Numerical Solutions Of Linear Boundary Value Problems By Multi-step Difference Methods

As we known, there are many subjects are involved in the field of differential equations to the boundary value problems, the method for solution ordinary differential equations has been the concern of scholars .However, ordinary differential equations, only some typical equation can be obtained primary solution (with the primary function of the solution), most of the demand equation is not the primary solution. In addition, although the primary solution can be, but because of the form is too complex not convenient to use. So the numerical solution of differential equations has important practical significance.The most effective method for solution boundary value problems of ordinary differential equations is finite difference method. Classic finite difference method is using difference quotient to replace derivative (numerical differential) or integral interpolation (numerical integration) approach to structural difference scheme. In order to structure with higher-order local truncated error of the difference scheme, many scholars use spline function or parameters spline function to approximate unknown function, construct some kind of difference scheme. However, the calculation of higher-order numerical differential formula and the high-order derivative of the high-degree spline function is more difficult, and it will caused a very large amount of calculation to structural difference scheme, and the accuracy of some method is not high, so these methods are not well adapted to the higher-order differential equationsThis paper based on the idea of [8], by the difference method, for the different differential equations of 4,5,6 order ,will does a further discussion. The appropriate format of the 6-order differential equations of boundary conditions and the estimate of the local truncated error are given. In addition, some examples are employed to evaluate the performance of the method. By comparing the results of other methods ,we find the numerical results show that it is superior to some methods presented by other papers.