Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using heat polynomials methods

The paper presents a new approximate method of solving one-dimensional inverse Cauchy–Stefan problems. We apply the heat polynomials method (HPM) for solving the one-dimensional inverse Cauchy–Stefan problem, where the initial and boundary data are reconstructed on a fixed boundary. The solution of the problem is presented in the form of linear combination of heat polynomials. We have studied the effects of accuracy and measurement error for different degree of heat polynomials. Due to ill-conditioning of the matrix generated by HPM, optimization techniques are used to obtain regularized solution. Therefore, the sensitivity of the method to the data disturbance has been checked. Theoretical properties of the proposed method, as well as numerical experiments, demonstrate that to reach accurate results, it is quite sufficient to consider only a few of polynomials.