Research On Several Specific Problems In Variable Limit Integral Method

Partial differential equations play an important role in many engineering applications.However,since the exact solutions of most partial differential equations are difficult to solve,the numerical solution of partial differential equations is particularly important.In this paper,we study several theory problems about the variable limit integral method,which is a relatively new numerical method to solve PDE.Some papers in recent years show that the variable limit integral method can be well applied to solve the numerical solution of multiple types of differential equations.However,the research of the method is still in the primary stage,and it needs to be improved and perfected.In this paper,the following innovative researches are present for the variable limit integral method:First by using the variable-limit integral method,we propose a numerical scheme for Laplace equation,which is more convenient for parallel operation according to the conclusion given in Lawrie Sameh algorithm.In order to get parallel numerical scheme,the function fitting is carried out in the discrete process with the help of the characteristics of the differential equation itself.This method can reduce the number of points and layer used in the discrete process,thus reducing the bandwidth of the coefficient matrix in the numerical scheme.Finally,we give the numerical scheme of the Laplace equation.It is worth mentioning that this construction method can be easily extended to construct parallel numerical scheme for other types of equations.Secondly,we improve the integral process of the method.In this paper,for the condition that the highest order of spatial derivative term in partial differential equation is the second order,the third order and the fourth order,we transform the third order,the seventh order and the fifteenth order of the corresponding multiple integrals into single integral with weight function,respectively.Then,the calculation result of single integral with weight function is verified to be equal to that of multiple integral for the case,which highest order of spatial derivative term is second order.That means the single integral with weight function is valid.Similarly,following the derivation steps in this paper,we can get the formula of transforming the multiple integrals into the single integral with weight function,where the spatial derivative term of the highest order is of any order.