Iteration Methods For Solving Two Types Of Inverse Heat Conduction Problems

Inverse problem is a very active research branch in recent decades which originated in mathematical physics problems,also called the inverse problem of mathematical physics,it has very important applications in geophysics,materials science,finance,industrial control,life science,pattern recognition,geology and environmental science,image processing,signal information and control and many other fields,so it has become a research focus by many math and science and technology workers at home and abroad.Because most inverse problems are ill-posed and cannot be solved analytically,numerical solutions play a key role in the study of inverse problems.With the development of calculation methods,numerical solution methods of inverse problems are becoming more and more perfect.Solving inverse problem refers to the determination of one or more unknowns in a mathematical physics problem using additional information.For heat conduction problems,these unknowns may be the material properties or thermal properties of heat-conducting materials,heat source terms,boundary size and shape,boundary heat flow and so on.Mathematically,inverse problems can be roughly classified into two categories,one of which is called function identification problem,which determines an unknown definite solution condition(initial value or boundary)in the problem,the unknown condition is usually a function of some variable,another class is called parameter identification problem,which determines some unknown coefficients in equations.In this paper,two kinds of inverse problems in heat conduction equations,that is,function identification problem of determining unknown boundary conditions and parameter identification problem of determining unknown coefficients in heat conduction,are studied,Firstly,this paper presents the inverse problems of unknown boundary temperature and boundary heat flow in one-dimensional heat conduction equation.We introduce the variational iteration method and analyze the existence of the solution of the two types of inverse problem.In the case of using only the initial conditions,using the variational iteration method,we find the convergence iterative sequences of solution,which convergence to the exact solution of this problem(if there is a exact solution),and this solution meets the other given conditions.This iterative approach is not need for variables discretization,so calculation process will not appear the discretization error,Finally,several typical data models are given to verify the effectiveness of this iterative method.Secondly,this paper studies the inverse problem of one-dimensional heat conduction equation with unknown coefficients.We first introduce the new iterative method.In the case of the existence of the solution of such inverse problem,first of all,using function transform techniques and the given additional information,the inverse problem can be converted to a heat conduction equation which has no unknown coefficients,then using new iteration method to solve this direct problem,so the solution of the original inverse problem and the unknown coefficients will be determined by another function transformation.This iterative method also doesn't need for variables discretization,so it will not appear discretization error in calculation process,Moreover,this iterative method has a fast convergence speed and high accuracy.Several typical data models illustrate the effectiveness of this method.Finally,by analyzing the results of data models,this paper finds out the similarities and differences between the two iterative methods,and obtains some conclusions.