Numerical simulations of 1D inverse heat conduction problems using overdetermined RBF-MLPG method

This paper proposes a numerical method to deal with the one-dimensional inverse heat conduction problem (IHCP). The initial temperature, a condition on an accessible part of the boundary and an additional temperature measurements in time at an arbitrary location in the domain are known, and it is required to determine the temperature and the heat flux on the remaining part of the boundary. Due to the missing boundary condition, the solution of this problem does not depend continuously on the data and therefore its numerical solution requires special care especially when noise is present in the measured data. In the proposed method, the time variable is eliminated by using finite differences approximation. The method uses a weak formulation of the problem to enjoy the stability condition. To avoid the numerical integration on the whole domain, the weak form equations are constructed on local subdomains. The approximate solution is assumed to be a linear combination of Multi Quadric (MQ) radial basis function (RBF) constructed on nodal points in the domain and on the boundary. Since the problem is known to be ill-posed, Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system.