Summary: We consider an inverse heat conduction problem (IHCP) in the quarter plane, where data are given at \(x = 1\). The problem is called a sideways parabolic equation and is severely ill-posed. Numerical methods such as Tikhonov and Fourier regularization methods have been developed. However, they contain an a priori bound of the solution in their parameter choice. A large estimate bound may cause bad numerical results. In this paper, we introduce a new class of iteration methods to solve the IHCP and prove that our methods are of order optimal under both a priori and a posteriori stopping rules. An appropriate selection of a parameter in the iteration scheme helps to reduce the iterative steps and get a satisfactory approximate solution. Furthermore, if we use the discrepancy principle, we can avoid the selection of the a priori bound.