Two regularization methods to identify time-dependent heat source through an internal measurement of temperature

This paper deals with an inverse problem for identifying an unknown time-dependent heat source in a one-dimensional heat equation, with the aid of an extra measurement of temperature at an internal point. Since this problem is ill-posed, two regularization solutions are obtained by employing a Fourier truncation regularization and a Quasi-reversibility regularization. Furthermore, the Hِlder type stability estimate between the regularization solutions and the exact solution, are obtained, respectively. Numerical examples show that these regularization methods are effective and stable.