Dynamic-similarity solver for PDEs

Symbolic search for scaling reductions from PDE → ODE

Similarity by construction: automate PDE → ODE reductions

Many transport and fluid problems hide a scaling symmetry. When it exists, a change of variables collapses the PDE to an ODE (diffusion, Blasius boundary layer, thin films).
This Julia tool automates that search: it tests a family of similarity ansätze, solves for exponents, reduces the PDE, and transforms boundary conditions.

What it does

  • Parse a restricted PDE syntax (fields, derivatives).
  • Search similarity variables
    ( \eta = x^{a_1} y^{a_2} t^{a_3} ), ( u = x^{b_1} y^{b_2} t^{b_3} f(\eta) ) (1–2D variants).
  • Solve exponent-balance equations so all PDE terms scale identically.
  • Reduce: substitute the ansatz, cancel factors, and return an ODE in (\eta).
  • Transform BCs/ICs into conditions on ( f(\eta) ).
  • Emit artifacts: exponents, similarity map, reduced ODE, transformed BCs as Julia expressions.

Built on Symbolics.jl; outputs feed directly into your ODE integrator.

Method sketch

  1. Tokenize and symbolically differentiate the PDE.
  2. Attach scaling weights to variables and derivatives.
  3. Equate exponents across terms → small linear system in (a_i,b_i).
  4. If solvable, substitute and simplify to an ODE in (\eta).
  5. Verify dependency only on (\eta); otherwise report “no scaling reduction.”

Example

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

```julia using SimilaritySolver # package namespace pde = @pde du/dt ~ κd2u/dx2 - λu bcs = [“u(±Inf,t)=0”] result = find_similarity(pde, bcs; vars=[:x,:t], field=:u)

References