A Stable Collocation Method For Solving Multi-point Boundary Value Problems

The multi-point boundary value problems of differential equations are widely used in engineering,but the multi-point boundary value problems of differential equations are generally difficult to obtain exact solutions.So it is very important to find a more stable and more valid method for obtaining numerical solutions of multi-point boundary value problems of differential equations.In this paper,a new stable collocation method based on a modification of Legendre multiwavelets is proposed to solve the multi-point boundary value problems.Firstly,the fractional derivative is employed to define a bounded linear operator,and the discrete Legendre multiwavelets of fourth order convergence in the square integrable space are mapped into the continuous Legendre multiwavelets of the same convergence in the reproducing kernel space.Secondly,the nonlinear equations are linearized by Newton iteration method.The expressions of numerical solutions for higher order linear and nonlinear multi-point boundary value problems are given by using the least square method.Simultaneously,we prove the stability of the method proposed in this paper by the fact that the condition number of coefficient matrix of normal equations is bounded.Finally,several numerical examples are compared with the results of the methods in the literature,such as the optimization homotopy method,Sinc-collocation method and some numerical solutions based on the reproducing kernel method,which illustrate the validity,stability and applicability of the method developed here.