Similarity solution

A similarity solution combines independent variables into scale-invariant coordinates and reduces a PDE to an ODE.

The transformation $\tau =\log t$, $\xi =x/\sqrt{t}$, $v=\sqrt{t}u$ gives $v_{\tau }=v_{\xi \xi }+(v-v^3)/2$. Thus the stationary half-line profile $v=\tanh (\xi /2)$ yields $u=t^{-1/2}\tanh [x/(2\sqrt{t})]$, with zero value at the origin and far-field value $t^{-1/2}$. For two bounded classical solutions, their difference has reaction coefficient $[1-(v_1^2+v_1{v}_2+v_2^2)]/2\le 1/2$. Multiplying it by $e^{-\tau /2}$ therefore produces a linear parabolic equation with nonnegative absorption. A positive growing quadratic spatial barrier and the maximum principle give uniqueness on the unbounded half-line. This proof identifies the finite-time boundedness or growth hypothesis needed at spatial infinity rather than assuming a supremum is automatically attained there.

The drift $-x{u}_x/(2t)$ exactly cancels the coordinate-motion term of the similarity solution $u=t^{-1/2}f(\eta )$, $\eta =x/\sqrt{t}$. Direct differentiation givesConsequently the nonautonomous partial differential equation reduces to the autonomous ordinary differential equation $f^{\prime \prime }=f^3-f/2$. The balance fixes the same length scale $x\sim \sqrt{t}$ as diffusion, but removes the first-derivative term that usually appears in the profile equation. A first-integral argument can then determine a whole profile family without solving a nonautonomous second-order equation.

A reduction obtained by substituting a scale-invariant ansatz into a differential equation.

A similarity ansatz assumes a profile invariant under a prescribed rescaling, commonly in the displayed power-law form. Substitution into a partial differential equation fixes compatible exponents and can reduce it to an ordinary differential equation for $F$. Boundary conditions and conserved quantities must also be compatible with the scaling; the ansatz alone does not establish existence of a solution.

A similarity profile is the time-independent function left after expressing a evolving field in its scaled coordinate, for example $\rho (z,t)={\rho }_{0}f(z/Z(t))$. Integrating such profiles converts a global conservation law into an algebraic relation among the time-dependent scales.