A Stefan problem for a non-classical heat equation with a convective condition
They consider the one-phase Stefan problem for a non-classical heat equation for a semi-infinite material with a convective free boundary condition. Also they assumed that the heat source depends on the temperature at the fixed face \(x=0\). The problem is as follows. They determined the temperature \(u=u(x,t)\) and the free boundary \(x=s(t)\) which satisfied the following conditions: (1) \(u_t-u_{xx}=-F(u(0,t))\) \(0<x<s(t)\), \(0<t<T\), (2) \(u_x(0,t)=g(t)[u(0,t)-f(t)]\), \(f(t)\geq0\), \(0\leq t<T\), (3) \(u(s(t),t)=0\), \(0<t<T\), (4) \(u_x(s(t),t)=-\dot s(t)\), \(0<t<T\), (5) \(u(x,0)=h(x)\geq0\), \(0\leq x\leq b\), (6) \(s(0)=b(b>0)\), where: \(F\) is the control function depends on the evolution of the temperature at the extremum \(x=0\), \(g(t)\) -- is the thermal transfer coefficient depending on the time, \(f(t)\) -- is the temperature of the external fluid. In the paper the author proved the existence and uniqueness, local in time of the solution to the problem (1)--(6). At first they proved that the non-classical Stefan problem (1)--(6) is equivalent to a system of two Volterra integral equations. In order to get it they used the Friedman-Rubinstein method. Next using the Banach contraction theorem they proved that a system of two Volterra integral equation has a unique local solution. At the end they obtained a comparison result of the solution of the considered problem with respect to the one corresponding with null source term. The consideration they performed in the class of continuous functions.
scientific article; zbMATH DE number 5836698
scientific article