New integral relations for analytic solutions of parabolic-type equations in noncylindrical domains. (Q1432362)

The author deals with the following boundary value problems \[ {\partial T\over\partial t}= a{\partial^2T\over\partial x^2}+ F(x,t),\quad 0< x< y(t),\quad y(0)= 0,\tag{1} \] \[ T(x,t)|_{t=0}= \phi_0(x),\quad 0\leq x\leq y(0),\tag{2} \] \[ \Biggl[\beta_{i_1}{\partial T\over\partial x}+ \beta_{i_2}T(x,t)\Biggr]_{\substack{ i=1,x=0\\ i=2,x= y(t)}}= \varphi(t).\tag{3} \] Here \(y(t)\) is a continuous function. Since the domain boundary varies with time, the author proposes a general method for solving (1)--(3), when \(y(t)= \ell+ vt\) \((\ell,v=\text{const})\).