Abstract: The governing equation of moving boundary problem is heat conductive equation. Its definite solution domain varies with time. Highly precision numerical algorithms on space-time domain were presented to solve 1+1 dimensional moving boundary problems. An initial moving boundary was given to form an irregular physical domain, and a regular region (a rectangular in Cartesian coordinate system) was chosen to cover the irregular physical domain. The heat equation was numerically computed with a barycentric interpolation collocation method (BICM) on space-time regular region with fixed and moving boundary conditions and initial condition to obtain numerical data in regular region. The data on moving boundary of physical domain were computed with barycentric interpolation. Then, the governing equation of moving boundary was solved with BICM to recover a new moving boundary. Repeat the process, numerical data of temperature and final moving boundary position were given. Numerical examples illustrate effectiveness and accuracy of the method.