Editing Stefan problem (section)

== Mathematical formulation ==

=== The one-dimensional one-phase Stefan problem ===
The one-phase Stefan problem is based on an assumption that one of the material phases may be neglected. Typically this is achieved by assuming that  a phase is at the phase change temperature and hence any variation from this leads to a change of phase. This is a mathematically convenient approximation, which simplifies analysis whilst still demonstrating the essential ideas behind the process. A further standard simplification is to work in [[Nondimensionalization|non-dimensional]] format, such that the temperature at the interface may be set to zero and far-field values to <math>+1</math> or <math>-1</math>.

Consider a semi-infinite one-dimensional block of ice initially at melting temperature <math>u=0</math> for <math>x \in [0;+\infty)</math>. The most well-known form of Stefan problem involves melting via an imposed constant temperature at the left hand boundary, leaving a region <math>[0;s(t)]</math> occupied by water.  The melted depth, denoted by <math>s(t)</math>, is an unknown function of time. The Stefan problem is defined by 
:* The heat equation: <math>\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad \forall (x,t) \in [0;s(t)] \times [0;+\infty]
</math>
:* A fixed temperature, above the melt temperature, on the left boundary: <math>u(0,t) = 1, \quad \forall t > 0
</math>
:* The interface at the melting temperature is set to <math>u \left(s(t),t \right) = 0
</math>
:* The Stefan condition: <math>\beta \frac{\mathrm{d}}{\mathrm{d}t} s(t) = -\frac{\partial}{\partial x} u \left(s(t), t \right)
</math> where <math>\beta
</math> is the Stefan number, the ratio of latent to ''specific'' [[sensible heat]] (where specific indicates it is divided by the mass). Note this definition follows naturally from the nondimensionalisation and is used in many texts <ref>{{Cite book|last=Davis, Stephen H., 1939-|title=Theory of solidification|isbn=978-0-511-01924-1|location=Cambridge|oclc=232161077}}</ref><ref>{{Cite book|last=Fowler, A. C. (Andrew Cadle), 1953-|title=Mathematical models in the applied sciences|date=1997|publisher=Cambridge University Press|isbn=0-521-46140-5|location=Cambridge|oclc=36621805}}</ref> however it may also be defined as the inverse of [[Stefan number|this]].
:* The initial temperature distribution: <math>u(x,0) = 0, \; \forall x \geq 0
</math>
:* The initial depth of the melted ice block: <math>s(0) = 0
</math>
:The Neumann solution, obtained by using self-similar variables, indicates that the position of the boundary is given by <math display="inline">s(t) = 2 \lambda \sqrt{t}
</math> where <math>\lambda
</math> satisfies the [[transcendental equation]] <math display="block"> \beta \lambda = \frac{1}{\sqrt{\pi}}\frac{\mathrm{e}^{-\lambda^2}}{\text{erf}(\lambda)}.
</math> The temperature in the liquid is then given by <math display="block">T=1-\frac{\text{erf}\left(\frac{x}{2\sqrt{t}}\right)}{\text{erf}(\lambda)}.
</math>