Several Methods Research Into Solving The Inverse Problem Of Heat Conduction Equation

Heat conduction equation has attracted people's attention since it was widely applied to daily life. It relates to space science and technology, water conservancy, geological exploration and other areas. Particularly, the inverse problem of heat conduction equation has become a point of people's attention. There are many methods to solve the inverse problem of heat conduction, such as the Adomian decomposition method, homotopy perturbation method, variational iteration method, finite-difference method and other methods. But each method is restricted to a certain extent.In this paper, we proposed several methods to solve the inverse problem of heat conduction equation. Firstly, we solved the equation which is simplified by the canonical representation formulation with the thought of finite difference method. However, the canonical representation formulation method only applies to well-posed problem.Secondly, the type of heat conduction equation is ill-posed. Therefore, we used the Tikhonov regularization method modified the canonical representation formulation method. The key is introduced Tikhonov regularization term when solving integral equation of formulation, and the ill-posed problem is transferred to well-posed problem. Furthermore, we used regularized-Gauss-Newton method for solving equation, and numerical result show that the method is stable to a certain extent.At last, we solved the equation which is simplified by the trace type functional formulation with the thought of finite difference method. At the same time, numerical results show that the method can be solved well with accurate data. When we add small noise disturbance to the inverse problem, the method still has better performance suppressing noise which is different to the fist method.Consider all, this paper puts forward several methods to solve inverse problem of heat conduction equation, when canonical representation formulation method and TTF formulation method invert the accurate data, those methods calculate quickly and we get well simulation result. The Modified of the canonical representation formulation method and TTF formulation method is stable to a certain extent. And we still get well simulation result when add the perturbation data to inverse problem.