Research On Numerical Solutions For Direct And Inverse Heat Conduction Problems

Heat conduction equation have played an important role in mathematical physics.It has proved to be a powerful tool for solving diffusion-type problems in many fields.The direct heat conduction problems and the inverse heat conduction problems are interdependent.Based on high precision numerical solutions of direct heat conduction problems,the feasible and effective numerical solutions for inverse heat conduction problems can be given.In this paper,the typical direct heat conduction problems and inverse heat conduction problems numerical solutions method are studied.The main research content as follows:(1)Due to the analytic solutions of direct heat conduction problems are improper integral or multiple integral,so it is difficult to calculate directly.Some high precision numerical solutions method for direct heat conduction problems are given by using Gauss numerical quadrature.(2)For solving the inverse heat conduction problems on the infinite domain,the direct heat conduction problems were solved by the fundamental solution method and Laplace transforms,respectively.Then the Fourier regularization method are introduced to solve the inverse heat conduction problems.(3)In order to solve the transcendental equation in the heat conduction problem,it is transformed into multimodal function optimization problem.Then the global-local mixed evolutionary algorithm is applied to solve optimization problem.It is shown that the algorithm has the advantages of high accuracy and strong practicality.Meanwhile,we have established the first kind Fredholm integral equation model for the inverse heat conduction problem,and the Tikhonov regularization method is applied to obtain the numerical solutions.(4)In the study of the inverse problem for sideways heat equation,the first kind Volterra integral equation is established,and the Tikhonov regularization and the truncated singular value decomposition are applied to obtain the numerical solutions,respectively.(5)In this paper,we study the inverse problem of heat source strength identification for the two-dimensional and three-dimensional heat conduction equations,the direct heat conductionproblems are solved by the eigenfunction expansion method and triple-integral transform,respectively.Then the first kind volterra integral equation of inverse problem was derived,the regularization method for identifying heat source strength is established and its efficiency is demonstrated by some numerical results.