ステファン問題
∂ u ∂ t = ∂ 2 u ∂ x 2 in { ( x , t ) : 0 < x < s ( t ) , t > 0 } , the heat equation , − ∂ u ∂ x ( 0 , t ) = f ( t ) , t > 0 , the Neumann condition at the left end of the domain describing the inlet heat flux , u ( s ( t ) , t ) = 0 , t > 0 , the Dirichlet condition at the water-ice interface: setting melting/freezing temperature , d s d t = − ∂ u ∂ x ( s ( t ) , t ) , t > 0 , Stefan condition , u ( x , 0 ) = 0 , x ≥ 0 , initial temperature distribution , s ( 0 ) = 0 , initial depth of the melted ice block {\displaystyle {\begin{aligned}{\frac {\partial u}{\partial t}}&={\frac {\partial ^{2}u}{\partial x^{2}}}&&{\text{in }}\{(x,t):0<x<s(t),t>0\},&&{\text{the heat equation}},\\-{\frac {\partial u}{\partial x}}(0,t)&=f(t),&&t>0,&&{\text{the Neumann condition at the left end of the domain describing the inlet heat flux}},&&\\u{\big (}s(t),t{\big )}&=0,&&t>0,&&{\text{the Dirichlet condition at the water-ice interface: setting melting/freezing temperature}},\\{\frac {\mathrm {d} s}{\mathrm {d} t}}&=-{\frac {\partial u}{\partial x}}{\big (}s(t),t{\big )},&&t>0,&&{\text{Stefan condition}},\\u(x,0)&=0,&&x\geq 0,&&{\text{initial temperature distribution}},\\s(0)&=0,&&&&{\text{initial depth of the melted ice block}}\end{aligned}}}