Collocation Methods For High Order Nonlocal Multipoint-Boundary Value Problem

In this paper, we main pay attention to the numerical solution methods forthe n order nonlocal multipiont boundary value problems(NMBVP). The main ideais solving differential equations converted into solving Volterra integral equation ofthe second kind at first, and then to choose a high efficient collocation method forVolterra integral equation of the second kind. As for the nolocal multipoint boundaryvalue conditions(NMBVC), we construct the corresponding node base functions,witch are polynomial of degree n-1such that their function values or correspondingderivative values equal to1at this node, but at the others equal to0. Using thesuperposition principle of the solutions of linear differential equation, the solutionof NMBVP can be consist of linear combination with n+1basic solutions. So wecan get two collocation methods with these theories: indirect collocation methodand direct collocation method. indirect collocation method is to work out the basisfunction with collocation method at first, and then to get linear combination withthe NMBVC. Direct collocation method is to put collocation method for functionsolution and NMBVC together. those collocation methods not only give a highprecise numerical solution, but also present numerical formulas for function and itsderived ones. As a result, we can get the m+1order convergence results(m is thenumber of collocation point in a cell). The numerical experimental results for initialvalue problem and NMBVP demonstrate the efficiency of the present methods.