Fix infinite loop when symbolic variables are missing from PDESystem

[ChrisRackauckas-Claude]

Fixes #475

Problem

When a user forgot to include a symbolic variable in the dependent variables list of PDESystem, the system transformation would enter an infinite loop with endless "Expanding derivatives" warnings.

For example:

@variables p(..) S(..)
eq = [S(t, v) ~ -p(t, v) - ∂ᵥ(p(t, v)), ∂ₜ(p(t, v)) ~ -∂ᵥ(S(t, v))]
@named sys = PDESystem(eq, bcs, domains, [t, v], [p(t, v)])  # Missing S(t, v)
discretization = MOLFiniteDifference([v => 0.5], t)
prob = discretize(sys, discretization)  # Infinite loop!

This would print endless warnings:

┌ Warning: Expanding derivatives in term Differential(v)(S(t, v)).
└ @ MethodOfLines ...

Solution

Added upfront validation in transform_pde_system! to detect unknown symbolic variables before the transformation loop begins. The implementation:

1. Scans all equations and boundary conditions for function-like calls
2. Identifies missing variables - finds variables that appear in equations but not in depvar_ops
3. Filters out built-in operators - ignores +, -, *, /, etc. and Differential/Integral
4. Throws a clear error immediately with the names of missing variables and helpful guidance:

ArgumentError: Found unknown symbolic variable(s): S. These variables appear in 
the equations or boundary conditions but were not included in the dependent 
variables list of the PDESystem. Please add them to the dependent variables, 
e.g.: PDESystem(eqs, bcs, domains, ivs, [existing_vars..., S])

Implementation Details

• Added find_unknown_variables() helper function that traverses symbolic expressions
• Check runs before the transformation loop to fail fast
• Kept fallback max iteration check (1000 iterations) for other potential infinite loops
• Added tracking of seen terms during expansion to detect repetition

Testing

• ✅ Tested with the exact example from issue Forgetting a variable in PDESystem results in infinite loop #475 - now throws clear error
• ✅ Verified correct systems (with all variables included) still work properly
• ✅ Ran existing test suite components to confirm no regressions
• ✅ Formatted code with JuliaFormatter using SciMLStyle

Example Before/After

Before:

┌ Warning: Expanding derivatives in term Differential(v)(S(t, v)).
└ @ MethodOfLines ...
[infinite loop, user has to kill process]

After:

ERROR: ArgumentError: Found unknown symbolic variable(s): S. These variables 
appear in the equations or boundary conditions but were not included in the 
dependent variables list of the PDESystem. Please add them to the dependent 
variables, e.g.: PDESystem(eqs, bcs, domains, ivs, [existing_vars..., S])

🤖 Generated with Claude Code

Co-Authored-By: Claude [email protected]