Integral Form of the Heat Transfer Equation With Arbitrarily Moving Boundary and Arbitrary Heat Source

For the first time, an integral form of one-dimensional heat transfer equation in a semi-infinite domain with a boundary, moving arbitrarily in time, and a heat source, depending arbitrarily on time and space location, is obtained. The obtained integral equation relates time histories of the temperature and its gradient at the boundary of the domain with the temperature at any given point inside or at the boundary of the domain. In the latter case, it delivers closed form integral equation for the rate of boundary movement in nonlinear problems where the time history of boundary movement is one of problem unknowns. The obtained equation accounts explicitly for the presence of an arbitrary heat source in the domain, while other existing methods do not allow a closed integral formulation to be obtained in such a case. The equation may be used for an analytical investigation of several types of boundary value problems (BVPs), as well as for numerical solution of such problems. Particular cases of this equation with a trivial heat source are known to demonstrate chaotic behavior. It is expected that the same is true for some nontrivial heat source functions, and this conjecture will be explored in subsequent publications.