This paper investigates the analytical approximate solutions of singular fourth order four-point boundary value problems using reproducing kernel method (RKM). The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve singular fourth order four-point boundary value p...

The Difference Potential Method (DPM) [1] proved to be a very efficient tool for solving boundary value problems (BVPs) in complex regions. It allows BVPs to be reduced to a boundary equation without the knowledge of Green’s functions. The method has been successfully used for solving very different problems related to the solution of partial differential equations [1]. However, it has mostly b...

This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green's functions are approximated by structured grid-based numerical solutions, which avoids the...

The aim of this paper is to apply Galerkin weighted residual method for solving eleventh-order BVPs. In this method, we exploit Bernstein and Legendre polynomials as basis functions which are modified into to a new set of basis functions to satisfy the corresponding homogeneous form of boundary conditions where the essential types of boundary conditions are mentioned. The method is formulated a...

Shape gradients of PDE constrained shape functionals can be stated in two equivalent ways. Both rely on the solutions of two boundary value problems (BVPs), but one involves integrating their traces on the boundary of the domain, while the other evaluates integrals in the volume. Usually, the two BVPs can only be solved approximately, for instance, by finite element methods. However, when used ...

Solving boundary value problems BVPs for the fourth-order differential equations by the reduction of them to BVPs for the second-order equations with the aim to use the achievements for the latter ones attracts attention from many researchers. In this paper, using the technique developed by ourselves in recent works, we construct iterative method for the second BVP for biharmonic-type equation,...

This paper investigates discrete boundary value problems (BVPs) involving secondorder difference equations and two-point boundary conditions. General theorems guaranteeing the existence and uniqueness of solutions to the discrete BVP are established. The methods involve a sufficient growth condition to yield an a priori bound on solutions to a certain family of discrete BVPs. The a priori bound...

A new one-step hybrid block method with two-point third derivatives is developed to solve the second-order boundary value problems (BVPs). The mathematical derivation of proposed based on interpolation and collocation methods. theoretical properties method, such as consistency convergence, are well analysed. Some BVPs different conditions solved demonstrate efficiency feasibility suggested meth...

In this article, we study five non-homogeneous multi-point boundary-value problems (BVPs) of second order differential equations with the onedimensional p-Laplacian. These problems have a common equation (in different function domains) and different boundary conditions. We find conditions that guarantee the existence of at least three positive solutions. The results obtained generalize several ...

This paper investigates discrete boundary value problems (BVPs) involving secondorder difference equations and two-point boundary conditions. General theorems guaranteeing the existence and uniqueness of solutions to the discrete BVP are established. The methods involve a sufficient growth condition to yield an a priori bound on solutions to a certain family of discrete BVPs. The a priori bound...