Order Two-point Boundary Value Problems Research Articles

Infinitely Many Nodal Solutions of Superlinear Third Order Two-Point Boundary Value Problems

Infinitely Many Nodal Solutions of Superlinear Third Order Two-Point Boundary Value Problems

Infinitely many positive solutions and Ulam–Hyers stability of fractional order two-point boundary value problems

Infinitely many positive solutions and Ulam–Hyers stability of fractional order two-point boundary value problems

Solving second order two-point boundary value problems accurately by a third derivative hybrid block integrator

This article deals with the development of an optimized third-derivative hybrid block method for integrating general second order two-point boundary value problems (BVPs) subject to different types of boundary conditions (BCs) such as Dirichlet, Neumann or Robin. A purely interpolation and collocation approach has been used in order to develop the method. A constructive approach has been applied in the development of the method to consider two off-step optimal points among an infinite number of possible choices in a two-step block corresponding to a generic interval of the form [xn,xn+2]. The obtained method simultaneously produces an approximate solution over the entire integration interval. Some numerical experiments have been presented that show the good performance of the presented scheme.

Iterative numerical method for fractional order two-point boundary value problems

In this paper we develop an iterative numerical method based on Bernstein splines for solving two-point boundary problems associated to differential equations of fractional order $\alpha\in\left( 0,1\right) $. The convergence of the method is proved by providing the error estimate and it is tested on a numerical example.

Addendum: Solvability for some fourth order two-point boundary value problems

Addendum: Solvability for some fourth order two-point boundary value problems

Discrete fractional order two-point boundary value problem with some relevant physical applications

The results reported in this paper are concerned with the existence and uniqueness of solutions of discrete fractional order two-point boundary value problem. The results are developed by employing the properties of Caputo and Riemann–Liouville fractional difference operators, the contraction mapping principle and the Brouwer fixed point theorem. Furthermore, the conditions for Hyers–Ulam stability and Hyers–Ulam–Rassias stability of the proposed discrete fractional boundary value problem are established. The applicability of the theoretical findings has been demonstrated with relevant practical examples. The analysis of the considered mathematical models is illustrated by figures and presented in tabular forms. The results are compared and the occurrence of overlapping/non-overlapping has been discussed.

Solvability for some fourth order two-point boundary value problems

Some fourth-order two-point boundary value problems are considered in this paper. Firstly, the Green's function is obtained by the use of the Laplace transform. Secondly, the first eigenvalue is given by using Ritz method. Then, by the use of the properties of self-adjoint operators and the fixed point index theory, the existence of positive solutions is obtained. Finally, an example is given to illustrate the main results.

Improved error estimate and applications of the complete quartic spline

Abstract In this paper an improved error bound is obtained for the complete quartic spline with deficiency 2, in the less smooth class of continuous functions. In the case of Lipschitzian functions, the obtained estimate improves the constant from Theorem 3, in J. Approx. Theory 58 (1989) 58-67. Some applications of the complete quartic spline in the numerical integration and in the construction of an iterative numerical method for fourth order two-point boundary value problems with pantograph type delay are presented.

Positive Solutions for Systems of Fourth Order Two-Point Boundary Value Problems with Parameter

This paper deals with the existence of positive solutions for a system of nonlinear singular fourth-order differential equations with a parameter $\lambda$ subject two-point boundary conditions. Our analysis relies on the Krasnoselskii fixed point theorem and under suitable conditions, we derive explicit eigenvalue intervals of $\lambda$ for the existence of at least one positive solution for the system.

Non-polynomial spline finite difference method for solving second order boundary value problem

Second -order two-point boundary value problems (BVPs) were solved based on a of non-polynomial spline general functions with finite difference method.In this paper we discusses a method depended on using special finite-difference approximations for derivatives and formatting a formula that can be deal with endpoints that exceed the usual finite-difference formula for derivatives. Convergence analysis of the method is discussed. The numerical description of the method is shown by four examples. So the results obtained by our method are very encouraging over other existing methods.

A Spectral Collocation Method for Systems of Singularly Perturbed Boundary Value Problems

Abstract We present a spectrally accurate method for solving coupled singularly perturbed second order two-point boundary value problems (BVPs). The method combines analytical coordinate transformations with a standard Chebyshev spectral collocation method; it is applicable to linear and to nonlinear problems. The method performs well in resolving very thin boundary layers. Compared to other methods which had been proposed for systems of BVPs this method is competitive in terms of accuracy, allows for different perturbation parameters in each of the equations, and does not require special properties of the coefficient functions.

Positive solutions for a third order two-point boundary value problem at resonance on the positive half line

Positive solutions for a third order two-point boundary value problem at resonance on the positive half line

Countably Infinitely Many Positive Solutions for Even Order Boundary Value Problems with Sturm-Liouville Type Integral Boundary Conditions on Time Scales

In this paper, we establish the existence of countably infinitely many positive solutions for a certain even order two-point boundary value problem with integral boundary conditions on time scales by using Holder’s inequality and Krasnoselskii’s fixed point theorem for operators on a cone.

PERFORMANCE OF MODIFIED NON-LINEAR SHOOTING METHOD FOR SIMULATION OF 2ND ORDER TWO-POINT BVPS

In this research article, numerical solution of nonlinear 2nd order two-point boundary value problems (TPBVPs) is discussed by the help of nonlinear shooting method (NLSM), and through the modified nonlinear shooting method (MNLSM). In MNLSM, fourth order Runge-Kutta method for systems is replaced by Adams Bashforth Moulton method which is a predictor-corrector scheme. Results acquired numerically through NLSM and MNLSM of TPBVPs are discussed and analyzed. Results of the tested problems obtained numerically indicate that the performance of MNLSM is rapid and provided desirable results of TPBVPs, meanwhile MNLSM required less time to implement as comparable to the NLSM for the solution of TPBVPs.

Even Number of Positive Solutions for the System of (p, q)-Laplacian Fractional Order Two-Point Boundary Value Problems

In this paper, we derive sufficient conditions for the existence of at least two positive solutions for a system of (p, q)-Laplacian fractional order two-point boundary value problems by using an Avery–Henderson functional fixed point theorem. We also establish the existence of at least 2m positive solutions to the boundary value problem for an arbitrary positive integer m.

Existence and multiplicity for a system of fractional higher-order two-point boundary value problem

The purpose of this paper is to establish some results on the existence of multiple positive solutions for a system of nonlinear fractional order two-point boundary value problem. The main tool is a fixed point theorem of the cone expansion and compression of functional type and five functional fixed point theorem. Some examples are also presented to illustrate the availability of the main results.

An effective method for numerical solution and numerical derivatives for sixth order two-point boundary value problems

In this paper, we study an effective quintic polynomial spline method for numerical solution, and first order to fifth order numerical derivatives of the analytic solution at the knots for a class of sixth order two-point boundary value problems. Our new method is based on a quintic spline interpolation problem. It is easy to implement and is able to provide sixth order accurate numerical results at the knots. Numerical tests show that our method is very practical and effective.

An effective method for numerical solution and numerical derivatives for sixth order two-point boundary value problems

An effective method for numerical solution and numerical derivatives for sixth order two-point boundary value problems

Efficient algorithms for fourth and sixth-order two-point non-linear boundary value problems using non-polynomial spline approximations on a geometric mesh

An efficient algorithm for the numerical solution of linear and non-linear higher (even)-order two-point boundary value problems has been developed. The method is third-order accurate and applicable to both singular and non-singular cases. We have used non-polynomial spline basis and geometric mesh finite differences for the generation of new scheme. The irreducibility and monotone property of the iteration matrix have been established, that justify the convergence property of the proposed method. Some computational experiments have been carried out to demonstrate the efficiency in terms of convergence order and maximum absolute error estimates. The numerical results justify the reliability and efficiency of the method both in terms of order and accuracy.

The Convergence of Geometric Mesh Cubic Spline Finite Difference Scheme for Nonlinear Higher Order Two-Point Boundary Value Problems

An efficient algorithm for the numerical solution of higher (even) orders two-point nonlinear boundary value problems has been developed. The method is third order accurate and applicable to both singular and nonsingular cases. We have used cubic spline polynomial basis and geometric mesh finite difference technique for the generation of this new scheme. The irreducibility and monotone property of the iteration matrix have been established and the convergence analysis of the proposed method has been discussed. Some numerical experiments have been carried out to demonstrate the computational efficiency in terms of convergence order, maximum absolute errors, and root mean square errors. The numerical results justify the reliability and efficiency of the method in terms of both order and accuracy.