Duhamel integral form for the interface heat flux between bubble and liquid

The solution of heat equation inside oscillating gas bubble with moving boundary was obtained by Fourier’s method. The integral formula for interface heat flux, containing theta-function in the integrand was derived. The kernel of the integral is represented by a series of exponential functions, and a simple analytic approximation obtained earlier is used for it with high accuracy. The asymptotic expression for the interface heat flux in the Duhamel integral form with rooted kernel was derived.The vapor bubbles were also considered. In this case the major problem is external heat problem in liquid. It is shown that the asymptotic expression for the heat flux at the interface in the case of gas bubbles has the similar structure as the heat flux from the vapor bubble surface to the liquid. In both cases it is Duhamel integral with rooted kernel.