The Numerical Solution Of An Inverse Two-phase Stefan Problem

Problems with free boundaries concern a large number of physical phenomena of which many can be encountered in thermal industrial processes,such as cast-ing,welding,purification by metal beams or laser machining by beams. In this pa-per,we consider an inverse two-phase Stefan problem which is from the following model.The melting of a thin block of ice occupying an interval b≤x≤L is described by the one-dimensional two-phase Stefan problemalong with boundary conditionsand initial datain which u₁,u₂ represent the state functions of process and s the position of free boundary,a₁, a₂,λ₁,λ₂ are positive constants and a₁≠a₂,λ₁≠λ₂. Given the initial distributions u₁⁰, v₂⁰ with compatible conditionsthe so-called inverse Stefan problem is to find,for a prescribed interface s,a time dependent function v such that the problem (1)—(8) has continuous solutions u₁, u₂ on 0≤x≤s(t), t≥0 and a(t)≤x≤L, t≥0, respectively.We have a lot of numerical methods to solve the inverse one-phase Stefan problem . In paper[2],the problem is reduced to a system of integral equation. Jochum considers the inverse Stefan problem as a problem of nonlinear approximation the-ory(see[3,4]).In paper[5],the author use Adomian decomposition method to solve this problem . In paper[6],for solution of one phase two-dimensional problems,authors use a complete family of solutions to the heat equation to minimize the maximal defect in the initial-boundary data.There are few numerical methods to solve the inverse two-phase Stefan problem. So the purpose of this paper is to construct a numerical method to the inverse two-phase Stefan problem. The problem (1)—(8) can be reduced to solve two heat -conduction equations. (10)is posed ,so we can use the difference approximation method to solve this equations. (9) is ill-posed and therefore intractable numerically. We first need to translate (9) into an integral equation,then use Tikhonov's method to regularize the integral equation.