Analytic closed solution for the heat conduction with time dependent heat convection coefficient at one boundary

A new solution method is proposed to develop the analytic closed form solution for the one dimensional heat conduction with one mixed type boundary condition and general time dependent heat convection coefficient for the first time. The solution method is the combination of an extension of the shifting function method developed by Lee and his colleagues and a series expansion. It is shown that the solution is simple and accurate. The convergence of the present analysis is very fast. One can find that when the dimensionless Fourier number is greater than 0.2, the error for the one term approximation solution of the infinite serious solution can be less than 2%. Examples are given to reveal the solution method. Numerical results are compared with those in the existing literature.