Existence And Uniqueness Of The Leakage Model With General Loss Functions

So far,the detection of the position,the function and the attenuation coefficient of the leaks in the pipeline net of gas(oil)is the most popular problem in the research of engineering project and theory of preservation and transportation.Especially,the position of leaks' ascertain is also a very interesting question for discussion.And at the same time it is a urgent problem in practice of the manufacture of the oil field.Many engineer and natural science workers in internal and overseas pay close attention to this difficult problem,and do their best to resolve it.During recent twenty to thirty years,there has been a great many of achievement,as well models and methods,which focus to find convenient and accurate way of detecting the position and function of the leaks in the pipeline net of gas(oil)and other relative quantity.One kind of the models which are being researched are the simple models set up on the base of stable condition,they have a larger error in detection,so they can not suffice to engineering.The other kind of the models are the nonlinear partial differential equation groups which set up from move equation,sequential equation,and condition equation.We used all kinds of numerical calculation means to seek the position and function of the leaks in pipeline net,but the result is not enough accurately,and the effects are not satisfactory.Many large-size petroleum establishments have developed some examination software,they keep their technology secret and it's very expensive.Therefore,it inspire strongly the confidence of countrymen to research the detection of the leaks in the pipeline net.On the basis of the former paper,this paper has described a nonlinear partial differential equation group of the move in the pipeline net of gas(oil)on this base,use a method to make it linear,and set up a linear partial differential equation model to describe the single-leak in the pipeline net of gas(oil)in the examine stage for the first time.And then use the theory of inverse problem of partial deferential equation,select some feasible means of measurement and data, added some additional conditions reasonably,at last compose a complete model of detection of leaks.This model's engineering concept is clear,the direct problem's analysis solution is easy to express,so it's very easy to construct the math-express formula which contain the parameter of the position and intensity of leaks.These express formulas are easy and convenient,especially easy to discuss the quality of the equation.For this reason,we discuss following questions on this base;First,from the sequential equation,move equation,and condition equation we give a nonlinear partial differential equation group which describes the move of the gas in pipeline net of gas(oil).Use KA linear method,predigesting it into a linear heat conduct equation.Then according to the characteristic of the gas(oil)in examine stage,we set up a linear partial differential equation model which describes the single-leak in the pipeline net of gas(oil).The model is a popularize model of ZhengQin,Yuan Darning,and Zhou xueshen',it is more directive and common.Second,we use Laplace integral transform to gain the analysis solution of the direct problem of linear partial differential equation model which describes the single-leak in the pipeline net of gas(oil)in examine stage.According to the characteristic of the move of gas(oil)in examine stage; we make two additional conditions to detect the position and function of the leaks.After then we obtain the math-express formula which contains the position l and function f(t)of the leaks.Third,we give and prove the theorem of the existence and uniqueness of the position and function of the leaks.Finally,according to the given model,we use two different additional conditions and many means to put up numerical calculation.The result indicate that it is tallies with the known data.In this paper,we have some new method.Firstly,we instead the leak form f = const,e^-kt(k = const),fe^-kt(f,k = const)for f(t),which is a limited and descending function,it is more common and more close to project.secondly,Using the transform of Laplace, we prove the theorem of the existence and uniqueness of the position l,function f(t)and it's foregone function of the leaks fast.And then elicit the existence and uniqueness of the position and function of the leaks,the method is novel and briefness.Lastly,we use two different additional conditions and many means to put up numerical calculation,according to the result of numerical calculation,it is tally and it owes the value of application and theory.Even though,the model that we have set up is also has some inadequacy,there are several difference between some supposing condition in the theory of the existence and uniqueness of the position and intensity of the leaks and the practical engineering problem.The analyses of the error which restrict and influence the position and intensity of the leaks are not enough,etc.we will make great effort continually to overcome the hard work of detecting the position and intensity of the leaks in the pipeline net of gas(oil).