The Theoretical Research Of Several Kinds Of Difference Equations And Their Application In Economic Control

As an important component of mathematical research,the theories and application of difference equation have been given a lot of attention in recent years.In this paper,two important branches of difference equation are studied,they are rational difference equation and fractional difference equation.Compared with fractional difference equation,the research history of rational difference equation is longer.In this paper,in the part of rational difference equation,the local stability of the equilibrium solution of Class I nonlinear rational difference equation as well as the asymptotic stability of Class I linear rational difference equation were studied.In the part of fractional difference equation,the existence and uniqueness of the solution of Class III fractional difference equation was studied,and the method to solve the equation was investigated.The main content of this paper consists of three parts as follows:In the first part,the stability of the solution of Class II rational difference equation was studied.First,the original rational difference equation was converted into balance equation and the equilibrium solution was calculated.According to the value range of the equilibrium solution(mainly the size relation between the absolute value of the equilibrium solution and 1),the asymptotic stability of the equilibrium solution was preliminarily judged.Secondly,the partial derivative of the difference equation at the equilibrium solution was calculated,and the linearized equation of the original rational difference equation was obtained.According to the linearized equation,characteristic equation and then characteristic root were obtained.Based on the existing relevant conclusions,according to the value range of the characteristic root and its relationship with the original difference equation,the theory properties of the rational difference equation were gotten.Finally,programming was carried out in the MATLAB environment,and the correctness of the conclusion was verified in aspect of numerical values.In the second part,the existence and uniqueness of the solution of Riemann-Liouvile fractional difference equation,Caputo fractional difference equation and sequential fractional difference equation was mainly studied.Based on the existing research theories of fractional difference equation,with forward difference as the starting point,referring to existing research methods and relevant conclusions of backward difference,using Z-transformation theory,some fractional Z transformation formulas suitable for backward difference and the method to seek inverse Z transform were derived.Then,using the convergence of two special functions,the existence and uniqueness of the solution of Class III fractional difference equation was studied,and the method to seek the solution of fractional difference equation was investigated.In the third part,the difference equation theory was used to study an application example in economic control.The regional macro-economic model has been built and proposed by many famous economists,but few of the economists have introduced monetary policies into the comprehensive model of economic growth.By introducing monetary policies into the macro-economic model,Smith's economic growth and monetary policy comprehensive model can be obtained,which was studied in this part mainly using the difference equation system theory and from the mathematical point of view.Finally,numerical examples were given to verify the model.In this paper,A class of second order rational difference equations and a class of third order rational difference equation were studied in the part of rational difference equation,respectively.The general method for studying the stability of the solution of the rational difference equation was given,in which the constant was replacing with the letter and educed the general conclusion.Backward difference can predict the current state according to the desired target state,and forward difference can predict the future target state from the current state.However,there have been a series of monographs with the theory of the fractional differential equation of the backward difference as the starting point,and the theory of fractional difference equation with the starting point of the forward difference is not found.In this paper,the uniqueness of the solutions of several fractional difference equations with forward difference as the starting point were obtained,and the explicit solutions of the fractional order difference equations were induced.A typical example was given in the economic control part,and the general method for the stability of the judgment model was derived.