Exact solutions for heat conduction in non-uniform mediums with general time-dependent boundary conditions

A new analytic solution method is developed, without integral transform, to find the exact solutions for the transient heat conduction in non-uniform mediums with general time-dependent boundary conditions. By introducing suitable shifting functions, the governing second-order differential equation with variable coefficients and time-dependent boundary conditions is transformed into a differential equation with homogenous boundary conditions. If the physic properties of the medium are in polynomial forms, the exact solution of the system can be developed. Finally, examples are given to illustrate the analysis. Limiting cases are studied and compared with those in the existing literature. The influence of physic parameters on the temperature distribution of the system is revealed.