Error estimates in interpolation methods for solving boundary value problems for ordinary differential equations

We estimate the error of the subdomain method, the multiple coincidence method, and the collocation method for two-point boundary value problems.     

Newton-like methods for two-point boundary value problems

A class of Newton-like methods for discrete two-point boundary value problems is constructed from the sum equation formulation of the problem. Each step of the Newton-like method can be described as first solving a system of linear algebraic equations. The solution vector of this system gives boundary values to a number of discrete boundary value problems which can be solved explicitly.     

Galerkin-wavelet methods for two-point boundary value problems

Summary Anti-derivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to two-point boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems. Numerical examples are given.This work was supported by National Science Foundation     

Finite difference methods for two-point boundary value problems involving high order differential equations

We discuss the construction of finite difference schemes for the two-point nonlinear boundary value problem:y ⁽²ⁿ⁾+f(x,y)=0,y ^(2j)(a)=A _2j ,y ^(2j)(b)=B _2j ,j=0(1)n–1,n2. In the case of linear differential equations, these finite difference schemes lead to (2n+1)-diagonal linear systems. We consider in detail methods of orders two, four and six for two-point boundary value problems involving a fourth order differential equation; convergence of these methods is established and illustrated by numerical examples.     

Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales

In this paper, by using critical point theory, we establish the existence of weak solutions of the two-point boundary value problem for a second-order dynamic equation on an arbitrary time scale T$\mathbb{T}$, so that the well known case of differential dynamic systems (T=R)$(\mathbb{T}=\mathbb{R})$ and the recently developed case of discrete dynamic systems (T=Z)$(\mathbb{T}=\mathbb{Z})$ are unified. To the best of our knowledge, this is the first time that boundary value problems of dynamic equations on time scales have been dealt with by using critical point theory.     

On the solvability of two-point,second-order boundary value problems

We gain solvability of a system of nonlinear, second-order ordinary differential equations subject to a range of boundary conditions. The ideas involve differential inequalities and fixed point methods. In particular, maximum principles are not employed.     

Solutions of two-point boundary value problems for even-order differential equations

The existence of solutions of the two-point boundary value problems consisting of the even-order differential equations

     

Multiderivative methods for non-linear second-order boundary value problems

Second-, fourth- and sixth-order methods are developed and analysed for the numerical solution of non-linear second order boundary value problems.The methods arise from a two-step recurrence relation involving exponential terms, these being replaced by Padé approximants.The methods are tested on two problems from the literature.     

Periodic boundary value problems for second-order impulsive integro-differential equations

By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for second-order nonlinear impulsive integro-differential equations of mixed type.     

Nonlocal boundary value problems for second-order functional differential equations

We study the existence of a solution to a nonlocal boundary value problem for a class of second-order functional differential equations with piecewise constant arguments. The equation studied includes terms depending on the derivative as well as delay arguments in the derivative.     

Positive solutions of two-point boundary value problems for systems of nonlinear second-order singular and impulsive differential equations

This paper is devoted to the study of multiple and single positive solutions of two-point boundary value problems for nonlinear second-order singular and impulsive differential systems. By constructing a cone _K1×_K2$K_1\times K_2$, which is the Cartesian product of two cones in the space C[0,1]$C[0,1]$, and computing the fixed-point index in the _K1×_K2$K_1\times K_2$, we establish the conditions for the existence of at least one or at least two positive solutions to two-point boundary value problems for systems of nonlinear second-order singular and impulsive differential equations.     

Solutions of periodic boundary value problems for second-order nonlinear differential equations via variational methods

By introducing a variational framework for a class of second order nonlinear differential equations with non-separated periodic boundary value conditions, some results on the existence of non-trivial, positive and negative solutions of the problems are obtained. Some results by Atici-Guseinov, Graef-Kong, etc. obtained by topological degree methods are extended. The resonant case of the problems where the nonlinearities are unbounded and satisfy Ahmad-Lazer-Paul type conditions is also considered.     

The convergence of shooting methods for singular boundary value problems

We investigate the convergence properties of single and multiple shooting when applied to singular boundary value problems. Particular attention is paid to the well-posedness of the process. It is shown that boundary value problems can be solved efficiently when a high order integrator for the associated singular initial value problems is available. Moreover, convergence results for a perturbed Newton iteration are discussed.

     

Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities

We consider boundary value problems for nonlinear second order differential equations of the form