The Application And Research Of Multiple Finite Variable Limit Integral Method

In the field of practical engineering,partial differential equations and their theory play an important role,but it is difficult to obtain the analytical solution of most partial differential equations.In practical applications,it is particularly important to use the approximate substitution of numerical solutions.A new numerical method for solving partial differential equations is proposed in this paper-the multiple finite variable limit integral method.In addition,a number of specific ways are given to construct numerical schemes with this method,it includes the different ways which are combined with the fitting methods such as Lagrange interpolation function,Taylor's expansion and Taylor formula.Compared with other numerical methods,the multiple finite variable integration method has the advantages of controllable accuracy,clear physical meaning,a scheme that can be constructed with conservation,scheme diversification.In addition,the process of constructing the numerical scheme is clear,it is an ideal numerical method to construct the numerical scheme according to the given accuracy.However,this method is still in the primary stage of research,and there are many aspects to be further studied.In this paper,the following innovative studies are made for the multiple finite variable limit integral method:First,a new method of function fitting-the Taylor's formula,is proposed in this paper.The advantage of this method is to analyze the error accuracy of the discrete format,and to construct a discrete scheme with any given n order accuracy.Secondly,in the process of structural format,each item of partial differential equation should be multiple integral,the calculation of integral will bring about a certain amount of work.In this paper,a simple formula for calculating multiple integrals of 7 times integral and 15 times integral is given for the case of partial differential equation containing three derivative and four derivative of space,which provides convenience for the construction of format.Then,the Taylor's formula method is applied to approximate the Sobolev type equations which are widely applied in engineering.This is a new attempt to construct the discrete scheme of Sobolev type equations.In this paper,the specific discrete process of the numerical discrete scheme for Sobolev type equations is given,and the existence and uniqueness of the solution of the numerical scheme are proved.Finally,some numerical examples are given,and the numerical experiments are carried out in five practical problems.Numerical experiments show that the multiple finite variable integration method is effective for discrete partial differential equations,and the numerical scheme can achieve the design accuracy.Finally,the Taylor formula method is given as the approximate integral value for calculating the multiple integral.The exact calculation of multiple integral is also studied in this paper.The exact weighted function integral formula is obtained for the three integral with two derivative term.