Higher Order Boundary Value Problems Research Articles

Upper and Lower Solutions Method for Higher Order Boundary Value Problems

Upper and Lower Solutions Method for Higher Order Boundary Value Problems

On the Existence of Positive Solution for nth Order Differential Equation for Boundary Value Problems

We considering the problem of solving a nonlinear differential equation in the Banach space of real functions and continuous on a bounded and closed interval. By means of the fixed point theory for a strict set contraction operator, this paper investigates the existence, nonexistence, and multiplicity of positive solutions for a nonlinear higher order boundary value problem.

Solvability for a system of nonlinear fractional higher-order three-point boundary value problem

Solvability for a system of nonlinear fractional higher-order three-point boundary value problem

Solvability for a system of nonlinear fractional higher-order three-point boundary value problem

Solvability for a system of nonlinear fractional higher-order three-point boundary value problem

Green function's new property and its application to the solution for a higher-order fractional boundary value problem

Green function's new property and its application to the solution for a higher-order fractional boundary value problem

Positive Solutions for Singular Semipositone Fractional Differential Equation Subject to Multipoint Boundary Conditions

Existence result together with multiplicity result of positive solutions of higher-order fractional multipoint boundary value problems is given by considering the integrations of height functions on some special bounded sets. The nonlinearity may change its sign and may possess singularities on the time and the space variables at the same time.

Green function's new property and its application to the solution for a higher-order fractional boundary value problem

Green function's new property and its application to the solution for a higher-order fractional boundary value problem

Symmetric positive solutions for the systems of higher-order boundary value problems on time scales

AbstractIn this paper, we discuss the existence of symmetric positive solutions for the systems of higher-order boundary value problems on time scales. Our results extend some recent work in the literature. The analysis of this paper mainly relies on the monotone iterative technique.

A class of Birkhoff–Lagrange-collocation methods for high order boundary value problems

A general procedure to determine collocation methods for high order boundary value problems is presented. These methods provide globally continuous differentiable solution in the form of polynomial functions and also numerical solution on a set of discrete points. Some special cases are considered. Numerical experiments are presented which support theoretical results and provide favorable comparisons with other existing methods.

A generalized Lyapunov inequality for a higher-order fractional boundary value problem

In the paper, we establish a Lyapunov inequality and two Lyapunov-type inequalities for a higher-order fractional boundary value problem with a controllable nonlinear term. Two applications are discussed. One concerns an eigenvalue problem, the other a Mittag-Leffler function.

A hybrid numerical method for solving system of high order boundary value problems

Abstract Higher order system of boundary value problems arise in several areas of applications. In this paper, we employ the Chebyshev wavelet finite difference method to solve such system of higher order boundary value problems. Numerical experiments are conducted to show the feasibility of the proposed method.

A Barycentric Interpolation Collocation Method for Linear Nonlocal Boundary Value Problems

This paper is devoted to the numerical treatment of a class of higher-order multi-point boundary value problem-s(BVPs). The method is proposed based on the Lagrange interpolation collocation method, but it avoids thenumerical instability of Lagrange interpolation. Numerical results obtained by present method compare with othermethods show that the present method is simple and accurate for higher-order multi-point BVPs, and it is eectivefor solving six order or higher order multi-point BVPs.

Optimal Spectral Schemes Based on Generalized Prolate Spheroidal Wave Functions of Order $$-1$$

We introduce a family of generalized prolate spheroidal wave functions (PSWFs) of order $$-1,$$ and develop new spectral schemes for second-order boundary value problems. Our technique differs from the differentiation approach based on PSWFs of order zero in Kong and Rokhlin (Appl Comput Harmon Anal 33(2):226–260, 2012); in particular, our orthogonal basis can naturally include homogeneous boundary conditions without the re-orthogonalization of Kong and Rokhlin (2012). More notably, it leads to diagonal systems or direct “explicit” solutions to 1D Helmholtz problems in various situations. Using a rule optimally pairing the bandwidth parameter and the number of basis functions as in Kong and Rokhlin (2012), we demonstrate that the new method significantly outperforms the Legendre spectral method in approximating highly oscillatory solutions. We also conduct a rigorous error analysis of this new scheme. The idea and analysis can be extended to generalized PSWFs of negative integer order for higher-order boundary value and eigenvalue problems.

Positive solutions of higher-order Sturm-Liouville boundary value problems with derivative-dependent nonlinear terms

We consider the Sturm-Liouville boundary value problem $$\left \{ \textstyle\begin{array}{@{}l} y^{(m)}(t)+ F (t,y(t),y'(t),\ldots,y^{(q)}(t) )=0, \quad t\in[0,1], y^{(k)}(0)=0,\quad 0\leq k\leq m-3, \zeta y^{(m-2)}(0)-\theta y^{(m-1)}(0)=0,\qquad \rho y^{(m-2)}(1)+\delta y^{(m-1)}(1)=0, \end{array}\displaystyle \right . $$ where $m\geq3$ and $1\leq q\leq m-2$ . We note that the nonlinear term F involves derivatives. This makes the problem challenging, and such cases are seldom investigated in the literature. In this paper we develop a new technique to obtain existence criteria for one or multiple positive solutions of the boundary value problem. Several examples with known positive solutions are presented to dwell upon the usefulness of the results obtained.

The application of block pulse functions for solving higher-order differential equations with multi-point boundary conditions

In this paper, the block pulse function method is proposed for solving high-order differential equations associated with multi-point boundary conditions. Although the orthogonal block pulse functions frequently have been applied to approximate the solution of ordinary differential equations associated with the initial conditions, the presented method provides the flexibility with respect to multi-point boundary conditions in separated or non-separated forms. This technique, which may be named the augmented block pulse function method, reduces a system of high-order boundary value problems of ordinary differential equations to a system of algebraic equations. The illustrated results confirm the computational efficiency, reliability, and simplicity of the presented method.

On higher-order nonlinear boundary value problems with nonlocal multipoint integral boundary conditions

In this paper, we study the existence of solutions for nonlinear nth-order ordinary differential equations and inclusions with nonlocal multipoint integral boundary conditions. Fixed point theorems due to Schaefer and Banach are employed to prove the existence results for the single-valued case, whereas the existence of solutions for the multivalued problem is established by means of a nonlinear alternative for Kakutani maps and Covitz–Nadler fixed point theorem. The obtained results are well explained by examples. We extend our discussion to some new problems.

B-spline and singular higher-order boundary value problems

In this paper, B-spline method is developed to find an approximate solution for singular linear and non-linear higher-order differential equation. Error analysis is presented. The method is then tested on linear and nonlinear examples. The numerical results reveal that B-spline method is very efficient and accurate.

Application of modifed optimal homotopy perturbation method to higher order boundary value problems in a nite domain

This research focuses on the solution of higher order boundary value problems by our proposed method Modified Optimal Homotopy Per- turbation Method (MOHPM). A homotopy with an embedding param- eter and Daftardar-Jafari polynomials are used. To control the conver- gence of solution, some auxiliary functions which depend upon variables and some constants are used. The proposed method is simple, rapid, ef- fective and accurate. The accuracy has been proved by comparing our results with the solutions of optimal homotopy perturbation method (OHPM), optimal homotopy asymptotic method (OHAM), variational iterative method (VIM),variational iteration method using He's poly- nomials (VIMHP), homotopy perturbation method (HPM), Adoman decomposition method (ADM) and Quintic B-Spline Galerkin's scheme.

Numerical Solutions of a Generalized Nth Order Boundary Value Problems Using Power Series Approximation Method

In this paper, a new approach called Power Series Approximation Method (PSAM) is developed for the numerical solution of a generalized linear and non-linear higher order Boundary Value Problems (BVPs). The proposed method is efficient and effective on the experimentation on some selected thirteen-order, twelve-order and ten-order boundary value problems as compared with the analytic solutions and other existing methods such as the Homotopy Perturbation Method (HPM) and Variational Iteration Method (VIM) available in the literature. A convergence analysis of PSAM is also provided.

Application of Modified Optimal Homotopy Perturbation Method to Higher Order Boundary Value problems in a Finite Domain

Application of Modified Optimal Homotopy Perturbation Method to Higher Order Boundary Value problems in a Finite Domain