Direct and inverse solutions of the two-dimensional hyperbolic heat conduction problems

A sequential method is proposed to estimate boundary condition of the two-dimensional hyperbolic heat conduction problems. An inverse solution is deduced from a finite difference method, the concept of the future time and a modified Newton–Raphson method. The undetermined boundary condition at each time step is denoted as an unknown variable in a set of non-linear equations, which are formulated from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. The example problem is used to demonstrate the characteristics of the proposed method. In the example, a well-known problem is used to demonstrate the validity of the proposed direct method and then the inverse solutions are evaluated. In the second example, the larger value of the relaxation time is implemented in the direct solutions and the inverse solutions. The close agreement between the exact values and the estimated results is made to confirm the validity and accuracy of the proposed method. The results show that the proposed method is an accurate and stable method to determine the boundary conditions in the two-dimensional inverse hyperbolic heat conduction problems.