a nonlinear third order boundary value problem (BVP). In this method, the given nonlinear third-order BVP will be transformed into two third-order initial value problems (IVPs), then spline function approximations are applied to both two IVP for finding the Spline solution and its derivatives up to third order of the given BVP. The study shows that the spline solution of the BVP is existent and unique, and the convergence order of the spline method is fourth with a local truncation error . The presented algorithm is designed for solving a general BVP, where it is applied to some types of nonlinear third-order differential equations. Comparisons of the results obtained by spline method with other methods show the efficiency and highly accurate of the proposed method.

Efficient Decomposition Shooting Method for Solving Third-Order Boundary Value Problems

Efficient Decomposition Shooting Method for Solving Third-Order Boundary Value Problems

Existence and Uniqueness Results for a Fully Third-Order Boundary Value Problem

Existence and Uniqueness Results for a Fully Third-Order Boundary Value Problem

A faster iterative method for solving nonlinear third-order BVPs based on Green’s function

In this article, we propose an iterative method, called the GA iterative method, to approximate the fixed points of generalized α-nonexpansive mappings in uniformly convex Banach spaces. Further, we obtain some convergence results of the new iterative method. Also, we provide a nontrivial example of a generalized α-nonexpansive mapping and with the example, we carry out a numeral experiment to show that our new iterative algorithm is more efficient than some existing iterative methods. Again, we present an interesting strategy based on the GA iterative method to solve nonlinear third-order boundary value problems (BVPs). For this, we derive a sequence named the GA–Green iterative method and show that the sequence converges strongly to the fixed point of an integral operator. Finally, the approximation of the solution for a nonlinear integrodifferential equation via our new iterative method is considered. We present some illustrative examples to validate our main results in the application sections of this article. Our results are a generalization and an extension of several prominent results of many well-known authors in the literature.

Applications of the Bernoulli wavelet collocation method in the analysis of MHD boundary layer flow of a viscous fluid

This study focuses on the flow of viscous, electrically conducting incompressible fluid over a stretching plate. The Falkner–Skan equation is a nonlinear, third-order boundary value problem. No closed-form solutions are available for this two-point boundary value problem. Here, we developed a new functional matrix of integration using the Bernoulli wavelet and also generated a new technique called Bernoulli wavelet collocation method (BWCM) to solve the nonlinear differential equation that arises in the fluid flow over a stretching plate. The boundary layer model is transformed to a nonlinear ordinary differential equation called the Falkner-Skan type equation using suitable transformation. Using BWCM, we have solved the unbounded governing equations of different types that arise in the MHD boundary-layer flow of a viscous fluid over a stretching plate. Several aspects of this problem are justified using the Haar wavelet and the previously obtained theoretical results. It is observed that the boundary-layer thickness decreases as the pressure gradient and magnetic field parameters increase. The overshoots and undershoots are observed for some particular parameters using BWCM. Furthermore, our research yields dual solutions for some physical parameters, which are investigated for the first time in the literature using the Bernoulli wavelet approach. The nature of the flow problem is discussed through the graphs by varying the physical parameters.

Green’s function and existence of solutions for a third-order boundary value problem involving integral condition

Green’s function and existence of solutions for a third-order boundary value problem involving integral condition

A Higher-order Block Method for Numerical Approximation of Third-order Boundary Value Problems in ODEs

In recent times, numerical approximation of 3rd-order boundary value problems (BVPs) has attracted great attention due to its wide applications in solving problems arising from sciences and engineering. Hence, A higher-order block method is constructed for the direct solution of 3rd-order linear and non-linear BVPs. The approach of interpolation and collocation is adopted in the derivation. Power series approximate solution is interpolated at the points required to suitably handle both linear and non-linear third-order BVPs while the collocation was done at all the multiderivative points. The three sets of discrete schemes together with their first, and second derivatives formed the required higher-order block method (HBM) which is applied to standard third-order BVPs. The HBM is self-starting since it doesn’t need any separate predictor or starting values. The investigation of the convergence analysis of the HBM is completely examined and discussed. The improving tactics are fully considered and discussed which resulted in better performance of the HBM. Three numerical examples were presented to show the performance and the strength of the HBM over other numerical methods. The comparison of the HBM errors and other existing work in the literature was also shown in curves.

Existence of positive solutions for a singular third-order two-point boundary value problem on the half-line

In this paper, we consider the following singular third-order two-point boundary value problem on the half-line of the form {x‴+ϕ(t)f(t,x,x′,x″)=0,0<t<+∞,x(0)=0,x′(0)=a1,x′(+∞)=b1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} x'''+\\phi (t)f(t,x,x',x'')=0, \\quad 0< t< +\\infty , \\\\ x(0)=0, \\qquad x'(0)=a_{1},\\qquad x'(+\\infty )=b_{1}, \\end{cases} $$\\end{document} where phi in C[0,+infty ), fin C([0,+infty )times (0,+infty )times mathbb{R}^{{2}},mathbb{R}) may be singular at x=0, and a_{1}, b_{1} are positive constants. Using the Leray–Schauder nonlinear alternative and the diagonalization method together with the truncation function technique, we obtain the existence and qualitative properties of positive solutions for the problem. As applications, an example is given to illustrate our result.

Existence of solutions for a third-order boundary value problem with integral boundary conditions

Bu çalışmada, yarı sonsuz aralık üzerinde tanımlı üçüncü mertebeden üç noktalı integral koşullu, η∈(0,∞)η∈(0,∞) sabit ve\R g:(0,∞)×R3⟶Rg:(0,∞)×R3⟶Rg:(0,∞)×\R3⟶\R Nagumo koşullarını sağlayan bir fonksiyon olmak üzere ϑ′′′(t)+r(t)g(t,ϑ(t),ϑ′(t),ϑ′′(t))=0, t∈(0,∞),ϑ‴(t)+r(t)g(t,ϑ(t),ϑ′(t),ϑ″(t))=0, t∈(0,∞),ϑ(0)=∫η0ϑ(s)ds, ϑ′(0)=A, ϑ′′(∞)=limt→∞ϑ(t)=Bϑ(0)=∫0ηϑ(s)ds, ϑ′(0)=A, ϑ″(∞)=limt→∞ϑ(t)=Bsınır değer probleminin sınırlı veya sınırlı olmayan çözümlerinin varlığı ispatlanmıştır. Schäuder sabit nokta teoremi ve alt ve üst çözümler yöntemi uygulanarak istenilen sonuca ulaşılmıştır. Problemimizde uygun ve yeterli koşullar belirlenerek problemin en az bir çözümünün varlığı gösterilmiştir. Yarı sonsuz aralık üzerinde çalışılması zor olduğundan yarı sonsuz aralık üzerinde integral koşullu bu çalışma bu konuda yapılacak çalışmalar için litaratüre katkı sağlamış olacaktır. Ayrıca, bu sınır değer probleminin çözümleri sınırsız olabilir.

Positive Solutions for Third-Order Boundary Value Problems with the Integral Boundary Conditions and Dependence on the First-Order Derivatives

In this paper, by the use of a new fixed point theorem and the Green function of BVPs, the existence of at least one positive solution for the third-order boundary value problem with the integral boundary conditions is considered,where there is a nonnegative continuous function. Finally, an example which to illustrate the main conclusions of this paper is given.

A new iteration method for the solution of third-order BVP via Green's function

Abstract In this study, a new iterative method for third-order boundary value problems based on embedding Green’s function is introduced. The existence and uniqueness theorems are established, and necessary conditions are derived for convergence. The accuracy, efficiency and applicability of the results are demonstrated by comparing with the exact results and existing methods. The results of this paper extend and generalize the corresponding results in the literature.

Eigenvalues of discontinuous third-order boundary value problems with eigenparameter-dependent boundary conditions

Regular third-order boundary value problems with eigenparameter-dependent boundary conditions and transmission conditions are investigated. It is obtained that the eigenvalues of the considered problems depend not only continuously but also smoothly on the parameters of the problems, i.e., the coefficients, the boundary conditions and transmission conditions. Moreover, the corresponding differential expressions for each parameters are given.