Linear Nonlocal Boundary Value Problems Research Articles

An optimal reproducing kernel method for linear nonlocal boundary value problems

Based on the piecewise polynomial reproducing kernels, the authors presented the reproducing kernel method(RKM) for nonlocal boundary value problems. In this paper, we will present an optimal RKM for linear nonlocal boundary value problems by combining the piecewise polynomial kernel with polynomial kernel. The method is of high accuracy. Also, the method can avoid reducing the inhomogeneous boundary conditions to homogeneous boundary conditions and constructing reproducing kernel satisfying corresponding homogeneous boundary conditions. Numerical examples are included to demonstrate the validity of the new 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.

Solving a class of linear nonlocal boundary value problems using the reproducing kernel

Recently, the reproducing kernel Hilbert space methods(RKHSM) (see Wang et al (2011) [2]; Lin and Lin (2010) [3]; Wu and Li (2011) [4]; Zhou et al. (2009) [6]; Jiang and Chen (2014) [7]; Wang et al. (2010) [8]; Du and Cui (2008) [9]; Akram et al. (2013) [10]; Lü and Cui (2010) [11]; Wang et al. (2008) [12]; Yao and Lin (2009) [13]; Geng et al. (2014) [14] ; Arqub et al. (2013) [15]) emerged one after the other. But, a lot of difficult work should be done to deal with multi-point boundary value problems(BVPs). Our work is aimed at giving a new reproducing kernel method for multi-point BVPs. We do not put the homogenization conditions into the reproducing kernel space which can avoid to compute the reproducing kernel satisfying boundary conditions and the orthogonal system. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate that new algorithm has the following advantages: small computational work, fast convergence speed and high precision.

A new reproducing kernel method for linear nonlocal boundary value problems

In the previous works, the authors developed the reproducing kernel method (RKM) for nonlocal boundary value problems. A key of the method is the construction of the reproducing kernel (RK) satisfying the homogenous boundary conditions (BCs) of the considered problems. However, it is very difficult to obtain the RK of a reproducing kernel space satisfying nonlocal BCs or nonlinear BCs. Even if the RK is found, its representation is also very complicated compared with the RK without any BCs. In this paper, we will present a new RKM for linear nonlocal boundary value problems. The method can avoid reducing the inhomogeneous BCs to homogeneous BCs and constructing RK satisfying homogeneous nonlocal linear BCs. Numerical examples are provided to show the effectiveness of the new method.

A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method

In this work, we present an algorithm for solving fourth-order multi-point boundary value problems (BVPs) based on the reproducing kernel method (RKM). In previous works, the RKM has been used to solve various two-point BVPs. However, it cannot be used directly to solve multi-point BVPs, since it is very difficult to obtain a reproducing kernel satisfying multi-point boundary conditions. The aim of this work is to fill this gap. A numerical example is given to demonstrate the efficiency of the present method. The results obtained show that the present method is quite reliable for linear fourth-order multi-point BVPs.

A numerical algorithm for solving a class of linear nonlocal boundary value problems

In order to solve a class of linear nonlocal boundary value problems, a new reproducing kernel space satisfying nonlocal conditions is constructed carefully. This makes it easy to solve the problems. Furthermore, the exact solutions of the problems can be expressed in series form. The numerical results demonstrate that the new method is quite accurate and efficient for solving fourth-order nonlocal boundary value problems.

Uniqueness of solutions of linear nonlocal boundary value problems

For n ≥ 3 and 1 ≤ k ≤ n − 1 , conditions are established for which the nonlocal boundary value problem, y ( n ) + ∑ i = 0 n − 1 a i ( x ) y ( i ) = 0 , y ( i − 1 ) ( x j ) = 0 , 1 ≤ i ≤ m j , 1 ≤ j ≤ k , y ( x k + 1 ) − y ( x k + 2 ) = 0 , has only the trivial solution, for all positive integers m 1 , … , m k such that m 1 + ⋯ + m k = n − 1 , and all a < x 1 < ⋯ < x k + 2 < b .