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Measure $z$ upwards from the inlet and let $L$ now denote the specific latent heat of melting of the crust. Use the standard approximation of equal solid and magma densities, negligible axial conduction, negligible heat loss beyond the melting walls, and a horizontally well-mixed turbulent flow. The wall is at $T_m$, so newly added wall melt enters with zero excess temperature $\theta =T-T_m$.

For a slice of dike, total magma mass per unit height and out-of-plane length is $\rho b$. Both melting walls add mass at rate $\rho b_t$. Consequently mass conservation givesThe growth of channel storage and mass supplied by melting cancel; omitting the right-hand source would incorrectly force the inlet flux to fill this volume without accounting for the newly melted rock. Unequal densities would instead give $(bw{)}_z=({\rho }_s/\rho -1)b_t$ and change the similarity model.

Let $q_w={\alpha }_T\rho c_{p}w\theta$ be heat transfer to each wall. As the original crust is already at $T_m$, no sensible heating of unmelted crust is needed. Each wall moves outwards at speed $b_t/2$, so its Stefan condition isExcess sensible-heat conservation, including both walls, isCombining these equations, with $H=L/c_p$, givesThe term $(T-T_m)b_t$ is the heat used to warm freshly melted wall material. It cannot simply be dropped from a conservative balance. This is melt-dilution heat balance in a dike.

At the inlet $bw=Q_0$ and $\theta ={\Delta }_0=T_0-T_m$. The wall balance there gives $b(0,t{)}^2=b_0^2+4{\alpha }_T{Q}_{0}t/S$, where $S=L/(c_p{\Delta }_0)$. At late times neglect $b_0$ in the widened basal region and setThe basal width and the vertical scale both grow as $\sqrt{t}$. Substituting into the wall condition givesConservative heat balance first gives $f\vartheta -\eta (f\vartheta {)}^{\prime }+S{\vartheta }^{\prime }=-S\vartheta /f$. Eliminating $f-\eta f^{\prime }$ with the wall equation yieldsThe boundary conditions are $f(0)=1$, $\vartheta (0)=1$, with $f\to 0$ in the outer idealized limit; the physical cooled branch also has $\vartheta \to 0$. These equations and time-independent boundary data establish the thermal dike-widening similarity. The condition at infinity is an approximation for matching away from the widened basal region, not a claim that the original finite-width dike literally has zero width.

The inlet is a regular singular point of the first equation. Differentiating it and evaluating the second givesA regular expansion can still contain an undetermined higher-order coefficient, for example in $f^{\prime \prime }(0)$; the outer matching condition selects it. Thus specifying the inlet values does not eliminate the boundary-value aspect of the problem. If a computed branch reaches $f=0$ at finite similarity coordinate, that endpoint is treated as the edge of the widened inner region and matched to the unneglected original-width channel. One must not continue $w=Q_0/(Bf)$ through zero width as a physical flow.

For numerical solution, formulate this as a nonlinear two-point boundary-value problem: start from a regular inlet expansion and adjust its free coefficient to meet the outer decay/matching condition, or solve for the profiles and boundary data together on a truncated or compactified similarity interval. Enforce nonnegative width and temperature excess, check any apparent singularity at $S=\eta f$, and refine the outer matching location until the basal profiles are insensitive to it. No specific algorithm is required.

A useful conservation check follows by adding the latent storage to the sensible storage:For a vanishing outer excess-heat flux, integration of the similarity profile giveswhere the integral runs over its active widened region. This checks the heat admitted at the base against latent and sensible storage. All factors of two refer to the two channel walls; $b$ is the full width.

With equal solid and liquid densities, magma entering a dike at constant flux $Q_0$ has $bw=Q_0$: melt added by the walls exactly supplies the increasing cross-sectional storage. If wall heat flux is ${\alpha }_T\rho c_{p}w(T-T_m)$, both walls give $b_t=2{\alpha }_T{c}_{p}w(T-T_m)/L$. Combining this with sensible-heat conservation admits $b=B(t)f(\eta )$, $\eta =2{\alpha }_{T}z/B$, $B^2=4{\alpha }_T{Q}_{0}t/S$. The shape equations are $f(f-\eta f^{\prime })=\vartheta$ and $(S-\eta f){\vartheta }^{\prime }=-(S+\vartheta )\vartheta /f$.