Numerical Algorithm Study On Two Kinds Of Backward Heat Conduction Problem

In this paper,two kinds of numerical algorithms for inverse heat conduction problems are discussed.The inverse heat conduction problems also have very important applications in practical engineering fields.Through relevant theoretical analysis,we conduct numerical simu-lation with the direct solving method and the Gradient iterative method.The numerical results are carried out to confirm the feasible and effective of the aforementioned method.This article is mainly divided into the following four chapters:The first chapter is the introduction part,which briefly describes some research back-grounds and valuable domestic and international research status of the inverse problems of partial differential equations,especially the research on the inverse problems of parabolic equa-tions.The second chapter mainly considers the inverse problem of reconstructing heat flux using the interior temperature measurement data.Problems of this type have been widely used in the field of engineering applications and characteristics of rail transit.The previous method mostly based on iterative algorithm,the iterative process is complex and convergence speed is very slow.Based on the exact analysis of forward problem,we presents a direct solving method.Firstly,we solve analytic solution(series solution)of the forward problem with the method of variable separation.Secondly,numerical solution is obtained with the implicit diference scheme,and compare differences between analytic solution and numerical solution.Then,by the expression of the analytic solution of forward problem,we construct the relationship ex-pression of between surface heat flux q(t)and the interior temperature measurement p(t),and the original problem is transformed into linear Volterra integral equation of the second kind and the uniqueness of the solution of integral equation is proved.Finally,the numerical integral method is used to discretize equation and the calculation formulation of integral equation is given.The numerical results are carried out to confirm the feasible and effective of the afore-mentioned method.The third chapter investigates an inverse problem of using the additional conditions to re-construct the first-order coefficient of the second order parabolic equation.Based on the optimal control framework,the existence of the minimize of the cost functional is proved and the nec-essary condition which must be satisfied by the minimize is deduced.We construct the Gradient iterative method,and iterative format and stopping criteria are given.The numerical results show that the algorithm designed in this paper is stable and that the coefficient is recovered very well.The four chapter is summary and outlook.Subsequent work mainly considers from two aspects:on the one hand,the research object of this paper is one-dimensional,and we will consider the two-dimensional and three-dimensional,high-dimensional situation;on the other hand,we will discuss the non-linear case and reconstruct the unknown coefficient with other algorithms.