Uncertain Heat Conduction Equation

In the classical partial differential equation theory,heat conduction equation plays an important role,whose heat source is a determinate function.However,heat source is often affected by the interference of noise in practice because of environment and the material itself.It is usually to use Wiener process to deal with the noise.Then heat conduction equation turns into stochastic conduction heat equation.But,it is reasonable to describe real heat conduction process via stochastic heat equation?This thesis gives a paradox of stochastic conduction heat equation,then uses Liu process to describe the noise of heat source and derives an uncertain heat conduction equation.This is a main problem of this thesis.Besides,this thesis also obtains the solution and the inverse uncertainty distribution of solution for a special uncertain conduction heat equation.For general uncertain heat conduction equation,an existence and uniqueness theorem and a stability theorem are proved.However,it is difficult to find analytic solutions of general uncertain heat conduction equations.This thesis shows that the solution of an uncertain heat conduction equation can be represented by a spectrum of partial differential equations,which supplies a possibility to solve the general uncertain heat conduction equations numerically.On the whole,the contributions of this thesis are:Defining a concept of uncertain partial differential equation and deriving an un-certain heat conduction equation;Obtaining the analytic solution and inverse uncertainty distribution of solution for a special uncertain heat conduction equation;Proving an existence and uniqueness theorem for general uncertain heat conduc-tion equation under linear growth condition and Lipschitz continuous condition;Defining the concept of stability for uncertain heat conduction equation,and proving a stability theorem under strong Lipschitz condition;Defining the ?-path of uncertain heat conduction equation,and showing that the solution of an uncertain heat conduction equation can be represented by a spectrum of partial differential equations;Getting the formulas of inverse uncertainty distribution,expected value and ex-treme value of solution,and designing some numerical methods to gain the in-verse uncertainty distribution,expected value and extreme value of solution.