A Nonclassical Stefan Problem with Nonlinear Thermal Parameters of General Order and Heat Source Term

Abstract

The analytic solution for a general form of the Stefan problem with nonlinear temperature-dependent thermal parameters and a heat source the term is obtained. We prove the existence and uniqueness of the solution to the problem in the absence of a heat source (β=0), and in the presence of a heat source β(x)=exp(−x2). Then, we establish lower and upper bounds for the solutions of the homogeneous equation and the nonhomogeneous equation, for different values of δi and γi. It was found that the lower bounds exhibit an excellent alignment with the numerical solutions of the homogeneous and nonhomogeneous equations, so the lower bounds can serve as approximate analytic solutions to the problem. This is a generalization to the open problem proposed by Cho and Sunderland in 1974 and also generalizes the problem proposed by Oliver and Sunderland in 1987, in addition to the problems investigated recently.

Keywords

• Stefan problem
• nonlinear thermal parameters
• modified error function
• lower- and upper-bound solutions