On the problem of control the composition of material in the binary alloy solidification process

Abstract

Abstract In a framework of classical one-dimensional binary alloy solidification problem with zero diffusion coefficient in the solid phase the inverse problem is posed and considered. The problem consists in finding of the temperature regime at the outer boundary of domain when the final distribution of admixture in solidified part is known. In the cases when the desired concentration is linear function of space coordinate the boundary-value problem for the system of equations can be disjoin and the inverse problem can be reduced to the successive solving of three problems. First one is a “supercooled” Stefan problem [Fasano, Primicherio, Quart. Appl. Math. 38: 439–460, 1981,Petrova, Tarzia, Turner, Advances in Math. Sciences and Applications, Gakkotosho 4: 35–50, 1994], the second one is an initial-boundary problem for heat equation in the domain with known moving boundary, and the last one is a non-characteristic Cauchy problem [Alifanov, Inverse Problems of Heat Exchange (in Russian), Nauka, 1988]. We investigate this last problem in two aspects – classical “exact” and extremal, as a problem of minimizing a functional. Besides, we consider the self-similar version of the inverse problem and construct the exact solution to the inverse problem.