The Optimization Method Research On The Grey Verhulst Model And Non-equidistant Power Model

The grey system theory was put forward by Professor Deng in 1982,the theory is mainly used to describe and deal with the "small sample,poor information" uncertainty problem in the objective world.Among them,the grey GM prediction model is the most important part of the application field,and has produced many valuable research results,it has been widely used in many fields such as petroleum,ecology,architecture,information,management and so on.It helps solving so many difficulties in production,life and engineering technology.Although the grey GM model has been applied in many fields such as natural sciences and social sciences,and has achieved relatively considerable research results,there are still many deficiencies in theory,and it needs to be developed and perfected.Therefore,many experts and scholars did some related theoretical researches in the aspects of time response,background value,grey derivative,parameter estimation and grey function.On the basis of the previous scholars,we mainly do two researches,the first one is studying the optimization method of the background value in the grey Verhulst model,and the second one is the solving method of the power exponent (?) in the non-equidistant power model.The main researches are as follows:(1)Studying the grey Verhulst model and its optimization method.Firstly,we analyzed the error source of the background value in the model.Secondly,we transformed the whitening equation of the model into a reciprocal transformation.Let the,get a new whitening equation,and the two sides of the new equation are integrated on the interval [k-1,k].The corresponding new grey differential equation is,and then combined with Mu Y to find the combination of coefficients to optimize the background value,to explore the appropriate combination of coefficient a so that,thus realizing the optimization of its background value.Finally,the optimization method of the background value is used to simulate the population data of China from 2007 to2011.The results show that the optimization method is better than the traditional method and improves the modeling accuracy of the model.(2)Studying the solution of the power exponent (?) in the non-equidistant power model.Firstly,we gave the relative definition of non-equidistant power model according to the definition of non-equidistant given by Wang Z X.Secondly,the twosides of the model whitening equation are divided by(1)(x(t))g,at the same time the derivation of t.Then the formula of the power exponent (?) in the model is deduced by using the difference quotient approximation instead of the derivative method.After determining the value of g according to this formula,the parameters of a and b are estimated by least squares method.Finally,the method is used to simulate the fatigue strength of titanium alloy.The result proves the practicability of this method.