Uncertain Wave Equation

Wave equation is one of the most typical hyperbolic partial differential equation,and it is always used to model the phenomena of vibration of elastic body and wave propagation.In this equation,the density function of external force is a determinate function.However,disturbances or noises exist almost everywhere in real life.Obviously,external force is disturbed by noises and not a determinate function.Hence,wave equation is not enough to describe the wave propagations.Then we should introduce a tool to the wave equation to deal with the noise.Before twentieth century,Wiener process is often used to model this dynamic noise.Following that,classical wave equation is transformed into stochastic wave equation.Inspired by stochastic wave equation,we introduce Liu process which is a special type of uncertain process to wave equation.Then we derive a new type of partial differential equation by analyzing vibrating string,which is called uncertain wave equation.Following that,this paper studies the Cauchy initial problem and the infinite half-boundary problem of an uncertain wave equation,obtains the solutions of corresponding problems,proves that the solutions possess the property of uniqueness,and finally gives the inverse uncertainty distribution of the solutions of corresponding problems.Furthermore,a general uncertain wave equation is discussed for solving vibrating string in a more complex case.For a general uncertain wave equation,the solution is obtained,and its existence and uniqueness are proved.Also,the inverse uncertainty distribution is investigated concerning the solution of a general uncertain wave equation.Then we give the definition of stability and conditions for an uncertain wave equation being stable.However,whether there is a solution is always the focus of attention.If an uncertain wave equation has no analytic solution,then numerical solution becomes more important.For obtaining numerical solution,we define a concept of ?-path and prove a fundamental formula by using ?-path which can make an uncertain wave equation be a class of uncertain wave equations.On the basis of that formula,we get the uncertainty and inverse uncertainty distributions of the solution.In addition,we get the expected value for a solution of a general uncertain wave equation.Summarily speaking,the main contributions of this thesis can be expressed as follows:? It derives uncertain wave equation by analyzing vibrating string,discusses the Cauchy initial value and infinite half-boundary problems of an uncertain wave equation,gives corresponding solutions of the couple problems and provides their inverse uncertainty distributions.? It proves the existence and uniqueness theorem of a general uncertain wave equation under Lipschitz and linear growth increasing conditions.? It defines stability and give conditions of an uncertain wave eqaution being stable.? It puts forward a concept of ?-path,and derives a fundamental formula for changing an uncertain wave equation into a class of wave equations;Based on the for-mula and ?-path,it obtains the inverse uncertainty distribution of solution,and gets the expected value for the solution.