Non‐linear asymptotic solution to Stefan‐like problems and the validity of the linear approximation

SUMMARY The movement of a phase‐change boundary in the Earth is determined by the solution of a non‐linear boundary value problem (Stefan‐like problem). It is equivalent to the solution of a non‐linear integral equation. Approximate linear and non‐linear asymptotic solutions to a variety of Stefan‐like problems are derived and compared. If the amplitude of the phase boundary motion is small, the integral equation is linearized and analytical solutions are obtained by Laplace transform techniques. Series expansions converge asymptotically to the exact solution at small the regardless of the amplitude of the phase boundary motion. Solutions were derived for three different boundary value problems describing the motion of a phase boundary in the lithosphere or at the lithosphere‐asthenosphere boundary: (i) motion of a phase‐change boundary under change in surface pressure, (ii) motion of a phase boundary under change in heat flow at the lithosphere asthenosphere boundary and (iii) change in temperature at the base of the lithosphere. The comparison shows that, for the first two problems, the linear and the non‐linear solutions are close at small values of t and the linear approximation remains valid even when the amplitude of the phase boundary motion is large.