Dynamic-similarity solver for PDEs

Automate PDE → ODE reductions via similarity

Before numerical solvers existed, physicists routinely tamed partial differential equations by finding a change of variables that collapsed them into ODEs. The diffusion equation, the Blasius boundary layer, thin-film spreading — all yield to this trick, called a similarity reduction. The catch: discovering the right change of variables requires pattern-matching by hand, and it is easy to miss one or get the exponents wrong.

This Julia package automates the search. You write the PDE; it finds the exponents.

The tool tests a family of power-law ansätze of the form [ \eta = x^{a_1} y^{a_2} t^{a_3}, \qquad u = x^{b_1} y^{b_2} t^{b_3} f(\eta), ] symbolically differentiates the PDE, attaches scaling weights to every term, and equates exponents to form a small linear system in the unknowns (a_i, b_i). If the system is consistent, it substitutes the ansatz, cancels common factors, and returns an ODE in (\eta) along with transformed boundary and initial conditions — ready to pass directly into your ODE integrator. If no scaling reduction exists, it says so.

Built on Symbolics.jl, it handles 1D and 2D variants and emits the exponents, similarity map, reduced ODE, and transformed boundary conditions as Julia expressions.

Heat equation with linear decay: [ u_t = \kappa\,u_{xx} - \lambda u,\qquad x\in\mathbb{R},~ t>0. ]

using SimilaritySolver
pde = @pde du/dt ~ κ*d2u/dx2 - λ*u
bcs = ["u(±Inf,t)=0"]
result = find_similarity(pde, bcs; vars=[:x,:t], field=:u)
# => η = x / sqrt(t),  u = exp(-λt) * f(η),  f'' + η/2 * f' = 0

The package recovers the standard Gaussian similarity solution and hands back an ODE that any Julia integrator can solve in milliseconds.