demathpy 0.0.2

PDE/ODE math backend

Demathpy

A Python library for parsing and safely evaluating symbolic Ordinary and Partial Differential Equations (ODEs/PDEs) on numerical grids.

This repository provides:

• A lightweight symbol normalizer that converts human-readable mathematical notation into valid Python expressions.
• A secure evaluation engine that evaluates PDE/ODE right-hand sides on NumPy grids without using raw eval on untrusted input.
• Built-in support for common differential operators and vector calculus notation.

Key Features

1. Symbol Normalization

The parser supports Unicode and mathematical shorthand, including:

• Greek letters: α, β, γ, λ, ε, φ, θ → alpha, beta, gamma, lam, epsilon, phi, theta
• Powers: u², v³ → u**2, v**3
• Implicit multiplication:
  • αu → alpha*u
  • 2u → 2*u
  • (u+1)(v+1) → (u+1)*(v+1)
• Absolute values: |u| → abs(u)
• Common functions:
  sin, cos, tan, exp, log, tanh, sech, sign, sqrt

2. Differential Operator Support

The parser recognizes and evaluates:

• First derivatives:
  • du/dt, u_t
• Second derivatives:
  • dxx(u), dzz(u)
• Laplacian:
  • ∇²u, lap(u)
• Gradient:
  • ∇u, grad(u)
• Divergence:
  • ∇·(A), div(A)

These are mapped to finite-difference operators implemented in NumPy.

3. PDE / ODE Parsing

The library provides:

• parse_pde(equation: str)
  Parses a PDE string and returns (lhs_variable, rhs_expression).

• evaluate_rhs(rhs: str, grids: dict, constants: dict, grid_dx: float)
  Safely evaluates the right-hand side on NumPy arrays representing fields and parameters.

Supports equations such as:

∂T/∂t = α ∇²T - σ T
u_t = D dxx(u) + f(u)
∂φ/∂t = ∇·((1 + T²) ∇φ)

4. Secure Evaluation Environment

Only a restricted set of functions and operators are exposed. No access to Python builtins, file I/O, or unsafe functions. All variables must come from: grids: NumPy arrays for fields (u, T, phi, etc.) constants: scalar parameters (alpha, beta, lambda, etc.)

5. Coordinate-Aware PDEs

The evaluator automatically provides spatial coordinate grids: x, z as NumPy arrays derived from grid shape and spacing. This enables anisotropic and spatially varying coefficients such as: α x dxx(T) + α z dzz(T) Typical Use:

from pde import parse_pde, evaluate_rhs

lhs, rhs = parse_pde("∂T/∂t = α ∇²T - σ T")

result = evaluate_rhs(
    rhs,
    grids={"T": T_grid},
    constants={"alpha": 0.1, "sigma": 0.01},
    grid_dx=0.01
)

6. Purpose

This project is intended as a safe mathematical expression parser for: Scientific computing PDE/ODE solvers Physics and engineering simulations Educational or sandboxed equation evaluation