Convergence of the line method approximation for a parabolic free boundary problem
The authors consider an implicit free boundary problem for the one- dimensional heat equation with a nonlinear source term \[ u_{xx}-u_ t=f(t,x,u,u_ x)\qquad \text{for}\qquad t>0,\;s(t)>x>0, \] \[ u(0,x)=0,\qquad au(t,0)+bu_ x(t,0)=\alpha(t),\qquad u(t,s(t))=0. \] The moving boundary is described by \(s=s(t)\) and is linked to \(u\) by the implicit equation \[ u_ x(t,s(t))=0,\qquad s(0)=0. \] Applying Rothe's horizontal line method, an approximate solution to this diffusion process is obtained. Rothe's method consists of solving a sequence of free boundary problems for second order ordinary differential equations, which arise when the time derivative is replaced by a difference quotient. This scheme of equations is discussed with the help of comparison results, stemming from the theory of differential inequalities.
scientific article; zbMATH DE number 8649